Daniele Angella

Florian William Maxence

Solène Jean-Marc Dominique

Fu-Hsuan Papa

Les flots de Monge-Ampère pluripotentiels

Keywords: Équations de Monge-Ampère complexes dégénérées, classe de cohomologie big, flot de Kähler-Ricci, flot de Pluripotential Monge-Ampère flows Degenerate complex Monge-Ampère equations, big cohomology class, Kähler-Ricci flow, Chern-Ricci flow, Kähler manifolds, Hermitian manifolds

Chern-Ricci, variétés kähleriennes, variétés hermitiennes.

Résumé : Ce travail de thèse s'intéresse aux équations de Monge-Ampère complexes dégénérées aux sens elliptiques et paraboliques sur les variétés kählériennes compactes et leurs applications géométriques, et leurs généralisations sur les variétés hermitiennes compactes. Dans le premier chapitre, en généralisant l'approche de Di Nezza-Lu au cadre de classes de cohomologie big, nous démontrons que les solutions des équations de Monge-Ampère complexes dégénérées sur les variétés kählériennes compactes sont continues sur un ouvert de Zariski. Cela nous permet de démontrer que les métriques de Kähler-Einstein singulières sur les variétés log-canoniques (lc) de type général (ou disons plutôt les paires lc) sont continues sur le lieu ample hors de la partie non-Kawamata log-terminale. Les deux chapitres suivants constituent le coeur de ce mémoire. Nous étudions d'abord les flots de Monge-Ampère com-plexes pluripotentiels dans les classes big sur les variétés kähleriennes compactes. Nous montrons, sous des hypothèses naturelles, que l'enveloppe des sous-solutions pluripotentielles est localement uniformément semi-concave en temps et continue en espace, et qu'elle est l'unique solution pluripotentielle avec une telle régularité. Nous appliquons cette théorie pour étudier les flots de Kähler-Ricci faibles sur les variétés kähleriennes compactes de type général ainsi que sur les variétés stables. Enfin, nous généralisons les flots de Monge-Ampère pluripotentiels à des contextes plus généraux et établissons également une régularité partielle de telles solutions sous des hypothèses supplémentaires sur les densités, ce qui nous permet de prouver l'existence et l'unicité du flot de Chern-Ricci faible sur les variétés complexes à singularités log-terminales.

nerate complex Monge-Ampère equations in both elliptic and parabolic senses on compact Kähler manifolds and geometric application and their generalizations on compact Hermitian manifolds. In the first part, extending Di Nezza-Lu's approach to the setting of big cohomology classes, we prove that solutions of degenerate complex Monge-Ampère equations on compact Kähler manifolds are continuous on a Zariski open set. This allows us to show that singular Kähler-Einstein metrics on log canonical varieties of general type have continuous potentials on the ample locus outside of the non-klt part. The heart of this dissertation is the two following chapters. We study pluripotential complex Monge-Ampère flows in big cohomology classes on compact Kähler ma-nifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential Kähler-Ricci flows on compact Kähler manifolds of general type as well as on stable varieties. Finally, we generalize the pluripotential flows to a more general setting and establish a partial regularity of such pluripotential solutions under some extra assumptions of the densities. This allows us to prove the existence and uniqueness of the weak Chern-Ricci flow on complex compact varieties with log terminal singularities.

Chapitre 1

Introduction (en français)

Cette partie introdutive se divise en trois sections. On commence par quelques rappels historiques sur les approches paraboliques, en particulier, le flot de Kähler-Ricci sur les variétés kähleriennes compactes et son application géométrique : chercher des métriques de Kähler-Einstein. Nous mentionnons également un flot plus récent sur les variétés complexes, appelé le flot de Chern-Ricci.

Nous rappelons brièvement dans une deuxième section quelques notions essentielles pour la formulation de nos résultats principaux : classe big, produit non-pluripolaire, classe d'énergie finie, etc.

La troisième section expose les résultats obtenus dans cette thèse. En tant que telle, elle se divise elle-même pricipalement en trois sous-sections, correspondant aux trois chapitres suivants dans lesquels nous présentons nos résultats principaux. Ces trois chapitres sont essentiellement distincts.

Motivation : approche parabolique et MMP

Métriques de Kähler-Einstein

Une variété kählerienne (X, ω) est une variété complexe de dimension finie n = dim C X munie d'une forme réelle fermée ω de type (1,1) (c-à-d ; dω = 0) s'exprimant dans un système de coordonnées holomorphes locales (z α ) 1≤α≤n sous la form ω = i ∑ α,β g α βdz α ∧ d zβ où (g α β) est une matrice hermitienne définie positive. La courbure de Ricci de (X, ω) est la (1,1)forme Ric(ω) donnée en coordonnées locales par : Ric(ω) := -i∂ ∂ log det(g α β).

On notera que la forme de Ricci Ric(ω) ne dépend que de la forme volume ω n . Elle appartient toujours à la classe de cohomologie c 1 (X) ∈ H 1,1 (X, R) ∩ H 2 (X, Z) égale à la premième classe de Chern-Weil de la métrique hermitienne sur le fibré en droites holomorphe anticanonique K -1 X = Λ n T X attachée à ω n multipliée par 2π. Autrement dit la classe de cohomologie [Ric(ω)] ∈ H 1,1 (X, R) ne dépend pas de la métrique ω. Définition 1.1.1. On dit que ω est une métrique de Kähler-Einstein s'il existe une constante λ ∈ R telle que Ric(ω) = λω. (KE λ )

On dit ainsi que X est une variété de Kähler-Einstein si X admet une telle métrique.

Le problème de l'existence de métriques de Kähler-Einstein est l'un des plus fondamentaux en géométrie différentielle et complexe (kählerienne). Le cas des surfaces de Riemann est connu depuis longtemps grâce au théorème d'uniformisation. Cependant, il est facile de voir qu'il y a des obstructions à l'existence de telles métriques. En effet, en prenant les classes de cohomologie de l'équation de Kähler-Einstein (KE λ ), nous obtenons λ[ω] ∈ c 1 (X). Il faut donc à l'avance supposer que la première classe de Chern c 1 (X) a un signe qui est celui de λ. Dès lors, une première condition apparaît : pour résoudre (KE λ ), il est nécessaire que la première classe de Chern soit nulle ou contienne une forme définie positive ou négative suivant que λ = 0, λ > 0 ou λ < 0. En fait dans ces deux derniers cas nous normalisons la métrique en supposant λ = ±1.

Nous avons alors des résultats concernant l'existence et l'unicité des métriques de Kähler-Einstein selon ces trois cas. Avant de les énoncer, nous allons expliquer la démarche pour les obtenir. L'idée est de ramener la recherche de telles métriques à la résolution d'une équations aux dérivées partielles non-linéaire scalaire, dite de Monge-Ampère complexe.

Soit ω ′ une forme kählerienne dans [ω]. Un résultat fondamental de géométrie kählerienne, le "lemme ∂ ∂", nous dit que Ric(ω) = λω + i∂ ∂ f pour une certaine fonction réelle f lisse sur X et que ω ′ = ω + i∂ ∂φ, avec φ réelle lisse. Comme Ric(ω ′ ) = Ric(ω) -i∂ ∂ log ω ′n ω n , il résulte du principe de maximum que résoudre (KE λ ) ramène à résoudre une équation du type (ω + i∂ ∂φ) n = e -λφ+ f ω n .

(MA λ )

Il sera commode par la suite de normaliser f en supposant ´X(e f -1)ω n = 0. Une des méthodes les plus efficaces pour résoudre des équations non-linéaires comme celle-là est la méthode de continuité, suggérée par E. Calabi. Cela permet de résoudre des équations de plus en plus proches de celle qui nous intéresse, et via l'établissement d'estimées a priori sur les solutions approchées, on peut espérer obtenir une solution en passant à la limite.

Dans le cas λ < 0, les conditions nécessaires et suffisantes d'existence de métriques de Kähler-Einstein sur une variété X compacte ont été élucidées par Aubin et indépendamment par Yau ; voir [START_REF] Aubin | Équations du type Monge-Ampère sur les variétés kählériennes compactes[END_REF][START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation[END_REF]. Lorsque λ = 0, i.e. K X est numériquement trivial, il existe une formulation plus forte de ce problème généralisant le cas de courbure nulle, qui a été démontrée par Yau [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation[END_REF].

Théorème 1.1.2 (Conjecture de Calabi). Soit [ω] ∈ H 1,1 (X, R) une classe kählerienne. Alors pour toute (1, 1)-forme Ω réelle, de classe c 1 (X), il existe une unique métrique kählerienne ω ′ dans [ω] telle que Ric(ω ′ ) = Ω.

Le cas λ > 0 est beaucoup plus délicat. Dans ce cas le fibré anticanonique est ample, on dit alors que X est une variété de Fano. Des obstructions à l'existence ont été découvertes par Matsushima [START_REF] Matsushima | Sur la structure du groupe d'homéomorphismes analytiques d'une certaine variété kählérienne[END_REF] (réductivité du groupe d'automorphisme) et Futaki [START_REF] Futaki | An obstruction to the existence of Einstein Kähler metrics[END_REF] (existence d'un invariant lié aux champs de vecteurs) donnant des contre-exemples dès la dimension 2 (P 2 éclaté en un point). Le cas des surfaces a été complètement résolu par G. Tian [START_REF] Tian | On Kähler-Einstein metrics on certain Kähler manifolds with C 1 (M) > 0[END_REF][START_REF] Tian | Kähler-Einstein metrics on complex surfaces with C 1 > 0[END_REF]. Une conjecture de Yau, Tian et Donaldson reliant l'existence des métriques de Kähler-Einstein sur les variétés de Fano à la propriété de K-stabilité de la variété en question, est restée ouverte pendant longtemps.

On ne définira pas la notion de K-stabilité pour une variété de Fano. Il suffit de dire ici que cette condition est purement algébro-géométrique et ressemble à une condition de stabilité au sens de la théorie géométrique des invariants (GIT), comme prédit originellement par Yau. La conjecture de Yau-Tian-Donaldson a été récemment résolue par Chen-Donaldson-Sun :

Flot de Ricci

En 1982, Richard Hamilton, dans son article fondateur [START_REF] Hamilton | Three-manifolds with positive Ricci curvature[END_REF], a introduit le flot de Ricci nommé d'après Gregorio Ricci-Curbastro. C'est une équation aux dérivées partielles (EDP) portant sur le tenseur métrique d'une variété riemannienne : on part d'une métrique g 0 , que l'on fait évoluer par : ∂g ∂t = -2Ric(g) où g désigne la métrique riemannienne et Ric la courbure de Ricci de la métrique. Hamilton a lancé un nouveau programme pour démontrer la conjecture de géométrisation de Thurston et, en particulier, la conjecture de Poincaré. Son approche est basée sur le flot de Ricci. G. Perelman, dans une série d'articles [START_REF] Perelman | The entropy formula for the Ricci flow and its geometric applications[END_REF][START_REF] Perelman | Ricci flow with surgery on three-manifolds[END_REF][START_REF] Perelman | Finite extinction time for the solutions to the Ricci flow on certain threemanifolds[END_REF], a construit un flot de Ricci avec chirurgie sur les 3variétés compactes et terminé le programme de Hamilton.

Dans le cadre complexe, il est connu que le flot de Ricci préserve la condition kählerienne dans le sens que la métrique évoluée est Kähler si la donnée initiale l'est (voir e.g. [START_REF] Hamilton | The formation of singularities in the Ricci flow[END_REF]p. 31]). Le flot est donc appelé le flot de Kähler-Ricci. Rappelons que sur une variété kählerienne compacte (X, ω 0 ), une solution du flot de Kähler-Ricci avec comme valeur initale ω 0 est la famille de métriques kähleriennes ω = ω(t) pour t ∈ [0, T) vérifiant ∂ω ∂t = -Ric(ω), ω| t=0 = ω 0 .

(1.1.1)

Hamilton [START_REF] Hamilton | Three-manifolds with positive Ricci curvature[END_REF] a démontré l'existence d'une solution en temps petit pour toute donnée initale lisse (voir [START_REF] Deturck | Deforming metrics in the direction of their Ricci tensors[END_REF] pour une preuve simple). Par des manipulations analoges à celles du paragraphe précédent, on se ramène à la résolution de l'équation de Monge-Ampère complexe parabolique     

(ω 0 -tRic(ω 0 ) + i∂ ∂φ) n = e ∂φ ∂t ω n 0 , ω 0 -tRic(ω 0 ) + i∂ ∂φ > 0, φ(0) = 0. Grâce au résultat de Hamilton, cette équation a une solution φ(t) pour t ∈ [0, T max [, où le temps d'existence maximal est caractérisé par T max = sup{t > 0 : [ω 0 ]tc 1 (X) est Kähler}, comme l'ont montré Tian-Zhang [START_REF] Tian | On the Kähler-Ricci flow on projective manifolds of general type[END_REF].

Afin d'étudier le problème (d'existence de métrique canonique) qui nous intéresse, une approche parabolique a été utilisée par H.D. Cao [START_REF] Cao | Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds[END_REF]. Il montre notamment que sur les variétés dont la première classe de Chern est nulle ou négative, l'équation d'évolution (1.1.1) a une solution pour tous les temps, i.e., T max = +∞. La question qui demeure est d'étudier ce qui se passe lorsque le paramètre d'évolution tend vers l'infini. Précisément, nous avons : Théorème 1.1.4 (Cao [Cao85]). Soit X une variété kählerienne compacte à courbure nulle ou négative, i.e. c 1 (X) = 0 ou c 1 (X) < 0. Alors le flot de Kähler-Ricci ω, à renormalisation près, converge de manière C ∞ vers une métrique de Kähler-Einstein ω KE lorsque t → ∞.

Programme du Modèle Minimal analytique : programme de Song-Tian

La classification des variétés algébriques est une question centrale en géométrie birationelle. Le programme du modèle minimal (MMP en anglais) est une approche de la classification qui a été introduite au début des années '80 par Kawamata, Kollár, Mori, Reid, Shokurov et beaucoup d'autres. Son but initial était de généraliser en toute dimension la classification des surfaces algébriques. Le travail dans le MMP et, en particulier, la découverte de son cadre naturel (celui des paires singulières) a produit une multitude d'outils extrêmement puissants, dont les applications ont largement dépassé le but initial de classifier les variétés selon la positivité de leur diviseur canonique.

Rappelons qu'une variété X est dite "bonne" si elle satisfait l'une des conditions suivantes :

• K X est nef, i.e. K X • C ≥ 0 pour toute courbe irréductible C. Dans ce cas, la variété X est appelée modèle minimal ;

• Il existe une application holomorphe π : X → Y à Y de dimension inférieure telle que la fibre générique X y = π -1 (y) est une variété de Fano (i.e. c 1 (X y ) > 0). Dans ce cas, la variété X est appelée une fibration de Fano ou de Mori.

Le MMP produit conjecturalement une suite des nouvelles variétés

X 1 , • • • , X k , des applica- tions birationelles f 1 , • • • , f k , X = X 0 f 1 X 1 f 2 • • • f k X k
telle que X k est bonne, i.e. ou bien X k est un modèle minimal ou bien X k est une fibration de Fano (ou de Mori).

Dans l'article fondateur [START_REF] Song | The Kähler-Ricci flow through singularities[END_REF], Song et Tian ont introduit un autre programme (connu comme l'analogue analytique à MMP) afin de classifier des variétés projectives (algébriques) complexes via le flot de Kähler-Ricci. Nous rappelons brièvement un schéma de ce programme : Étape 1. Nous commençons par une métrique ω 0 dans la classe d'un diviseur H sur une variété X. Nous considérons ensuite la solution ω(t) du flot de Kähler-Ricci sur X à partir de ω 0 . Le flot existe donc sur l'intervalle maximale [0, T) où T = sup{t > 0 : H + tK X > 0}.

Étape 2. Si T = ∞, alors K X est nef et le flot de Kähler-Ricci existe pour tout temps. Le flot ω(t), à normalisation près, devrait converger vers une métrique de Kähler-Einstein généralisée canonique sur X lorsque t → ∞. Étape 3. Si T < ∞, le flot de Kähler-Ricci déforme X, lorsque t → T, sur (Y, ω Y ) où la métrique ω Y est éventuellement singulière.

1. Si dim X = dim Y et Y est de la forme X \ S où S est une sous-variété de X. Alors il existe un flip

X X + Y π (π + ) -1 •π π +
où π + est un morphisme birationnel et X + est π + -ample. On répète alors l'étape 1 avec (X + , (π + ) * ω Y ).

2. Si 0 < dim Y < dim X, X admet alors une fibration de Fano sur Y. On retourne alors l'étape 1 en remplaçant (X, ω 0 ) par (Y, ω Y ).

3. Si dim Y = 0, il faudrait avoir c 1 (X) > 0, X est Fano. De plus, à renormalisation près, la solution (X, ω(t)) du flot de Kähler-Ricci converge vers (X ′ , ω ′ ) où X ′ est éventuellement différent de X et ω ′ soit une métrique de Kähler-Einstein soit un soliton de Kähler-Ricci (i.e., Ric(ω ′ ) = ω ′ + L V (ω ′ ) pour un champ de vecteurs holomorphe V).

Autrement dit, le flot de Kähler-Ricci peut construire une suite des variétés X 1 , • • • , X k du MMP, où X k est nef (modèle minimal) ou bien une fibration de Fano. Nous désirons que le flot de Kähler-Ricci converge, à renormalisation près, vers une métrique canonique. En outre, nous voudrions démontrer que la procédure ci-dessus est continue dans la topologie de Gromov-Haussdorff : le flot effectue des chirurgies géométriques dans la topologie Gromov -Hausdorff à chaque temps singulier et remplace la variété projective initale par un "meilleur" modèle.

Flot de Chern-Ricci

Cette partie est un résumé très bref du chapitre sur le flot de Chern-Ricci, et nous ne ferons que donner la notion du flot et ses applications géométriques. La question naïve naturelle est la suivante : soit X une variété complexe, le flot de Ricci préserve-t-il la condition hermitienne ? Malheuresement la réponse est NON. Il est raisonable de chercher un autre flot géométrique de métriques hermitiennes qui se spécialise au flot de Ricci lorsque la valeur initale est Kähler.

Soit X une variété complexe compacte, muni d'une métrique hermitienne ω 0 . Le flot de Chern-Ricci est une famille (ω t ) t∈[0,T] (où t ∈]0, ∞]) de métriques hermitiennes vérifiant ∂ω ∂t = -Ric(ω), ω| t=0 = ω 0 , (1.1.2) où Ric(ω) est la forme de Chern-Ricci de ω (localement définie par Ric(ω) = -i∂ ∂ log ω n ) qui coïncide avec le tenseur de courbure d'une connexion de Chern de ω. Le flot de Chern-Ricci a d'abord été étudié par Gill [START_REF] Gill | Convergence of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds[END_REF] dans un cas spécifique et developpé en général par Tosatti-Weikove [START_REF] Tosatti | On the evolution of a Hermitian metric by its Chern-Ricci form[END_REF][START_REF] Tosatti | The Chern-Ricci flow on complex surfaces[END_REF]. De même, ce flot peut être ramené à l'équation scalaire de Monge-Ampère complexe parabolique. Comme précédemment, nous espérons que le flot de Chern-Ricci peut être utilisé comme un outil dans la classification des variétés hermitiennes. Dans [START_REF] Tosatti | The Chern-Ricci flow on complex surfaces[END_REF], Tosatti et Weinkove ont démontré que le flot de Chen-Ricci, à partir d'une métrique de Gauduchon, doit avoir une singularité en temps fini qui est nécessairement volume non effondré, au sens que Vol(X, ω) ≥ c > 0 lorsque t → ∞. Ce flot contracte un nombre fini des (-1) courbes E i aux points et converge de manière C ∞ (X \ ∪ i E i ), vers lorsque t → T -. Il a été démontré par Nie [START_REF] Nie | Weak solution of the Chern-Ricci flow on compact complex surfaces[END_REF] et Tô [START_REF] Tô | Regularizing properties of complex Monge-Ampère flows II : Hermitian manifolds[END_REF] indépendamment que le flot de Chern-Ricci peut continuer de manière unique sur la variété "blow-down" et l'on obtient la convergence de Gromov-Hausdorff en arrière dans le temps.

Nous finissons cette partie par mentionner que beaucoup de flots préservant la propriété hermitienne ont été proposés et étudiés ; voir par exemple [START_REF] Streets | A parabolic flow of pluriclosed metrics[END_REF][START_REF] Streets | Hermitian curvature flow[END_REF][START_REF] Streets | Regularity results for pluriclosed flow[END_REF][START_REF] Liu | Geometry of Hermitian manifolds[END_REF].

Notions préliminaires

Dans cette partie, nous allons parler brièvement des notions préliminaires qui vont nous intéresser dans la suite du mémoire.

Singularités des paires

Avant de présenter nos résultats et d'expliquer pourquoi on va essayer de développer la théorie des flots de Monge-Ampère (pluripotentiel) dans le cadre des variétés (ou des paires) singulières, il est raisonable de redonner quelques définitions concernant les singularités (auxquelles on va se resteindre). Le lecteur pourra consulter le très joli texte [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF].

Dans essentiellement toute la suite, Y va désigner une variété complexe normale. En particulier, le lieu des singularités Y sing de Y a pour codimension au moins 2, on désigne également Y reg le lieu régulier de Y. D'après le théorème de Hironaka, Y admet une résolution des singularités, c-à-d qu'il existe une variété lisse X et un morphisme birationel surjectif π : X → Y. En outre, on peut supposer que π est un isomorphisme au dessus de Y reg , et plus précisément que π est une composition d'un nombre fini d'éclatements de centres lisses (inclus dans Y sing donc).

Du point de vue de la géométrie birationnelle, on va considérer une classe importante des variétés Y pour lesquelles le diviseur de Weil K Y est un diviseur de Q-Cartier, autrement dit qu'il provient d'un fibré en droites. De telles variétés sont dites Q-Gorenstein. Une paire (Y, ∆) est la donnée d'une variété normale (projective) Y et d'un Q-diviseur de Weil (effectif) ∆ sur Y tels que K Y + ∆ soit Cartier. On tire en arrière K Y + ∆ par (n'importe quelle résolution) π : X → Y, ce qui nous donne

K X + D = π * (K Y + ∆) + ∑ i a i E i
où D est le transformé strict de ∆ dans X, les E i sont des diviseurs exceptionnels, et a i ∈ Q (appelé la discrépance) ne dépend pas du choix de π. Nous donnons alors la définition suivante : Définition 1.2.1. Une paire (Y, ∆) est dite canonique, (resp. Kawamata log terminale (klt), ou log canonique (lc)) si pour toute résolution π : X → Y, les coefficients a i vérifient a i ≥ 0, (resp. a i > -1, et a i ≥ -1).

Il est facile de voir qu'il suffit de vérifier ces conditions sur une résolution. Nous disons également que la variété Y est canonique, (resp. klt ou lc) si la paire (Y, 0) l'est. Il est intéressant de garder à l'esprit que la condition "klt" est une condition de volume fini pour une certaine forme de volume naturrelle attaché à la paire (Y, ∆), cf. [START_REF] Eyssidieux | Singular Kähler-Einstein metrics[END_REF].

Exemple 1.2.2. Soient S une surface algébrique complexe normale, p ∈ S une singularité isolée. Alors

• p est canonique ⇐⇒ c'est une singularité quotient C 2 /G, G ⊂ SL 2 (C) un sous-groupe fini ;

• p est klt ⇐⇒ c'est une singularité quotient C 2 /G, où G ⊂ GL 2 (C) est un sous-groupe fini.

Exemple 1.2.3. En dimension supérieure, les singularités quotient sont encore log terminales.

Soient n ≥ 2, et H ⊂ CP n+1 une hypersurface lisse de degré d. Alors, le cône affine sur H n'a que des singularités canoniques ssi d ≤ n + 1. En particulier, le point double ordinaire z 2 0 + z 2 1 + z 2 2 + z 2 3 = 0 n'a que des singularités canoniques, mais il n'est pas une singularité quotient.

Afin de mieux comprendre la relation entre le flot de Kähler-Ricci (ainsi que la métrique canonique) et le Programme des modèles minimaux analytiques (qui a été proposé par Song-Tian), nous travaillons sur les variétés, et plus généralement sur les paires à singularités modérées.

Si les singularités sont modérées, on peut encore donner du sens aux objets d'étude clés : fibré canonique, courbure de Ricci, etc. La recherche de flots géométriques conduit à étudier certaines équations de Monge-Ampère complexes paraboliques dégénérées. La tentative de résoudre ces équations se heurte à des difficultés analytiques, car la solution n'est plus lisse. Il faut introduire de nouveaux outils, étudier les solutions faibles, et établir une régularité partielle de ces dernières. C'est l'objectif principal de cette thèse.

Courants et fonctions quasi-plurisousharmoniques

Courants positifs

Un courant T de dimension q sur une variété réelle lisse M de dimension m est une forme continue sur l'espace des q-formes différentielles lisses à support compact dans M. On note ⟨T, θ⟩ le crochet de dualité entre une q-forme lisse θ et un courrant T de dimension q. Notons qu'un courant T de dimension q peut aussi se concevoir comme une forme de degré (mq) à coefficients distributions. Le courant dT est le courant de dimension q -1 défine par ⟨dT, θ⟩ := (-1) m-q+1 ⟨T, dθ⟩, pour toute (q -1) forme test θ ; on dira que T est fermé si dT = 0.

Soit maintenant X une variété complexe de dimension n. Alors, les formes différentielles se décomposent selon leurs bidegrés en les coordonnées locales dz j , d zk . Un courant de bidimension (p, q) est une forme linéaire continue sur l'espace des formes différentielles lisses de bidegré (p, q) à support compact dans X. On dit également que le courant est de bidegré (np, nq) ; nous nous intéresserons particulièrement aux courants de bidegré (1,1), un tel courant T peut être représenté en coordonnées locales comme une forme différentielle T = ∑ 1≤j,k≤n T jk idz j ∧ d zk où les coefficients T jk sont des distributions. On dira que T est positif si la distribution ∑ a j āk T jk est positive pour tout a ∈ C n ; dans ce cas les T jk sont des mesures.

Fonctions quasi-plurisousharmoniques

On définit l'opérateur réel d c = i 2 ( ∂ -∂) tel que dd c = i∂ ∂. Rappelons qu'une fonction semicontinue supérieurement (scs) φ sur un ouvert Ω de C n à valeur dans [-∞, +∞[ est dite plurisousharmonique (psh en abrégé) si le courant T := dd c φ est un courant positif fermé de bidegré (1,1) sur Ω ; on impose de plus à φ de ne pas être identiquement -∞ sur chaque composante de Ω. L'avantage de définir ainsi les fonctions psh est qu'on voit que le tiré en arrière d'une fonction psh par une application holomorphe est encore psh. En particulier, la notion de fonction psh s'étend à une variété complexe X.

Même si la notion de fonction psh a un sens global, on s'intéresse le plus souvent au comportement local de telles fonctions. Cependant, sur une variété (kählerienne) compacte, le principe du maximum nous dit qu'il n'y a aucune fonction psh non constante. Un moyen simple de contourner ce problème est de rendre la notion locale : soit X une variété complexe compacte, une fonction scs φ : X → [-∞, +∞[ est dite quasi-plurisousharmonique (quasi-psh en abrégé) si elle s'écrit localement comme somme d'une fonction psh et d'une fonction lisse sur X. Définition 1.2.4. Soit θ une (1,1)-forme lisse réelle sur une variété complexe X. On dit qu'une fonction est θ-plurisousharmonique (θ-psh) si elle est quasi-psh et θ + dd c φ ≥ 0 au sens des courants. On notera PSH(X, θ) l'ensemble des fonctions θ-psh qui ne sont pas identiquement -∞.

Nous revoyons à [START_REF] Demailly | Complex analytic and differential geometry. freely available from the author's web site[END_REF][START_REF] Guedj | Degenerate complex Monge-Ampère equations[END_REF] pour les propriétés fondamentaux concernant les fonctions quasi-psh et les courants positifs.

Nombres de Lelong

Un des invariants pour l'étude ponctuelle des singularités d'une fonction (quasi-)psh est le nombre de Lelong. En réalité, il a été défini directement pour des courants positifs fermés de bidegré (p, p), mais en ce qui nous concerne, nous considérerons le cas du bidegré (1,1), c'est-àdire des fonctions (quasi-)psh.

Soient X une variété (kählerienne) complexe et φ une fonction quasi-psh sur X. Le nombre de Lelong de φ en un point x ∈ X est défini par

ν(φ, x) := sup{γ ≥ 0 : φ(z) ≤ γ log |z -x| + O(1) près de x}.
En particulier, si φ = log | f | pour une fonction holomorphe f , alors on trouve facilement que ν(φ, x) est l'ordre d'annulation de f en x. On peut donc aussi parler des nombres de Lelong d'un (1,1)-courant positif fermé.

En fait, les nombres de Lelong d'une fonction quasi-psh mesurent les singularités analytiques. Soit φ une fonction quasi-psh sur une variété complexe X. Pour chaque c > 0, on définit l'ensemble de niveau de nombres de Lelong de φ comme suit :

E c (φ) := {x ∈ X : ν(φ, x) ≥ c}.
Un résultat fondamental dû à Siu [START_REF] Siu | Analyticity of sets associated to Lelong numbers and the extension of closed positive currents[END_REF] affirme alors que E c (φ) est un sous-ensemble analytique fermé de X. On renvoie à [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF] pour une preuve plus simple.

Positivité des classes de cohomologie

Soit X est une variété kählerienne compacte et ω une métrique kählerienne fixée. On introduit des notions de positivité pour des classes de cohomologie. Définition 1.2.5. Soit α ∈ H 1,1 (X, R). On dit que • α est une classe kählerienne ssi elle peut être présentée par une forme kählerienne, i.e., une

(1,1)-forme fermée lisse définie positive ;

• α est une classe nef (numériquement effective) ssi pour tout ε > 0 il existe une (1,1)-forme fermée lisse θ ε ∈ α telle que θ ε ≥ -εω ;

• α est une classe big ssi elle peut être présentée par un courant kählerienne, i.e., un (1,1)courant fermé T tel que T ≥ εω pour ε > 0 assez petit ;

• α est une classe pseudoeffective (psef) ssi elle peut être représentée par un (1,1)-courant positif fermé.

Ces définitions ne dépendent pas du choix de la métrique kählerienne ω.

Exemple 1.2.6. Soient π : X → P 2 l'éclatement de P 2 en un point p, et

E := π -1 (p) le diviseur exceptionel. Si α ∈ H 1,1 ( X, R) alors elle est de la forme α = aπ * {ω FS } + b{E}, où a, b ∈ R. Par définition, nous obtenons que α est kählerienne ssi a > -b > 0, nef ssi a ≥ -b ≥ 0, big ssi a > 0, a > -b, et psef ssi a ≥ 0, a ≥ -b.

Comparaison des singularités

Si φ et φ ′ sont deux fonctions quasi-psh sur X, on dit alors que φ est moins singulière que φ ′ s'il existe une constante C > 0 telle que φ ′ ≤ φ + C sur X.

Fixons θ une (1,1)-forme lisse représentant une classe (psef) α. Une fonction θ-psh est dite avoir des singularités minimales si elle est moin singulière que toutes les autres. De telles fonctions à singularités minimales toujours existent. En effet, l'enveloppe supérieure V θ := sup{φ ∈ PSH(X, θ) : φ ≤ 0 sur X} définit une fonction θ-psh à singularités minimales. Si θ est kählerienne (ou semi-positive) alors V θ ≡ 0.

Etant donnés un courant positif T = θ + dd c φ et son potentiel global φ, on dit qu'ils ont des singularités analytiques s'il existe un réel c > 0 tel que

φ = c log N ∑ j=1 | f j | 2 + v, localement sur X, où v est lisse et f 1 , . . . , f N sont des fonctions holomorphes.
Grâce au théorème de régularisation de Demailly [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF], il est possible d'approximer un courant kählerien dans sa classe de cohomologie par des courants à singularités analytiques. En particulier, si la classe α est big, alors elle contient de nombreux courants kähleriens à singularités analytiques.

Définition 1.2.7. Si α est une classe big, on définit son lieu ample Amp(α) comme l'ensemble des points x tels qu'il existe un courant kählerien T ∈ α à singularités analytiques qui est lisse au voisinage de x.

Le lieu ample Amp(α) est par définition un ouvert de Zariski, et il est non-vide grâce au théorème de régularisation de Demailly. En effet, il résulte de [Bou04, Theorem 3.17] qu'il existe un courant kählerien T 0 ∈ α, ou une fonction θ-psh φ 0 à singularités analytiques dont son lieu régulier (lisse) est exactement Amp(α). Par conséquent, on peut voir facilement qu'une fonction θ-psh à singularités minimales, e.g., V θ , est localement bornée sur le lieu ample Amp(α) puisque elle est moins singulière que φ 0 .

Produit de Monge-Ampère non-pluipolaire

Dans le papier [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF], Boucksom-Eyssidieux-Guedj-Zeriahi ont définit le produit nonpluripolaire de tout (1,1)-courant positif fermé en suivant [START_REF] Bedford | Fine topology, Šilov boundary, and (dd c ) n[END_REF][START_REF] Guedj | The weighted Monge-Ampère energy of quasiplurisubharmonic functions[END_REF]. On rappelle brièvement la construction du produit non-pluripolaire d'une fonction quasi-psh. Soient X une variété kählerienne compacte et α = {θ} ∈ H 1,1 (X, R) une classe de cohomologie big. Etant donnée une fonction θ-psh φ à singularités minimales, grâce à Bedford-Taylor [START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF], la mesure de Monge-Ampère (θ + dd c φ) n est bien définie comme une mesure de Borel positive sur Amp(α) ; elle est de plus de masse finie (voir [BEGZ10, Sect. 1.2]). On étend donc trivialement cette mesure sur tout X, et la mesure obtenue est appelée la mesure de Monge-Ampère non-pluripolaire de φ. Pour une fonction θ-psh quelconque φ, on va considérer ses approximants canoniques φ j := max(φ, V θj) qui sont à singularités minimales. On observe que la suite des mesures de Borel

µ j := 1 {φ>V θ -j} (θ + dd c φ j ) n
est croissante, puisqu'on a même µ j = 1 {φ>V θ -j} µ j+1 , vu que φ j = φ j+1 sur {φ > V θ -j}. Comme la masse de µ j est finie la limite croissante des µ j définit alors une mesure de Radon sur X, qu'on note MA θ (φ) ou (θ + dd c φ) n , la mesure de Monge-Ampère non-pluripolaire de φ. On notera que la mesure non-pluripolaire MA(φ) ne charge pas les pluripolaires grâce à sa construction. On définit maintenant le volume de la classe big α comme la masse totale de V θ ,

Vol(α) := ˆX MA θ (V θ ).
Observons que pour une fonction θ-psh quelconque φ, la masse totale de sa mesure non-pluripolaire MA θ (φ) est inférieure ou égale à Vol(α).

Définition 1.2.8. On dit que φ ∈ PSH(X, θ) est de masse de Monge-Ampère maximale si sa mesure non-pluripolaire MA θ (φ) est de masse Vol(α). On note E (X, θ) l'ensemble des telles fonctions.

Notons que les fonctions θ-psh à singularité minimales est de masse de Monge-Ampère maximale ; voir [BEGZ10, Theorem 1.16].

Comme on l'a vu plus haut, pour tout φ ∈ PSH(X, θ) de masse Monge-Ampère maximale MA θ (φ) est une mesure de masse Vol(α) ne chargeant pas les pluripolaires. Réciproquement, on a le suivant : Théorème 1.2.9 ([GZ07, BEGZ10]). Soient X une variété kählerienne compacte et α = {θ} ∈ H 1,1 (X, R) une classe de cohomologie big. Soit µ une mesure positive ne chargeant pas les pluipolaires et vérifiant

µ(X) = Vol(α). Alors, il existe une unique φ ∈ E (X, θ) telle que MA θ (φ) = µ normalisée par sup X φ = 0. Deplus, lorsque µ = f dV avec f ∈ L p (X, dV) pour un certain p > 1, alors il existe une constante C > 0 ne dépendant que de θ, dV, p et ∥ f ∥ L p telle que φ ≥ V θ -C.
La partie existence étend le résultat principal de [START_REF] Guedj | The weighted Monge-Ampère energy of quasiplurisubharmonic functions[END_REF], qui correspond exactement au cas où α est une classe kählerienne. En fait, la preuve de ce théorème consiste à réduire au cas kählerien via les décompositions de Zariski (approchées). La démonstration de [START_REF] Guedj | The weighted Monge-Ampère energy of quasiplurisubharmonic functions[END_REF] consiste d'ailleurs à réguliser µ de manière à pouvoir lui appliquer le résultat fondamental de Yau [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation[END_REF]. On pourra utliser l'approche variationelle établie dans [START_REF] Berman | A variational approach to complex Monge-Ampère equations[END_REF] pour une autre preuve.

L'unicité peut être obtenue en adaptant la démonstration de S. Dinew [START_REF] Dinew | Uniqueness in E (X, ω)[END_REF] (qui traite aussi le cas kählerien).

La dernière partie du théorème est démontrée en adaptant la méthode de Kołodziej [START_REF] Kołodziej | The complex Monge-Ampère equation[END_REF] à notre cas où la forme de référence n'est que big. Il a besoin de généraliser la notion de capacité et d'estimer la décroissance de capacité des sous-ensembles de niveau {φ < V θ -t} lorsque t → +∞.

Présentation des résultats

On s'est rendu compte il y a quelques décennies qu'il est nécessaire de traiter des variétés algébriques complexes (ou des paires) modérément singulières afin d'étudier la classification birationnelle des variétés complexes lisses en dimension ≥ 3. Il est donc souhaitable de construire des métriques canoniques ainsi que des flots de Kähler (Chern)-Ricci (en un sens faible) sur des variétés à singularités modérées.

Continuité des potentiels de Monge-Ampère dans les classes big

Étant donné une paire kählerienne (Y, ∆), sur laquelle on introduit la notion suivante de métrique de Kähler-Einstein (dû à [START_REF] Berman | Kähler-Einstein metrics on stable varieties and log canonical pairs[END_REF]) :

Définition 1.3.1. On dit qu'un courant positif fermé ω KE ∈ c 1 (K Y + ∆) sur Y est une métrique de Kähler-Einstein singulière à courbure negative pour la paire (Y, ∆) si 1. Le produit non-pluripolaire ω n KE définit une mesure (localement) absolument continue sur Y reg par rapport à dz ∧ d z et log(ω n KE /dz ∧ d z) ∈ L 1 loc (Y reg ), où z = (z i ) sont des coordonnées holomorphes locales ; 2. Ric(ω KE ) = -ω KE + [∆] sur Y reg ; 3. ´Yreg ω n KE = vol(K Y + ∆).
La condition sur ω n KE nous permet de donner un sens à la courbure de Ricci au sens des courants en posant Ric(ω KE ) := -i∂ ∂ log ω n KE . La condition de bord que ω n KE ne porte pas de masse sur Y sing est assez naturelle, ainsi les potentiels locaux vont vivre dans des espaces d'énergie pour lesquels la théorie pluripotentialiste est très efficace. Cette condition de masse (ainsi que la condition de KE) est intrinsèque sur Y reg , et donne une notion de métrique de Kähler-Einstein assez naturelle à manipuler.

Dans [START_REF] Berman | Kähler-Einstein metrics on stable varieties and log canonical pairs[END_REF], Berman-Guenancia ont défini une "bonne" notion de métrique de Kähler-Einstein sur la paire (semi)-log canonique (Y, ∆) telle que K Y + ∆ est ample (la variété n'est nécessairement pas normale). Cette métrique a un sens habituel sur le lieu Y reg \ Supp(∆), i.e., sur lequel ω est lisse.

Peut-on construire une métrique de KE sur Y qui soit au moins lisse sur le lieu ample de K Y + ∆ ? La réponse est affirmative si K Y + ∆ est additionellement nef, cf. [START_REF] Tsuji | Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type[END_REF] (sur des variétés minimales) ou [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF] (du point de vue des équations de Monge-Ampère dans des classes big).

A ce jour, la question de la régularité dans le cas big est encore ouverte. Nous montrons la continuité de telles métriques.

Théorème A. Supposons que (Y, ∆) est une paire log canonique telle que le fibré canonique K Y + ∆ est big. Alors, il existe une unique métrique de Kähler-Einstein ω sur Y vérifiant

Ric(ω) = -ω + [∆] au sens de courants et ´Y ω n = Vol(K Y + ∆). De plus, ω a des potentiels continus sur Amp(K Y + ∆) ∩ (Y, ∆) reg \ ⌊∆⌋. Ici, (Y, ∆) reg désigne le lieu où la paire (Y, ∆) est log lisse et ⌊∆⌋ désigne la partie entière de ∆, i.e. si ∆ = ∑ i d i ∆ i alors ⌊∆⌋ = ∑⌊d i ⌋∆ i .
Il se trouve qu'une telle définition est bien intrinsèque, et équivalente à la définition (chronologiquement postérieure) présentée précédente en terme de masse, étant donné le travail fondateur de [START_REF] Eyssidieux | Singular Kähler-Einstein metrics[END_REF]

(θ + i∂ ∂φ) n = f dV, sup X φ = 0. (1.3.2)
Supposons que f ≲ e -ϕ g où g ∈ L p (dV), avec p > 1 et ϕ est une fonction quasi-plurisousharmonique. Si ϕ est localement bornée sur un ouvert de Zariski U ⊂ Amp(θ), alors φ est continue sur U.

Nous rappelons que comme µ = f dV est une mesure non-pluripolaire, i.e. elle ne charge pas sur tout l'ensemble pluripolaire, l'équation (1.3.2) admet alors une unique solution (normalisée) φ ∈ E (X, θ) ; voir [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF]. Comme avant, E (X, θ) désigne l'ensemble des fonctions θ-psh de masse de Monge-Ampère maximale, au sens de [START_REF] Guedj | The weighted Monge-Ampère energy of quasiplurisubharmonic functions[END_REF][START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF]. La condition nécessaire est donc ´X f dV = Vol({θ}).

Notons aussi que l'existence d'une unique solution à l'équation de Monge-Ampère complexe ci-dessus a été produite dans [START_REF] Berman | Kähler-Einstein metrics on stable varieties and log canonical pairs[END_REF] en adaptant l'approche variationelle (qui est developpée dans [START_REF] Berman | A variational approach to complex Monge-Ampère equations[END_REF]). Il reste à étudier la continuité de cette solution. Pour cela, on va adapter la méthode de Di Nezza-Lu [START_REF] Di Nezza | Complex Monge-Ampère equations on quasi-projective varieties[END_REF] pour établir une estimée uniforme et ensuite on utilise le résultat de Berman [START_REF] Berman | From Monge-Ampère equations to envelopes and geodesic rays in the zero temperature limit[END_REF] pour construire une suite décroissante de fonctions continues sur le lieu ample de θ, qui converge vers φ. Enfin, nous montrons que cette convergence est uniforme.

Les flots de Monge-Ampère dans les classes big

Suite au programme de Song-Tian, on a besoin de reprendre l'étude et la définition du flot dans des situations plus dégénérées, ou encore sur une nouvelle variété singulière. On est ramené à l'étude d'équations de Monge-Ampère complexes paraboliques avec des données initiales dégénérées. Cela nécessite de développer une théorie des solutions faibles à ces équations.

Dans [START_REF] Song | The Kähler-Ricci flow through singularities[END_REF], Song et Tian ont démontré l'existence des solutions faibles à certaines équations de Monge-Ampère complexe parabolique dégénérées avec des données initiales continues. Cela nous permet de démarrer le flot de Kähler-Ricci (en un sens faible) à partir d'un courant avec des potentiels continus sur les variétés projectives à singularités log terminales. Eyssidieux, Guedj et Zeriahi [START_REF] Eyssidieux | Weak solutions to degenerate complex Monge-Ampère flows II[END_REF][START_REF] Eyssidieux | Convergence of weak Kähler-Ricci flows on minimal models of positive Kodaira dimension[END_REF] ont développé une approche de la viscosité parabolique, il faut néanmoins que la densité soit continue. Cela nous permet de traiter les singularités canoniques. Récemment, Guedj, Lu, et Zeriahi dans leurs travaux [START_REF] Guedj | The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows[END_REF][START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF], développent une théorie pluripotentielle parabolique afin de traiter le cas fondamental des paires kähleriennes à singularités Kawamata log terminales (klt). Cette théorie est connue comme l'analogue parabolique à celle de Bedfor-Taylor. Du point de vue analytique, les singularités klt nous conduisent à traiter les densités qui appartiennent à L p , pour certains p > 1.

On va étendre la théorie des solution faibles aux flots de Monge-Ampère complexes au cadre des classes de cohomologie big.

Avant de présenter nos résultats, nous donnons les notations et hypothèses que nous utiliserons dans la suite de ce paragraphe.

Soit X une variété kählerienne compacte de dimension complexe n. On suppose que (ω t ) t∈]0,T[ où T ∈]0, +∞[, est une famille lisse de (1,1) formes big vérifiant qu'il existe une (1,1) forme réelle big θ telle que

g(t)θ ≤ ω t , ∀ t ∈]0, T[, (1.3.3)
où g : [0, T] → R + est une fonction de classe C 1 positive et croissante. On désigne par Ω le lieu ample de θ. Supposons ainsi que

• 0 ≤ f ∈ L p , pour p > 1 et f est presque partout strictement positive ; • F : [0, T] × X × R → R est continue ;
• la fonction r → F(•, •, r) est croissante en r ;

• F est uniformément Lipschitz en (t, r), i.e. il existe une constante κ F > 0 telle que pour tout

t, t ′ ∈ [0, T], r, r ′ ∈ R, et x ∈ X, |F(t, x, r) -F(t ′ , x, r ′ )| ≤ κ F (|t -t ′ | + |r -r ′ |); • la fonction (t, r) → F(t, •, r) est convexe.
Nous considérons maintenant l'équation de Monge-Ampère complexe parabolique du type

dt ∧ (ω t + dd c φ t ) n = e φt +F(t,•,φ t ) f dV ∧ dt (CMAF)
sur X T :=]0, T[×X, où dV est une forme volume sur X. Remarquons que les deux côtés de l'équation ci-dessus peuvent être interprétés comme des mesures sur ]0, T[×Ω.

Nous utiliserons la méthode de Perron, inspirée par [START_REF] Guedj | The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows[END_REF]. Sous des hypothèses naturelles sur les données de Cauchy, nous montrons que l'enveloppe des sous-solutions pluripotentielles est semi-concave en temps et continue en espace.

Théorème C. Soit φ 0 une fonction ω 0 -psh à singularités minimales. Alors l'enveloppe supérieure U des sous-solutions pluripotentielles à (CMAF) avec la donnée initale φ 0 est une solution à (CMAF) qui est localement uniformément Lipschitz et localement uniformément semi-concave en t ∈]0, T[. De plus, si ωt ≤ Aω t pour une constante A > 0, alors U est continue sur (0, T) × Ω.

La preuve de ce théorème se divise en quelque étapes. Nous montrons 1. premièrement que l'enveloppe des sous-solutions pluripotentielles est localement uniformément Lipschitz en temps et qu'elle-même est une sous-solution pluripotentielle ;

2. ensuite qu'elle est localement uniformément semi-concave ;

3. qu'elle est une solution pluripotentielle en appliquant le processus de balayage ;

4. qu'elle a des singularités minimales et enfin qu'elle est continue sur ]0, T[×Ω (une conséquence directe de [START_REF] Dang | Continuity of Monge-Ampère Potentials in Big Cohomology Classes[END_REF]).

Nous démontrons que la solution U est unique en établissant le principe de comparaison suivant :

Théorème D. Soit φ (resp. ψ) une sous-solution (resp. sur-soulution) pluripotentielle de (CMAF) avec la valeur initale φ 0 (resp. ψ 0 ). On suppose que ψ est localement uniformément semi-concave en t ∈]0, T[ et continue sur ]0, T[×Ω. On suppose de plus que pour chaque t, ψ t est à singularités minimales. Si φ 0 ≤ ψ 0 , on a alors φ ≤ ψ sur [0, T[×X.

Nous obtenons alors deux applications géométriques grâce aux résultats précédents. La première est l'existence du flot pout tout temps dans le cas de type général.

Il est connu que le temps d'existence maximal du flot de Kähler-Ricci est infini à condition que le fibré canonique K X soit nef. Dans [FIK03, Section 10], Feldman, Ilmanen et Knopf ont posé une question : peut-on définir et construire des solutions faibles du flot de Kähler-Ricci au-delà du temps singulier ? Tô [START_REF] Tô | Convergence of the Weak Kähler-Ricci Flow on Manifolds of General Type[END_REF] a utilisé la théorie de la viscosité pour montrer que le flot de Kähler-Ricci existe pour tout temps au sens de la viscosité et converge (sur Amp(K X )) vers l'unique métrique de Kähler-Einstein singulière dans la classe c 1 (K X ) construite dans [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF], lorsque t → +∞. Nous appliquons les outils ci-dessus pour établir l'analogue pluripotentiel du résultat principal de [START_REF] Tô | Convergence of the Weak Kähler-Ricci Flow on Manifolds of General Type[END_REF] :

Théorème E. Soit X

Les flots de Chern-Ricci pluripotentiels

Dans cette partie, nous présentons brièvement les résultats de [START_REF] Dang | Pluripotential Chern-Ricci Flows[END_REF], qui seront détaillés ensuite au chapitre 4. C'est une généralisation des résultats de Guedj-Lu-Zeriahi [GLZ20] et de l'auteur [START_REF] Dang | Pluripotential Monge-Ampère flows in big cohomology classes[END_REF] (qui ont été introduits dans le paragraphe précédent).

Cette partie consiste à définir le flot de Chern-Ricci sur une variété complexe modérément singulière, motivée par le programme de Song-Tian [START_REF] Song | The Kähler-Ricci flow through singularities[END_REF]. Cela nécessite d'étendre les théories des solutions faibles à certaines équations de Monge-Ampère complexes paraboliques dégénérées.

Étant donnée une variété complexe compacte X, on considère dans ce paragraphe l'équation de Monge-Ampère complexe parabolique dégénérée du type suivant On introduit une notion de solutions pluripotentielles de telles équations, un analogue parabolique de la théorie développée par Bedford et Taylor dans leurs articles fondateurs [START_REF] Bedford | The Dirichlet problem for a complex Monge-Ampère equation[END_REF][START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF].

dt ∧ (ω t + dd c φ t ) n = e ∂ t φ t +F(t,x,φ t ) f (x)dV(x) ∧ dt (CMAF) sur X T :=]0, T[×X,
On peut interpréter l'équation ci-dessus sur X comme une équation aux dérivées partielles (EDP) d'ordre 2 sur la variété X T de dimension réelle 2n + 1. Les deux côtés de (CMAF) sont bien définis commes des mesures de Radon positives. On introduit donc P (X T , ω) l'ensemble des potentiels paraboliques ; voir Définition 4.1.6 pour plus de détails.

Nous régularisons (CMAF) par des flots de Monge-Ampère complexes lisses (nous régularisons les formes ω t en des formes positives et nous régularisons les membres droites de l'équation) et établissons des estimées a priori afin de démontrer le résultat suivant : Théorème G. Soit φ 0 une fonction ω 0 -psh bornée. Alors il existe un potentiel parabolique φ ∈ P (X T , ω) tel que

• (t, x) → φ(t, x) est localement bornée sur [0, T[×X ; • φ vérifie (CMAF) ; • (t, x) → φ(t, x) est continue dans ]0, T[×(X \ {ρ = -∞}) ; • t → φ t est localement uniformément semi-concave dans ]0, T[×X ;
• φ t → φ 0 au sens faible et ponctuellement, lorsque t → 0 + . De plus, la fonction φ est l'unique solution pluripotentielle avec une telle régularité.

Il s'avère que t → φ t (x)n(t log tt) + Ct est croissante pour C > 0 fixée. La convergence en temps zéro est donc plutôt forte : ie. uniforme si φ 0 est continue.

La propriété de la semi-concavité de la solution φ construite dans le Théorème G est un outil crucial pour la procédure d'approximation. La continuité de φ est une conséquence immédiate dans la théorie elliptique ; cf [GL21, Theorem 3.7]. L'unicité de la solution avec une telle régularité découle du principe de comparaison ; voir Section 4.3.3 pour le détail. Nous désignons par Φ(ω t , F, f , φ 0 ) l'unique solution de (CMAF) avec des données initiales ci-dessus ω t , F, f et φ 0 .

Nous étudions ensuite la régularité partielle de la solution pluripotentielle ci-dessus. Théorème H. On suppose de plus que F est lisse sur [0, T] × X × R. Supposons ainsi que la densité f est de la forme f = e ψ + -ψ -où • ψ ± sont quasi-psh sur X ;

• sup X ψ ± ≤ C, et ∥e -ψ -∥ L p ≤ C pour une constante quelconque C > 0 ;

• ψ ± sont (lisse) localement bornées sur un ouvert de Zariski U de X \ {ρ = -∞}.

Alors la solution pluripotentielle Φ(ω t , F, f , φ 0 ) de (CMAF) est lisse sur ]0, T[×U.

La forme de la densité f est très naturelle, en lien avec la nature des singularités que l'on traite : Kawamata log terminales (klt).

L'approche pluripotentielle actuelle nous permet de définir une bonne notion du flot de Chern-Ricci faible sur les variétés à singularités log terminales (ou plus généralement sur les paires klt). Nous introduissons la définition suivante : Définition 1.3.2. Soient Y une variéte Q-Gorenstein complexe à singularités log terminales et θ 0 une métrique hermitienne vérifiant

T max := {t > 0 : ∃ ψ ∈ C ∞ (Y) telle que θ 0 -tRic(θ 0 ) + dd c ψ > 0} > 0.
Fixons S 0 = θ 0 + dd c ϕ 0 un (1,1)-courant positif avec ϕ 0 une fonction θ 0 -psh bornée. On dit qu'une famille (θ t ) t∈[0,T max [ des courants semi-positifs sur Y est une solution du flot de Chern-Ricci pluri- potentiel à partir de S 0 si les conditions suivantes sont vérifiées

1. θ t = θ 0 -tRic(θ 0 ) + dd c φ t pour φ t ∈ PSH(Y, θ 0 -tRic(θ 0 )) ; 2. θ t → S 0 lorsque t → 0 + ; 3. (θ t ) t∈]0,T max [ se restreint à une famille des formes hermitiennes sur Y reg vérifiant ∂θ t ∂t = -Ric(θ t ).
Nous obtenons un résultat sur l'existence et l'unicité du flot de Chen-Ricci pluripotentiel sur les variétés à singularités log terminales. En outre, on établit une régularité partielle pour le flot si la donnée initiale n'est pas lisse. On obtient le résultat suivant.

Théorème I. Soit Y une variété complexe à singularités log terminales. Supposons que θ 0 est une métrique hermitienne sur Y et soit

T max := {t > 0 : ∃ ψ ∈ C ∞ (Y) telle que θ 0 -tRic(θ 0 ) + dd c ψ > 0},
et S 0 = θ 0 + dd c ϕ 0 un (1,1)-courant positif avec ϕ 0 une fonction θ 0 -psh bornée. Alors il existe une solution unique (θ t ) t∈[0,T max [ du flot de Chern-Ricci pluripotentiel à partir de S 0 .

Introduction

Finding canonical metrics on complex varieties is a fundamental problem of complex geometry. As evidenced by recent developments in Kähler geometry in connection with the Minimal Model Program, it is natural and necessary to allow the varieties Y in question to be singular. Working on a desingularization π : X → Y is led to consider degenerate complex Monge-Ampère equations of the form

(θ + dd c φ) n = e λφ f dV, (2.1.1)
where θ is a smooth closed real (1, 1)-form representing a big cohomology class α on X, λ ∈ {0, ±1}, and f is a density of the form f = e ψ + -ψ -, with ψ + , ψ -being quasi-plurisubharmonic functions on X. The integrability properties of f depend on the singularities of Y.

Finding a Kähler-Einstein metric on a stable variety Y boils down to solving (2.1.1) on X for λ = 1 and f ∈ L 1-δ (X, dV) for all δ ∈ (0, 1). Building on the variational approach developed in [START_REF] Berman | A variational approach to complex Monge-Ampère equations[END_REF], Berman and Guenancia [START_REF] Berman | Kähler-Einstein metrics on stable varieties and log canonical pairs[END_REF] have proved that the equation (2.1.1) admits a unique finite energy solution φ, in the sense of [START_REF] Guedj | The weighted Monge-Ampère energy of quasiplurisubharmonic functions[END_REF][START_REF] Berman | A variational approach to complex Monge-Ampère equations[END_REF]. When {θ} is additionally nef, they established the smoothness of φ on a Zariski open set. As in the classical case of Yau [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation[END_REF], the main difficulty lies in establishing an a priori C 0 -estimate. Unfortunately the (normalized) solution φ to (2.1.1) is in general unbounded, so a natural idea is to try and bound such solution from below by a reference quasi-plurisubharmonic function.

This motivates the following general question : for which densities f ≥ 0 is the solution φ locally bounded in some Zariski open subset of X ? The main result of this paper is the following : Theorem 2.1.1. Let X be a compact Kähler manifold of dimension n and fix a smooth closed real (1, 1)form θ which represents a big cohomology class. Let φ ∈ E (X, θ) be the unique normalized solution to

(θ + dd c φ) n = f dV, sup X φ = 0.
(2.1.2)

Assume that f ≤ e -ϕ for some quasi-plurisubharmonic function ϕ on X. Then φ is continuous on Amp(θ) \ E 1 (ϕ), where Amp(θ) is the ample locus of θ and E 1 (ϕ) = {x ∈ X : ν(ϕ, x) ≥ 1}, with ν(ϕ, x) being the Lelong number of ϕ at x.

Let us recall that E 1 (ϕ) which is called the Lelong super-level set of ϕ, is an analytic subset of X by Siu's result [START_REF] Siu | Analyticity of sets associated to Lelong numbers and the extension of closed positive currents[END_REF]. We refer the reader to Section 2.2.3 for the definition of the ample locus of a big cohomology class.

Since µ = f dV is non-pluripolar, it is known [BEGZ10, Section 3] that there exists a unique normalized solution φ ∈ E (X, θ), so the point is to study its regularity. The idea of the proof is that we first use Demailly's equisingular approximation [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF][START_REF] Demailly | On the cohomology of pseudoeffective line bundles[END_REF] (see Theorem 2.2.1) to replace ϕ by a quasi-psh function ϕ 1 which has analytic singularities with polar locus Z contained in the set of points where the Lelong number of ϕ is greater than or equal to 1. We then adapt the approach of Di Nezza and Lu [START_REF] Di Nezza | Complex Monge-Ampère equations on quasi-projective varieties[END_REF] (see Theorem 2.3.2) to prove the continuity of φ in the complement of E 1 (ϕ) in the ample locus of θ. We also prove a slightly stronger version of Theorem 2.1.1 valid for more singular densities (see Theorem 2.3.1).

When the density f is smooth in a Zariski open set (i.e. outside an analytic subset), one expects the solution φ to be smooth in a Zariski open set (see [START_REF] Dinew | Open problems in pluripotential theory[END_REF]Question 21,22]), but we are unable to prove this for the moment. When f belongs to L p (X) for some p > 1, the Hölder continuity of φ on the ample locus was shown in [DDG + 14]. The smoothness of φ when f is smooth is largely open.

Under the extra assumption that the class {θ} is nef, the regularity properties for the solutions for the degenerate Monge-Ampère equation (2.1.2) have been studied by many authors (see [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF][START_REF] Berman | Kähler-Einstein metrics on stable varieties and log canonical pairs[END_REF][START_REF] Di Nezza | Complex Monge-Ampère equations on quasi-projective varieties[END_REF] and the references therein). The strategy in these papers is that one first establishes a relative uniform estimate which allows to adapt classical ideas of Yau [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation[END_REF] and Siu [START_REF] Siu | Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics[END_REF] to obtain locally uniform estimates for the Laplacian, and one finally uses Evans-Krylov's general regularity theory to conclude. In the above case functions in E (X, θ) have zero Lelong numbers (see [GZ07, Corollary 1.8], [DDL18b, Theorem 1.1]). Using this property Di Nezza and Lu [START_REF] Di Nezza | Complex Monge-Ampère equations on quasi-projective varieties[END_REF] have generalized Kołodziej's approach [START_REF] Kołodziej | The complex Monge-Ampère equation[END_REF] to establish a relative uniform estimate. Let us mention that in the general case of a big class even the "least singular" potential V θ may have positive Lelong numbers. To overcome this difficulty we exploit fine properties of quasi-plurisubharmonic envelopes inspired by [START_REF] Darvas | The metric geometry of singularity types[END_REF][START_REF] Lu | Complex Hessian equations with prescribed singularity on compact Kähler manifolds[END_REF].

Our approach allows us to deal with non-nef data. As an application we prove that the unique singular Kähler-Einstein metric obtained in [START_REF] Berman | Kähler-Einstein metrics on stable varieties and log canonical pairs[END_REF] is continuous on some Zariski open subset. More precisely, we have the following : Corollary 2.1.2. Let (Y, ∆) be a projective log canonical pair of general type, i.e. the canonical line bundle K Y + ∆ is big. Then there is a unique singular Kähler-Einstein metric ω on Y such that

Ric(ω) = -ω + [∆]
in the weak sense of currents, and such that ´Y ω n = vol(K Y + ∆). Furthermore ω has continuous potentials on Amp(K Y + ∆) ∩ (Y, ∆) reg \ ⌊∆⌋. Here, (Y, ∆) reg denotes the locus of points p ∈ Y where the pair (Y, ∆) is log smooth at p, i.e. Y is smooth and ∆ has simple normal crossing (snc) support on a neighborhood of p, and ⌊∆⌋ denotes the integral part of

∆, i.e. if ∆ = ∑ d i ∆ i , then ⌊∆⌋ = ∑⌊d i ⌋∆ i .
Recall that the ample locus of a big line bundle L, denoted by Amp(L) may be defined as the ample locus of its first Chern class α = c 1 (L) (see Definition 2.2.2).

If K Y + ∆ is additionally nef, then the potential of ω was already known to be smooth on Amp(K Y + ∆) ∩ (Y, ∆) reg \ ⌊∆⌋, as follows from the combined arguments of Boucksom-Eyssidieux-Guedj-Zeriahi [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF] (see also [START_REF] Berman | From Monge-Ampère equations to envelopes and geodesic rays in the zero temperature limit[END_REF]) and Berman-Guenancia [START_REF] Berman | Kähler-Einstein metrics on stable varieties and log canonical pairs[END_REF].

Organization of the chapter

The chapter is organized as follows. In Section 2.2 we recall basic pluripotential theory that will be needed later on. The proof of Theorem 2.1.1 is given in Section 2.3.1, while Corollary 2.1.2 is proved in Section 2.3.2.

Notations

In the whole chapter we fix • X a n-dimensional compact Kähler manifold,

• dV a smooth volume form on X,

• α ∈ H 1,1 (X, R) a big cohomology class, and θ a smooth representative of α • a Kähler form ω so that ω ≥ θ.

Preliminaries

The purpose of this section is to recall some essential materials in pluripotential theory which will be used later.

Quasi-psh functions

Recall that an upper semi-continuous function φ : X → R ∪ {-∞} is called quasi-plurisubharmonic (quasi-psh for short) if it is locally the sum of a smooth and a plurisubharmonic (psh for short) function. We say that φ is θ-plurisubharmonic (θ-psh for short) if it is quasi-psh, and θ + dd c φ ≥ 0 in the sense of currents, where d c is normalized so that dd c = i π ∂ ∂. By the dd c -lemma any closed positive (1, 1)-current T cohomologous to θ can be written as T = θ + dd c φ for some θ-psh function φ which is furthermore unique up to an additive constant.

We let PSH(X, θ) denote the set of all θ-psh functions which are not identically -∞. This set is endowed with the L 1 (X)-topology. By Hartogs' lemma φ → sup X φ is continuous in this weak topology. Since the set of closed positive currents in a fixed cohomology class is compact (in the weak topology), it follows that the set of φ ∈ PSH(X, θ), with sup X φ = 0 is compact.

Quasi-psh functions are in general singular, and a convenient way to measure their singularities is the Lelong numbers. Let x 0 ∈ X. Fixing a holomorphic chart x 0 ∈ V x 0 ⊂ X, the Lelong number ν(φ, x 0 ) of a quasi-psh function φ at x 0 ∈ X is defined as follows :

ν(φ, x 0 ) := sup{γ ≥ 0 : φ(z) ≤ γ log ∥z -x 0 ∥ + O(1), on V x 0 }.
We remark here that this definition does not depend on the choice of local charts. In particular, if φ = log | f | in a neighborhood V x 0 of x 0 , for some holomorphic function f , then ν(φ, x 0 ) is equal to the vanishing order ord x 0 ( f ) := sup{k ∈ N : D γ f (x 0 ) = 0, ∀ |γ| < k}. We can also define the Lelong super-level sets, for c > 0,

E c (φ) := {x ∈ X : ν(φ, x) ≥ c}.
We also use the notation E c (T) for a closed positive (1, 1)-current T. A well-known result of Siu [START_REF] Siu | Analyticity of sets associated to Lelong numbers and the extension of closed positive currents[END_REF] asserts that the Lelong super-level sets E c (φ) are analytic subsets of X. We refer the reader to [Dem92, Remark 3.2] for a simple proof.

Demailly's equisingular approximation

We next recall the basic result on the approximation of psh functions by psh functions with analytic singularities. For details about this, we refer the reader to [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF][START_REF] Demailly | On the cohomology of pseudoeffective line bundles[END_REF].

Following Demailly [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF], a closed positive (1, 1)-current T = θ + dd c φ and its global potential φ are said to have analytic singularities if there exists c > 0 such that

φ = c log N ∑ j=1 | f j | 2 + v,
locally on X, where v is a smooth function and the f j 's are holomorphic functions.

Thanks to dd c -Lemma, the problem of approximating a positive closed (1, 1)-current is reduced to approximating a quasi-psh function. The following result of Demailly [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF][START_REF] Demailly | On the cohomology of pseudoeffective line bundles[END_REF] on the equisingular approximation of a quasi-psh function by quasi-psh functions with analytic singularities is crucial : Theorem 2.2.1 (Demailly's equisingular approximation). Let φ be a θ-psh function on X. There exists a decreasing sequence of quasi-psh functions (φ m ) such that 1. (φ m ) converges pointwise and in L 1 (X) to φ as m → +∞, 2. φ m has the same singularities as 1/2m times a logarithm of a sum of squares of holomorphic functions, 3. θ + dd c φ m ≥ -ε m ω, where ε m > 0 decreases to 0 as m → +∞, 4. ´X e 2m(φ m -φ) dV < +∞ ; 5. φ m is smooth outside the analytic subset E 1/m (φ).

Proof. We refer the reader to [Dem15, Theorem 9, Lemma 2] for a proof.

Big cohomology classes

A cohomology class α ∈ H 1,1 (X, R) is big if it contains a Kähler current, i.e. there is a positive closed current T ∈ α and ε > 0 such that T ≥ εω. Theorem 2.2.1 enables us in particular to approximate a Kähler current T inside its cohomology class by Kähler currents T m with analytic singularities, with a very good control of the singularities. A big class therefore contains plenty of Kähler currents with analytic singularities. Definition 2.2.2. We let Amp(α) denote the ample locus of α, i.e. the Zariski open subset of all points x ∈ X for which there exists a Kähler current T x ∈ α with analytic singularities such that T x is smooth in a neighborhood of x.

It follows from the work of Boucksom [Bou04, Theorem 3.17 (ii)] that one can find a single Kähler current T 0 ∈ α with analytic singularities such that Amp(α) = X\Sing(T 0 ).

In particular T 0 is smooth in the ample locus Amp(α).

Given φ, ψ ∈ PSH(X, θ), we say that φ is less singular than ψ, and denote by ψ ⪯ φ, if there exists a constant C such that ψ ≤ φ + C on X. We say that φ, ψ have the same singularity type, and denote by φ ≃ ψ if φ ⪯ ψ and ψ ⪯ φ. Definition 2.2.3. A θ-psh function is said to have minimal singularities if it is less singular than any θ-psh function. Such a function is not unique in general, only its class of singularities is. Following Demailly, one defines the extremal function

V θ := sup{φ ∈ PSH(X, θ) : φ ≤ 0}.
It is a θ-psh function with minimal singularities. By the analysis above V θ is locally bounded on the ample locus Amp(α). Of course we have V θ ≡ 0 if θ is semi-positive.

Non-pluripolar Monge-Ampère operator

Let φ 1 , • • • , φ n ∈ PSH(X, θ) with minimal singularities. Then they are locally bounded on the ample locus Amp(α). Following the construction of Bedford-Taylor [START_REF] Bedford | The Dirichlet problem for a complex Monge-Ampère equation[END_REF][START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF] in the local setting, it has been shown in [BEGZ10, Section 1.2] that the product

(θ + dd c φ 1 ) ∧ • • • ∧ (θ + dd c φ n )
is well-defined as a positive Radon measure on Amp(α) and it has finite total mass. One can then extend it trivially on the whole X.

In particular, if φ 1 = • • • = φ n = φ then this procedure defines the (non-pluripolar) Monge-Ampère measure of a function φ ∈ PSH(X, θ) with minimal singularities. For a general φ ∈ PSH(X, θ), its canonical approximants φ j := max(φ, V θj), j > 0 have minimal singularities. One can show that the sequence of Borel positive measures 1 {φ>V θ -j} (θ + dd c φ j ) is increasing in j. Its (strong) limit

MA θ (φ) = (θ + dd c φ) n := lim j→+∞ ↗ 1 {φ>V θ -j} (θ + dd c φ j ) n
is the non-pluripolar Monge-Ampère measure of φ. The volume of a big class α = {θ} is given by the total mass of the non-pluripolar Monge-Ampère measure of V θ , i.e. Vol(α) := ˆX MA θ (V θ ).

We say that φ ∈ PSH(X, θ) has full Monge-Ampère mass if ´X MA θ (φ) = Vol(α). We let

E (X, θ) := φ ∈ PSH(X, θ) : ˆX MA θ (φ) = Vol(α)
denote the set of θ-psh functions with full Monge-Ampère mass. Note that θ-psh functions with minimal singularities have full Monge-Ampère mass (see [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF]Theorem 1.16] for more details), but the converse is not true.

We recall here the plurifine locality of the non-pluripolar product, which will be used several times in this chapter. Lemma 2.2.4. Assume that φ, ψ are θ-psh function such that φ = ψ on an open set U in the plurifine topology. Then

1 U MA θ (φ) = 1 U MA θ (ψ).
We stress in particular that sets of the form {u < v}, where u, v are quasi-psh functions, are open in the plurifine topology.

Proof. The proof for locally bounded functions can be found in [BT87, Corollary 4.3] or [BEGZ10, Section 1.2]. For the general case we write φ (resp. ψ) as the decreasing limits of its canonical approximants φ t := max(φ, V θt) (resp. ψ t := max(ψ, V θt)). We observe that φ t (resp. ψ t ) is locally bounded on the ample locus Amp(θ). By the result for locally bounded functions we have

1 U∩(φ>V θ -t) MA θ (φ t ) = 1 U∩(ψ>V θ -t) MA θ (ψ t ).
Letting t → +∞, we conclude the proof.

Capacities

The Monge-Ampère capacity

For the convenience of the reader we recall here a few facts contained in [START_REF] Guedj | Degenerate complex Monge-Ampère equations[END_REF]. Let K be a Borel subset of X. The Monge-Ampère capacity is

Cap ω (K) := sup ˆK(ω + dd c u) n : u ∈ PSH(X, ω), -1 ≤ u ≤ 0 .
Lemma 2.2.5. Let ν = gdV be a Radon positive measure with 0 ≤ g ∈ L p (dV) for some p > 1. Then there exists B > 0 depending on n, p ω, dV and ∥g∥ L p (dV) such that, for all Borel subsets K of X,

ν(K) ≤ B • Cap(K) 4 .
Proof. This result was proved by Kolodziej ; see [Koł98, Section 2.5]. We refer the readers to [EGZ09, Proposition 3.1] for an alternative proof.

The generalized capacity

We present here a generalization of this notion introduced by Di Nezza and Lu [START_REF] Di Nezza | Complex Monge-Ampère equations on quasi-projective varieties[END_REF][START_REF] Di Nezza | Generalized Monge-Ampère capacities[END_REF] (see also [START_REF] Darvas | Monotonicity of nonpluripolar products and complex Monge-Ampère equations with prescribed singularity[END_REF]Section 4.1]). Definition 2.2.6. Let ψ ∈ PSH(X, θ). We define the ψ-relative capacity of a Borel subset K ⊂ X by

Cap θ,ψ (K) := sup ˆK MA θ (u) : u ∈ PSH(X, θ), ψ -1 ≤ u ≤ ψ .
Note that when θ is Kähler, a related notion of capacity has been studied in [START_REF] Di Nezza | Generalized Monge-Ampère capacities[END_REF][START_REF] Di Nezza | Complex Monge-Ampère equations on quasi-projective varieties[END_REF]. The (generalized) Monge-Ampère capacity plays a vital role in establishing uniform estimates for complex Monge-Ampère equation (see e.g. [EGZ09, BEGZ10, DL15, DL17a] and the references therein). We shall use the ψ-capacity Cap θ,ψ in the proof of Theorem 2.3.2.

The following results are important for the sequel.

Lemma 2.2.7. Fix φ ∈ E (X, θ) and ψ ∈ PSH(X, θ). Then the function

H(t) := Cap θ,ψ ({φ < ψ -t}), t ∈ R,
is right-continuous and H(t) → 0 as t → +∞.

Proof. The proof is almost the same as the one of [DL17a, Lemma 2.6] in the Kähler case, i.e. θ = ω is Kähler. For the reader's convenience, we give the proof here. The right-continuity is straightforward . For the second statement, we first assume that ψ ≤ V θ . Fix u ∈ PSH(X, θ) such that ψ -1 ≤ u ≤ ψ. The generalized comparison principle ([BEGZ10, Corollary 2.3]) yields ˆ{φ<ψ-t}

MA θ (u) ≤ ˆ{φ<u-t+1} MA θ (u) ≤ ˆ{φ<u-t+1} MA θ (φ) ≤ ˆ{φ<V θ -t+1} MA θ (φ).
The last term goes to zero as t → +∞ as φ ∈ E (X, θ). This finishes the proof.

Lemma 2.2.8. Let ψ be a quasi-psh function such that θ + dd c ψ ≥ δω for some δ ∈ (0, 1). Then for any Borel set K ⊂ X,

Cap ω (K) ≤ 1 δ n Cap θ,ψ (K).
Proof. Let u be a ω-psh function such that -1 ≤ u ≤ 0. We then have that φ := ψ + δu is a candidate defining Cap θ,ψ . It follows that

δ n ˆK(ω + dd c u) n ≤ ˆK(θ + dd c ψ + dd c (δu)) n ≤ Cap θ,ψ (K),
and taking the supremum over all u we get the desired estimate.

Generalizing [Koł98, Ineq. (2.3.2)], we have the following result which is a simple consequence of the generalized comparison principle ([BEGZ10, Corollary 2.3]).

Lemma 2.2.9. Let ψ ∈ PSH(X, θ) and φ ∈ E (X, θ). Then for all t > 0 and 0 < s ≤ 1 we have

s n Cap θ,ψ ({φ < ψ -t -s}) ≤ ˆ{φ<ψ-t} MA θ (φ).
Proof. Let u be a θ-psh function such that ψ -1 ≤ u ≤ ψ. We then have

{φ < ψ -t -s} ⊂ {φ < su + (1 -s)ψ -t} ⊂ {φ < ψ -t}. Since s n MA θ (u) ≤ MA θ (su + (1 -s)ψ
) and φ has full Monge-Ampère mass, it follows from the generalized comparison principle ([BEGZ10, Corollary 2.3]) that

s n ˆ{φ<ψ-t-s} MA θ (u) ≤ ˆ{φ<ψ-t-s} MA θ (su + (1 -s)ψ) ≤ ˆ{φ<su+(1-s)ψ-t} MA θ (su + (1 -s)ψ) ≤ ˆ{φ<su+(1-s)ψ-t} MA θ (φ) ≤ ˆ{φ<ψ-t} MA θ (φ).
Since u was taken arbitrarily as a candidate in the definition of Cap θ,ψ , the proof therefore finishes.

Quasi-psh envelopes

For a Borel function h, we let P θ (h) denote the largest θ-psh function lying below h :

P θ (h) := (sup{φ ∈ PSH(X, θ) : φ ≤ h on X}) * .
Proposition 2.2.10. Fix φ ∈ E (X, θ). Then for any b > 0, P ω (bφ -bV θ ) is a ω-psh function with full Monge-Ampère mass.

The proof given below is inspired by [DDL21b, Lemma 4.3].

Proof. We first show that the function P ω (bφ -bV θ ) ̸ ≡ -∞ for all b > 0.

For each j ∈ N we set φ j := max(φ, V θj) and ψ j := P ω (bφ j -bV θ ). We observe that (ψ j ) is a decreasing sequence of ω-psh functions, and ψ j ≥ -jb for each j. Therefore the proof would follow if we could show that lim j ψ j is not identically -∞. We let for each j, D j := {ψ j = bφ j -bV θ } denote the contact set. Observe that the sets D j are non-empty for j large enough. Fix t > 0. We see that

{ψ j ≤ -t} ∩ D j = {φ j ≤ V θ -t/b} ⊂ {φ ≤ V θ -t/b}.
Set ω := 1 b + 1 ω. By Lemma 2.2.11 below and plurifine locality we have for j > t/b,

ˆ{ψ j ≤-t} (ω + dd c ψ j ) n = ˆ{ψ j ≤-t} 1 D j (ω + dd c ψ j ) n ≤ b n ˆ{ψ j ≤-t} 1 D j ( ω + dd c φ j ) n ≤ b n ˆ{φ≤V θ -t/b} ( ω + dd c φ j ) n = b n ˆX( ω + dd c φ j ) n - ˆ{φ>V θ -t/b} ( ω + dd c φ) n , (2.2.1) since φ j = φ on {φ > V θ -t/b} for j > t/b.
Suppose by contradiction that sup X ψ j → -∞ as j → +∞. It thus follows that {ψ j ≤ -t} = X for j large enough, t being fixed. Hence, for j > 0 large enough, (2.2.1) becomes

ˆX ω n ≤ b n ˆX( ω + dd c φ j ) n - ˆ{φ>V θ -t/b} ( ω + dd c φ) n .
Letting j → +∞, we obtain

ˆX ω n ≤ b n ˆX( ω + dd c φ) n - ˆ{φ>V θ -t/b} ( ω + dd c φ) n , (2.2.2)
where we have used that

( ω + dd c φ j ) n = n ∑ k=0 n k ( ω -θ) k ∧ (θ + dd c φ j ) n-k → ( ω + dd c φ) n
in the weak sense of measures on X, thanks to [DDL18a, Theorem 2.3, Remark 2.5]. Finally, letting t → +∞, in (2.2.2) we obtain a contradiction. Consequently, ψ j decreases to a ω-psh function, we infer that P ω (bφ -bV θ ) is a ω-psh function for any b > 0.

It remains to show that P ω (bφ -bV θ ) has full Monge-Ampère mass. Observe that P ω (bφ

- bV θ ) ≥ b A P ω (Aφ -AV θ ) for A > b.
Using monotonicity of mass (see e.g. [WN19, Theorem 1.2]), we obtain

ˆX (ω + dd c P ω (bφ -bV θ )) n ≥ 1 - b A n ˆX ω n + b A n ˆX (ω + dd c P ω (Aφ -AV θ )) n .
Letting A → +∞ we thus finish the proof.

Lemma 2.2.11. Let b > 0. Assume that φ and P ω (bφ

-bV θ ) ∈ PSH(X, ω). Then the measure (ω + dd c P ω (bφ -bV θ )) n is supported on the contact set D := {P ω (bφ -bV θ ) = bφ -bV θ }, and
1 D (ω + dd c P ω (bφ -bV θ )) n ≤ b n 1 D 1 + 1 b ω + dd c φ n .
Proof. We refer the readers to [DDL21b, Lemma 4.4] for a proof of the first statement.

For the second one, set u = 1 b P ω (bφ -bV θ ) + V θ . Then u is a ω := 1 b + 1 ω-psh function, and u ≤ φ. It follows from [GZ17a, Corollary 10.8] that

1 {u=φ} ( ω + dd c u) n ≤ 1 {u=φ} ( ω + dd c φ) n .
(2.2.3) Furthermore, the measure (ω + dd c P ω (bφ

-bV θ )) n is supported on the contact set D = {bφ - bV θ = P ω (bφ -bV θ )} = {u = φ}, hence 1 b n (ω + dd c P ω (bφ -bV θ )) n = 1 {u=φ} 1 b n (ω + dd c P ω (bφ -bV θ )) n ≤ 1 {u=φ} ( ω + dd c u) n .
(2.2.4) Combining (2.2.3) and (2.2.4) we obtain the desired estimate.

Regularity of solutions

Proof of the Main Theorem

In this section we prove Theorem 2.1.1. The key ingredient is an adaptation of Di Nezza-Lu's approach [START_REF] Di Nezza | Complex Monge-Ampère equations on quasi-projective varieties[END_REF] (see also [START_REF] Di Nezza | Generalized Monge-Ampère capacities[END_REF]).

Given a non-negative Radon measure µ whose total mass is Vol(α), we consider the Monge-Ampère equation

MA θ (φ) = µ. (2.3.1)
The systematic study of such equations in big cohomology classes has been initiated in [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF]. It has been shown there that (2.3.1) admits a unique normalized solution φ ∈ E (X, θ) if and only if µ is a non pluripolar measure on X.

Our goal is to prove the following :

Theorem 2.3.1. Assume ν = gdV to be a Radon measure, with 0 ≤ g ∈ L p (dV) for some p > 1.

Let µ be a non-pluripolar measure such that µ(X) = Vol(α). Assume that µ = f dν, with f ≤ e -ϕ for some quasi-psh function ϕ on X. Let φ ∈ E (X, θ) be the unique normalized solution to (2.3.1). Then φ is continuous on Amp(α) \ E 1/q (ϕ), where q denotes the conjugate exponent of p.

Note that Theorem 2.1.1 in the introduction is a particular case of Theorem 2.3.1. We first establish this result under an extra assumption. More precisely, we have the following theorem, which is closely similar to [DL17a, Theorem 3.1] in the case that θ is Kähler.

Theorem 2.3.2. Let φ ∈ E (X, θ) be normalized by sup X φ = 0. Assume that MA θ (φ) ≤ e -ϕ gdV, for some quasi-psh function ϕ on X, and 0 ≤ g ∈ L p (dV), with p > 1. Assume that ϕ is locally bounded on an open set U ⊂ Amp(α). Then φ is continuous on U.

Proof of Theorem 2.3.2. We fix a θ-psh function ρ 0 on X such that θ + dd c ρ 0 ≥ 2δ 0 ω, for some small constant δ 0 > 0. Replacing ρ 0 by ρ 0sup X ρ 0 , we can always assume that ρ 0 ≤ V θ . Moreover, we can choose ρ 0 such that it is smooth in the ample locus, with analytic singularities thanks to [Bou04, Theorem 3.17 (ii)].

We will divide the proof in three steps.

Step 1. We prove that φ is locally bounded on U.

We pick a > 0 so small that aϕ belongs to PSH(X, δ 0 ω). Set ψ := ρ 0 + aϕ. We thus have θ + dd c ψ ≥ δ 0 ω, and ψ ≤ V θ + aϕ. We claim that

φ ≥ ψ -A, (2.3.2)
for A > 0 depending only δ 0 , p, dV, ∥g∥ L p (dV) , and ´X e -2P ω (a -1 φ-a -1 V θ ) gdV.

We remark that by Proposition 2.2.10, for any b > 0, P ω (bφ -bV θ ) is a ω-psh function with full Monge-Ampère mass, hence it has zero Lelong numbers (see [GZ07, Corollary 1.8]). Therefore, Skoda's integrability theorem [START_REF] Skoda | Sous-ensembles analytiques d'ordre fini ou infini dans C n[END_REF] ensures that e -P ω (bφ-bV θ ) belongs to L q (dV) for all q < +∞.

In particular, ´X e -2P ω (bφ-bV θ ) gdV is finite for any b > 0.

The proof of the claim above follows the approach of Di Nezza and Lu [START_REF] Di Nezza | Complex Monge-Ampère equations on quasi-projective varieties[END_REF][START_REF] Di Nezza | Generalized Monge-Ampère capacities[END_REF]. To see this, fix s ∈ [0, 1], t > 0. Set dν = gdV, b = a -1 . Using Lemma 2.2.9 and the assumption on MA θ (φ) we have

s n Cap θ,ψ ({φ < ψ -t -s}) ≤ ˆ{φ<ψ-t} MA θ (φ) ≤ ˆ{φ<ψ-t} e b(ψ-φ) e -ϕ dν ≤ ˆ{φ<ψ-t} e -(bφ-bV θ ) dν ≤ ˆ{φ<ψ-t} e -P ω (bφ-bV θ ) dν, (2.3.3)
where we have used that bψ ≤ bV θ + ϕ in the third inequality. Using Hölder's inequality we have ˆ{φ<ψ-t}

e -P ω (bφ-bV θ ) dν ≤ (ν({φ < ψ -t})) 1/2 ˆX e -2P ω (bφ-bV θ ) dν 1/2 . (2.3.4)
By Lemma 2.2.5, one can find a constant B > 0 depending on n, p, ω, dV, and ∥g∥ L p (dV) such that

ν(•) 1/2 ≤ B(Cap ω (•)) 2 . Since θ + dd c ψ ≥ δ 0 ω it follows from Lemma 2.2.8 that Cap ω ≤ δ -n 0 Cap θ,ψ , hence ν(•) 1/2 ≤ Bδ -2n 0 Cap θ,ψ (•) 2 .
(2.3.5) By (2.3.3), (2.3.4) and (2.3.5) we thus get

s n Cap θ,ψ ({φ < ψ -s -t}) ≤ C Cap θ,ψ ({φ < ψ -t}) 2 , (2.3.6)
where C depends on ω, δ 0 , n, p, ∥g∥ L p (dV) , and ´X e -2P ω (bφ-bV θ ) dν. Set

H(t) := Cap θ,ψ ({φ < ψ -t}) 1/n , t > 0.
By the estimate (2.3.6) we get

sH(t + s) ≤ C 1/n H(t) 2 .
It follows from Lemma 2.2.7 that the function H is right-continuous and H(+∞) = 0. We can thus apply [EGZ09, Lemma 2.4] which yields H(t 0 + 2) = 0, where t 0 > 0 is such that

H(t 0 ) < 1 2C 1/n .
Therefore, for A = t 0 + 2 we have φ ≥ ψ -A on X\P for some Borel subset P such that Cap θ,ψ (P) = 0. By Lemma 2.2.8 we have Cap ω (P) = 0 so P is a pluripolar set. Hence φ ≥ ψ -A everywhere.

Using Hölder's inequality it follows from (2.3.3) (take s = 1) that

H(t) n ≤ ˆX e -2P ω (bφ-bV θ )) dν 1/2 ˆ{φ<ψ-t+1} dν 1/2 ≤ ˆX e -2P ω (bφ-bV θ ) gdV 1/2 1 t -1 ˆX |ψ -φ|gdV 1/2 .
The last integral is bounded by a uniform constant : using Hölder's inequality again we have ´X |ψ -φ|gdV ≤ ∥g∥ L p (dV) ∥ψ∥ L q (dV) + ∥φ∥ L q (dV) with q = p/(p -1). Since φ belongs to the compact set of θ-psh functions normalized by sup X φ = 0, its L q norm is bounded by an absolute constant only depending on θ, dV and p. Consequently, we can choose t 0 > 0 to be only dependent on dV, p, ∥g∥ L p (dV) , and an upper bound for ´X e -2P ω (bφ-bV θ ) dν.

Step 2. There exists a sequence of functions φ j ∈ PSH(X, θ) ∩ C 0 (Amp(α)) which decreases to φ.

For convenience, we normalize φ so that sup X φ = -1. Let 0 ≥ h j be a sequence of smooth functions decreasing to φ. Then the sequence of θ-psh functions φ j := P θ (h j ) decreases to φ as j → +∞. Indeed, since the operator P θ is monotone, the sequence φ j is decreasing to a θ-psh function u and since φ j ≥ φ for all j we have u ≥ φ. Moreover, u(x) ≤ φ j (x) ≤ h j (x), ∀ x ∈ X, for all j, hence u(x) ≤ φ(x), as claimed. Furthermore, we have that φ j is continuous in Amp(α) for each j. In fact, [Ber19, Ineq. (1.2)] gives an upper bound on the Monge-Ampère measure of φ j :

MA θ (φ j ) ≤ 1 D MA θ (h j ), D = {φ j = h j }.
The equality also holds following the work of Di Nezza and Trapani [START_REF] Di Nezza | Monge-Ampère measures on contact sets[END_REF]. In particular MA θ (φ j ) has an L ∞ -density, hence it follows from [DDG + 14, Theorem D] that φ j is Hölder continuous in Amp(α), as claimed.

Step 3. We show that the decreasing convergence φ j → φ is locally uniform on U and finish the proof.

Fix λ ∈ (0, 1). For any j ∈ N, set

ψ j := λψ + (1 -λ)φ j -(A + 2)λ,
where

A > 0 is the uniform constant so that φ ≥ ψ -A (see (2.3.2)). If we pick ε j ≤ λ 2(1-λ) δ 0 for j > 0 big enough, then θ + dd c ψ j ≥ λ 2 δ 0 ω. We observe by definitions that φ j ≤ V θ , hence ψ j ≤ λψ + (1 -λ)V θ ≤ V θ + λaϕ. Set H j (t) := Cap θ,ψ j ({φ < ψ j -t}) 1/n , t > 0.
For any s ∈ [0, 1], t > 0, we can argue as above to obtain

sH j (t + s) ≤ C 1/n H j (t) 2 ,
for C > 0 only depending on p, dV, ∥g∥ L p (dV) , δ 0 , λ, and ´X e -2P ω (cφ-cV θ ) gdV, with c = (λa) -1 . Let χ be an increasing convex weight such that χ(0) = 0, χ(-∞) = -∞, and φ has finite χ-energy

(see [BEGZ10, Proposition 2.11]). Since φ j ≥ φ ≥ ψ -A we have ψ j ≤ φ j -2λ. It follows from Lemma 2.2.9 (take s = λ, ψ = ψ j + λ) that λ n Cap θ,ψ j {φ < ψ j } ≤ ˆ{φ<ψ j +λ} MA θ (φ) ≤ ˆ{φ<φ j -λ} MA θ (φ) ≤ 1 -χ(-λ) ˆX(-χ • (φ -φ j )) MA θ (φ).
The latter converges to 0 as j → +∞ since φ j decreases to φ, namely H j (0) goes to 0 as j → +∞.

We thus take j > 0 so big that H j (0)

≤ 1/(2C 1/n ). It thus follows from [EGZ09, Remark 2.5] that H j (t) = 0 if t ≥ t 0 where t 0 ≤ 4C 1/n H j (0). We then have φ ≥ λψ + (1 -λ)φ j -(A + 2)λ -4C 1/n H j (0). (2.3.7)
We have H j (0) → 0 as j → ∞. Letting j → +∞, we thus obtain

lim j→+∞ inf K (φ -φ j ) ≥ -λ(sup K |ψ| + A + 2), (2.3.8)
for any compact subset K of U. We observe by definition that ψ = ρ 0 + aϕ is bounded on K.

Finally, letting λ → 0 in (2.3.8) we obtain that the convergence φ j → φ is locally uniform on U, hence φ is continuous on U.

Proof of Theorem 2.3.1. By rescaling, we may assume without loss of generality that ϕ is a ω-psh function. By Theorem 2.2.1, we can find a quasi-psh function ϕ m with analytic singularities such that ω + dd c ϕ m ≥ -ε m ω for ε m > 0 decreasing to 0 as m → +∞, and ˆX e 2m(ϕ m -ϕ) dV < +∞.

Note that ϕ m is smooth outside the analytic set E 1/m (ϕ) ⊂ X. We see that

MA θ (φ) ≤ e -ϕ m e (ϕ m -ϕ) gdV. Set now g ′ = e (ϕ m -ϕ) g. We choose m = [q]
, where [q] denotes the integer part of q. We have 2m > q, hence there is a constant

p ′ > 1 such that 1 p ′ = 1 p + 1 2m . It follows from Hölder's inequality that g ′ ∈ L p ′ (dV). Observe that ϕ m is smooth in the complement X \ E 1/m (ϕ) of the analytic set E 1/m (ϕ) ⊂ X, in particular it is locally bounded on Amp(α) \ E 1/m (ϕ) ⊃ Amp(α) \ E 1/q (ϕ).
We can thus apply Theorem 2.3.2 to complete the proof.

In particular if p = ∞ then we can choose m = 1 to conclude the proof of Theorem 2.1.1.

Kähler-Einstein metrics on log canonical pairs of general type

Log canonical singularities

A pair (Y, ∆) is by definition a connected complex normal projective variety Y and an effective Weil Q-divisor ∆. We will say that the pair (Y, ∆) has log canonical singularities if K Y + ∆ is Q-Cartier, and if for some (or equivalently any) log resolution π : X → Y of (Y, ∆), we have

K X = π * (K Y + ∆) + ∑ i a i E i ,
where E i are either exceptional divisors or components of the strict transform of ∆, and the coefficients a i ∈ Q satisfy the inequality a i ≥ -1 for all j. The divisor ∑ i E i has simple normal crossing support. We denote the singular set of Y by Y sing and let

Y reg := Y \ Y sing .
Let m be a positive integer such that m

(K Y + ∆) is Cartier. If we choose σ a local generator of m(K Y + ∆) defined on an open subset U of Y, then (i mn 2 σ ∧ σ) 1/m defines a smooth volume form on U ∩ (Y reg \ Supp(∆)). If f i is a local equation of E i around a point π -1 (U), then we can see that π * i mn 2 σ ∧ σ 1/m = ∏ i | f i | 2a i dV (2.3.9)
locally on π -1 (U) for some local volume form dV.

The previous construction leads to the following adapted measure which is introduced in [EGZ09, Sect. 6] : Definition 2.3.3. Let (Y, ∆) be a pair and let h be a smooth hermitian metric on the

Q-line bundle O Y (K Y + ∆). The corresponding adapted measure µ Y,h on Y reg is locally defined by choosing a nowhere zero section σ of O Y (m(K Y + ∆)) over a small open set U and setting µ Y,h := (i mn 2 σ ∧ σ) 1/m |σ| 2/m h m .
The point of the definition is that the measure µ Y,h does not depend on the choice of σ. This measure can be extended by zero across Y sing ∪ Supp(∆). Remark that the restriction σ| Y reg can be viewed as a meromorphic form with a pole of order md i on ∆ i where

∆ = ∑ i d i ∆ i is the decomposition of ∆ into prime divisors (see [EGZ09, Sect. 6.3]). The Lelong-Poincaré formula yields -dd c log µ Y,h = [∆] -iΘ h (K Y + ∆)
on Y reg , where [∆] is the integration current on ∆.

Kähler-Einstein metrics

Let (Y, ∆) be a log canonical pair, with

n = dim C Y. Assume that K Y + ∆ is a big Q-line bundle.
We next recall the notion of (negatively curved) Kähler-Einstein metric attached to a pair (Y, ∆). There are various equivalent definitions for such an object (e.g. in [EGZ09, BEGZ10, BG14]), we choose here the following definition in the sense of [START_REF] Berman | Kähler-Einstein metrics on stable varieties and log canonical pairs[END_REF]Sect. 3]. Definition 2.3.4. We say that a closed positive current

ω KE ∈ c 1 (K Y + ∆) on Y is a (singular) Kähler-Einstein metric (KE for short) with negative curvature for (Y, ∆) if 1. The non-pluripolar product ω n KE defines a (locally) absolutely continuous measure on Y reg with respect to dz ∧ d z and log(ω n KE /dz ∧ d z) ∈ L 1 loc (Y reg ), where z = (z i ) are local holo- morphic coordinates ; 2. Ric(ω KE ) = -ω KE + [∆] on Y reg ; 3. ´Yreg ω n KE = vol(K Y + ∆).
The condition (1) allows us to define (on Y reg ) the Ricci curvature of ω KE by setting Ric(ω KE ) := -dd c log(ω n KE ). Another way of thinking of this is to interpret the positive measure ω n KE | Y reg as a singular metric on -K Y reg whose curvature is then Ric(ω KE ) by definition.

Let h be a smooth hermitian metric on K Y + ∆ with curvature η. Finding a singular Kähler-Einstein metric is equivalent to solving the following Monge-Ampère equation for an η-psh ϕ with full Monge-Ampère mass

(η + dd c ϕ) n = e ϕ+c µ Y,h , (2.3.10)
for some c ∈ R. Indeed, we set ω := η + dd c ϕ. Since ϕ is locally integrable ω satisfies the condition (1) (see Eq. (2.3.9)). We also have

Ric(ω) = -dd c log ω n = -dd c ϕ -dd c log µ Y,h = -dd c ϕ -η + [∆]
on Y reg . Condition (3) is clearly satisfied.

We now prove Corollary 2.1.2 in the introduction. Assume the initial pair (Y, ∆) is lc of general type, i.e. the canonical bundle K Y + ∆ is big. We consider a log resolution π : (X, D) → (Y, ∆) of the pair, with

K X + D = π * (K Y + ∆).
Here, D = ∑ a i D i is a R-divisor with simple normal crossing support (snc for short) on X, consisting of π-exceptional divisors with coefficients in (-∞, 1], and of the strict transforms of the components of ∆ with coefficients in (0, 1]. The (singular) Kähler-Einstein metric ω KE for (X, D), or equivalently the pull-back of the (singular) KE metric for (Y, ∆) by π can be written as

ω KE = θ + dd c φ where θ = π * η is a smooth representative of c 1 (K X + D)
and φ is a θ-psh function (with full Monge-Ampère mass) solving the following Monge-Ampère equation It is convenient to differentiate the "klt part" of D from its "non-klt part", so we set g = ∏ a i <1 |s i | -2a i ∈ L p for some p > 1, and ϕ = ∑ a i =1 2 log |s i |. We see that ϕ is smooth outside of the non-klt locus D nklt , in particular it is locally bounded on Amp(θ) \ D nklt . Since φ is bounded from above, we can therefore apply Theorem 2.3.2 to complete the proof.

MA θ (φ) = e φ dV ∏ i |s i | 2a i . ( 2
With the notations above, since {θ} is a pull-back by π of a big class, we have

Amp(θ) = π -1 (Amp(K Y + ∆)) \ Exc(π)
which projects onto Amp(K Y + ∆) ∩ (Y, ∆) reg . We next observe that the projection of the non-klt locus

D nklt is contained in (Y, ∆) sing ∪ ⌊∆⌋, with ⌊∆⌋ = ∑ d i =1 ∆ i .
The proof of Corollary 2.1.2 thus follows.

Introduction

The primary goal of this chapter is to study pluripotential complex Monge-Ampère flows motivated by the Minimal Model Program (MMP for brevity) in algebraic geometry, whose aim is the (birational) classification of projective manifolds. In a recent celebrated work, Birkar-Cascini-Hacon-Mckernan [START_REF] Birkar | Existence of minimal models for varieties of log general type[END_REF] showed the existence of minimal models for a large class of varieties which are called varieties of general type. J. Song and G. Tian [START_REF] Song | Canonical measures and Kähler-Ricci flow[END_REF][START_REF] Song | The Kähler-Ricci flow through singularities[END_REF] have recently proposed an analytic analogue making use of (twisted) Kähler-Ricci flows on compact Kähler manifolds.

Let X be a compact Kähler manifold of dimension n equipped with a Kähler form ω. The (normalized) Kähler-Ricci flow on X starting at ω is the solution to the following evolution equation

∂θ t ∂t = -Ric(θ t ) -λθ t , θ| t=0 = ω, (3.0.1)
where the sign of λ ∈ R depends on that of the first Chern class c 1 (K X ). Solving the normalized Kähler-Ricci flow (3.0.1) turns out to be equivalent to solving the scalar complex Monge-Ampère flow

(ω t + dd c φ t ) n = e ∂ t φ t +λφ t +h(t,x) dV ω t + dd c φ t > 0,
where h is a smooth density, and

ω t ∈ {θ t } ∈ H 1,1 (X, R) is fixed.
Since the MMP requires one to work on singular varieties, it is necessary to develop a fine theory dealing with weak solutions. One has indeed to deal with similar complex Monge-Ampère 31 flows with various degeneracies : the reference forms ω t are no longer Kähler and the densities h is no longer smooth, with integrability properties that depend on the type of singularities. A parabolic viscosity approach has been developed recently in [START_REF] Eyssidieux | Weak solutions to degenerate complex Monge-Ampère flows II[END_REF][START_REF] Eyssidieux | Convergence of weak Kähler-Ricci flows on minimal models of positive Kodaira dimension[END_REF], which requires the densities to be continuous hence has a limited scope of applications. The first elements of a parabolic pluripotential theory has been laid down in [START_REF] Guedj | The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows[END_REF][START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF] which are the parabolic analogues of the pioneering work of Bedford and Taylor in the local case [START_REF] Bedford | The Dirichlet problem for a complex Monge-Ampère equation[END_REF][START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF]. We extend here this theory so as to be able to deal with big cohomology classes.

Assumptions and Notations.

Before going further and stating the main results of the paper, let us fix some notations. Let X be a compact Kähler manifold of dimension n. We let X T := (0, T) × X denote the real (2n + 1)dimensional manifold with T ∈ (0, +∞]. We focus mostly on finite time intervals i.e. T < +∞. The parabolic boundary of X T is denoted by

∂X T := {0} × X.
We fix θ a smooth closed (1, 1)-form representing a big cohomology class. We let Ω denote the ample locus of θ,

Ω := Amp(θ)
which is a non empty Zariski open subset of X. We also set Ω T := (0, T) × Ω.

We assume that (ω t ) t∈[0,T) is a smooth family of closed (1, 1)-forms on X such that

g(t)θ ≤ ω t , ∀ t ∈ [0, T),
where g(t) is an increasing smooth positive function on [0, T].

Throughout the article we assume that there exists a Kähler form Θ such that -Θ ≤ ω t , ωt , ωt ≤ Θ.

(3.0.2)

We let dV denote a smooth volume form on X. We shall always assume that • 0 ≤ f ∈ L p (X, dV) for some p > 1, and f is strictly positive almost everywhere ;

• F : [0, T] × X × R → R is continuous on [0, T] × X × R ; • the function r → F(•, •, r) is increasing in r ; • the function F is uniformly Lipschitz in (t, r) ∈ [0, T] × R, i.e. there exists a constant κ F > 0 such that for all t, t ′ ∈ [0, T], x ∈ X, r, r ′ ∈ R, |F(t, x, r) -F(t ′ , x, r ′ )| ≤ κ F (|t -t ′ | + |r -r ′ |); • the function (t, r) → F(t, •, r) is convex.
With the assumptions above, we consider the complex Monge-Ampère flow :

dt ∧ (ω t + dd c φ t ) n = e φt +F(t,•,φ t ) f dV ∧ dt (CMAF)
on X T . Note that the equation (CMAF) should be understood in the weak sense of measures in (0, T) × Ω (see Section 3.1.3). The existence of the weak Kähler-Ricci flow is often proved by using approximation arguments and a priori estimates (cf. [START_REF] Song | The Kähler-Ricci flow through singularities[END_REF][START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF]). Big cohomology classes can not be approximated by Kähler ones so this approach breaks down in our case. We shall instead use the Perron method, inspired by [START_REF] Guedj | The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows[END_REF], considering the upper envelope U of all pluripotential subsolutions to the Cauchy problem. We prove that this upper envelope is locally uniformly semi-concave in time :

Theorem A. Let φ 0 be a ω 0 -psh function with minimal singularities. Then the upper envelope U of all subsolutions to (CMAF) with initial data φ 0 is a pluripotential solution to (CMAF) which is locally uniformly Lipschitz and locally uniformly semi-concave in t ∈ (0, T).

We prove Theorem A by following the arguments of [START_REF] Guedj | The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows[END_REF] in the local context :

• we first show that the upper envelope of all subsolutions is locally uniformly Lipschitz in t (Theorem 3.2.7) and that it is itself a pluripotential subsolution ;

• we then show that the envelope is locally uniformly semi-concave (Theorem 3.2.13) ;

• we finally apply a balayage process and use the analogue result in the local context [START_REF] Guedj | The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows[END_REF] to conclude the proof.

We prove in Theorem 3.3.6 that the envelope U in Theorem A has minimal singularities and is continuous in (0, T) × Ω under an extra assumption :

ωt ≤ Aω t , t ∈ [0, T), (3.0.3)
for some positive constant A. We also show that U is the unique pluripotential solution with such regularity by establishing the following comparison principle :

Theorem B. Let φ (resp. ψ) be a pluripotential subsolution (resp. supersolution) to (CMAF) with initial data φ 0 (resp. ψ 0 ). We assume that ψ is locally uniformly semi-concave in t ∈ (0, T) and ψ is continuous in (0, T) × Ω. We assume moreover that for each t, ψ t has minimal singularities. Then

φ ≤ ψ on [0, T) × X if φ 0 ≤ ψ 0 .
The assumption that ψ t has minimal singularities means that for each t ∈ (0, T), there exists a constant C t such that |ψ t -V ω t | is bounded by C t , where V ω t is the largest negative ω t -psh func- tion. The proof of Theorem B is provided in Section 3.3.2, generalizing some ideas from [START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF].

Starting from a Kähler form ω 0 , it follows from [Cao85, Tsu88, TZ06] that the (smooth) normalized Kähler-Ricci flow exists in [0, T) where

T := sup{t > 0 : e -t {ω 0 } + (1 -e -t )c 1 (K X ) is Kähler}.
The maximal existence time T is finite unless K X is nef (numerically effective).

It is an interesting question to know how to define the flow for t > T. This was formulated in [FIK03, Section 10, Question 8] and a precise conjecture was made in [START_REF] Boucksom | Semipositivity of relative canonical bundles via Kähler-Ricci flows (Potential theory and fiber spaces)[END_REF]. Note that, if X is of general type, i.e. K X is big, then for any t > T the cohomology class e -t {ω 0 } + (1e -t )c 1 (K X ) remains big but are no longer nef, thus one can not hope to make sense of the flow in the classical one. It was proved in [START_REF] Tô | Convergence of the Weak Kähler-Ricci Flow on Manifolds of General Type[END_REF] that the flow can be continued through T in the viscosity sense and it eventually converges to the unique singular Kähler-Einstein metric on Amp(K X ). Using the tools developed above, we establish the pluripotential analogue of the main result of [START_REF] Tô | Convergence of the Weak Kähler-Ricci Flow on Manifolds of General Type[END_REF] :

Theorem C. Let X be a compact n-dimensional Kähler manifold of general type. Then the normalized pluripotential Kähler-Ricci flow emanating from a Kähler metric ω 0 ,

∂θ t ∂t = -Ric(θ t ) -θ t ,
exists for all time. It coincides with the smooth flow on [0, T) and deforms ω 0 towards the unique singular Kähler-Einstein metric ω KE on Amp(K X ), as t → +∞.

We actually establish a more general result allowing to run the flow from an arbitrary closed positive current with bounded potential ; (see Theorem 3.4.1). We can also continue the pluripotential Kähler-Ricci flow for all time when K X is pseudoeffective (see Section 3.4.2).

In the last part of the paper we study pluripotential Kähler-Ricci flows on Kähler varieties X with semi-log canonical singularities (the most general class of singularities appearing in the log MMP) and ample canonical line bundle.

It has been shown by R. Berman and H. Guenancia [START_REF] Berman | Kähler-Einstein metrics on stable varieties and log canonical pairs[END_REF] that X admits a unique Kähler-Einstein current ω KE in the class c 1 (K X ) which is smooth in the regular locus X reg . We apply our theory to run the pluripotential (normalized) Kähler-Ricci flow on X and recover the canonical metric ω KE as the long time limit of the flow. More precisely, we have the following : Theorem D. Let X be a projective complex algebraic variety with semi-log canonical singularities such that K X is ample. Then the normalized pluripotential Kähler-Ricci flow emanating from a Kähler metric ω 0 ,

∂θ t ∂t = -Ric(θ t ) -θ t ,
exists for all time. It deforms ω 0 towards the unique singular Kähler-Einstein metric ω KE on X reg , as t → +∞.

Again we actually show that the flow can be run from an arbitrary positive closed current with bounded potentials (see Theorem 3.4.10).

For varieties of general type with log terminal singularities the pluripotential Kähler-Ricci flow (with non continuous data) was constructed in [GLZ20, Section 5.1]. A similar result has been obtained in the recent work [START_REF] Chau | The Kähler-Ricci flow with Log Canonical Singularities[END_REF], where the authors have extended the approach of Song-Tian [START_REF] Song | The Kähler-Ricci flow through singularities[END_REF] to the case of Q-factorial projective varieties with log canonical singularities : establishing higher order a priori estimates, they obtain a good notion of weak Kähler-Ricci flow which is smooth in the regular locus of variety.

Organization of the chapter

In Section 3.1 we provide some backgrounds on pluripotential theory in big cohomology classes. In Section 3.2 we study the regularity properties of the envelope of pluripotential subsolutions. In Section 3.3 we shall prove Theorem A and Theorem B. We study in Section 3.4 the pluripotential normalized Kähler-Ricci flow on compact Kähler manifolds of general type (resp. stable varieties) and prove Theorem C (resp. Theorem D).

Preliminaries

In this section we recall necessary definitions and backgrounds. Let X be a compact Kähler manifold of complex dimension n, and Θ be a Kähler metric on X. We let H 1,1 (X, R) denote the Bott-Chern cohomology of d-closed real (1, 1)-forms (or currents) modulo ∂ ∂-exact ones.

Monge-Ampère operators in big cohomology classes

Big cohomology classes

Let θ be a smooth real closed (1, 1)-form on X. An upper semi-continuous function φ : X → [-∞, +∞) is called θ-plurisubharmonic (θ-psh for short) if in any local holomorphic coordinates φ can be written as the sum of a psh and a smooth function, and θ + dd c φ ≥ 0, in the weak sense of currents, where d = ∂ + ∂ and d c = i 2π ( ∂ -∂). We let PSH(X, θ) denote the set of all θ-psh functions on X which are not identically -∞. This set is endowed with the weak topology which coincides with the L 1 -topology. By Hartogs' lemma φ → sup X φ is continuous in the L 1 -topology.

By the dd c -lemma any closed positive (1, 1)-current T cohomologous to θ can be written as T = θ + dd c φ for some θ-psh function φ which is moreover unique up to an additive constant.

If T and T ′ are two closed positive (1, 1)-currents on X which are cohomologous, then T is said to be less singular than T ′ if their global potentials satisfy φ ′ ≤ φ + O(1) (then we also say that φ is less singular than φ ′ ). A positive current T is now said to have minimal singularities if it is less singular than any other positive current in its cohomology class. Definition 3.1.1. A θ-psh function φ is said to have minimal singularities if it is less singular than any other θ-psh function on X.

Such θ-psh functions always exists, one can consider, following Demailly, the upper envelope

V θ := sup{φ : φ ∈ PSH(X, θ), and φ ≤ 0}. Observe that V * θ is a θ-psh function satisfying V * θ ≤ V θ , hence V θ = V * θ is a θ-psh function with minimal singularities.
The cohomology class α = {θ} ∈ H 1,1 (X, R) is said to be big if there exists a closed (1, 1)current

T + = θ + dd c φ + ,
cohomologous to θ such that T + is strictly positive i.e T + ≥ ε 0 Θ for some constant ε 0 > 0.

A function u has analytic singularities if it can locally be written as

u = c 2 log N ∑ j=1 | f 2 j | + h,
where the functions f j are holomorphic, h is smooth and c is a positive constant.

In the sequel we always assume that the class α = {θ} is big. By Demailly's regularization theorem [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF], any θ-psh function u can be approximated from above by a sequence of (θ + ε j ω)-psh functions (u j ) with analytic singularities. Applying this to the potential φ + of a Kähler current T + = θ + dd c φ + , one can moreover assume that the function φ + has analytic singularities. Such a current T + is then smooth on a Zariski open subset, this motivates the following : Definition 3.1.2. The ample locus Amp(α) of α is the set of x ∈ X such that there exists a Kähler current with analytic singularities which is smooth around x.

It follows from the Noetherian property of closed analytic subsets that Amp(α) is a Zariski open set. Note that any θ-psh function φ with minimal singularities is locally bounded on the ample locus Amp(α) since it has to satisfy φ + ≤ φ + O(1). Moreover, φ + does not have minimal singularities unless α is a Kähler class (cf. [Bou04, Proposition 2.5]).

By the above analysis, there exists a θ-psh function χ on X with analytic singularities such that, for some δ 0 > 0, θ + dd c χ ≥ 2δ 0 Θ.

(3.1.1) Subtracting a large constant, we can always assume that χ ≤ 0, thus χ ≤ V θ . Moreover, χ is smooth in the ample locus Amp(α), and χ(x) → -∞ as x → ∂Amp(α) (cf. [Bou04, Theorem 3.17]).

Full Monge-Ampère mass

In [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF], the authors defined the non-pluripolar product T → ⟨T n ⟩ of any closed positive (1, 1)-current T ∈ α, which is shown to be well-defined as a positive measure on X putting no mass on pluripolar sets. In particular given a θ-psh function φ, one can define its non-pluripolar Monge-Ampère product by MA θ (φ) := ⟨(θ + dd c φ) n ⟩. From now we denote the non-pluripolar Monge-Ampère product (θ + dd c φ) n instead of ⟨(θ + dd c φ) n ⟩. By definition the total mass of MA(φ) is less than or equal to the volume Vol(α) of the class α :

ˆX MA(φ) ≤ Vol(α) := ˆX MA(V θ ).

A particular class of θ-psh functions that appears naturally is the one for which the last inequality is an equality. We will say that such functions (or the associated currents) have full Monge--Ampère mass. For example, θ-psh functions with minimal singularities have full Monge-Ampère mass (cf. [BEGZ10, Theorem 1.16]), but the converse is not true. We let c 1 be the normalizing constant such that 2 n e c 1 f dV has total mass equal to Vol(α). We have the following : Theorem 3.1.3 ([BEGZ10, Theorem 4.1]). There exists a unique θ-psh function ρ with full Monge-Ampère mass such that

(θ + dd c ρ) n = 2 n e c 1 f dV, (3.1.2)
and normalized by sup X ρ = 0. Moreover, there exists a constant M > 0 only depending on θ, dV, and p > 1 such that

ρ ≥ V θ -M∥ f ∥ 1/n p .

Parabolic potentials

In this section we define the parabolic pluripotential objects in big cohomology classes necessary for our study. These are mainly taken from [GLZ21a, GLZ20] but we need to be more precise when dealing with unbounded functions. Let ω = (ω t ) t∈[0,T) be a smooth family of closed real (1, 1)-forms satisfying the assumptions in Introduction.

Definition 3.1.4. We let P (X T , ω) denote the set of functions φ :

X T → [-∞, +∞) such that • φ is upper semi-continuous on X T and φ ∈ L 1 loc (X T ) ; • for each t ∈ (0, T) fixed, the slice φ t : x → φ(t, x) is ω t -psh on X ;
• for any compact subinterval J ⊂ (0, T), there exists a positive constant κ = κ J (φ) such that

∂ t φ ≤ κ -κ(ρ + χ), (3.1.3)
in the sense of distributions on J × Ω, where ρ, χ are defined in (3.1.2), (3.1.1).

We would like to have an interpretation of the last condition. For any compact subset K ⋐ Ω, there exists a constant C = C(K) > 0 such that

∂ τ (t, x) ≤ C, ∀ (t, x) ∈ J × K.
Hence for every x ∈ K, the function t → φ(t, x) -Ct is decreasing in J, so the partial derivative ∂ t φ exists for almost everywhere t ∈ J (see e.g. [KK96, Theorem 2.1.8]).

Lemma 3.1.5. Let φ 0 be an ω 0 -psh function and φ ∈ P (X T , ω).

If φ t → φ 0 in L 1 (X) as t → 0, then the extension φ : [0, T) × X → [-∞, +∞) is upper semi-continuous in [0, T) × X.
Proof. It suffices to prove that the extension φ is upper semi-continuous at (0, x 0 ) for any x 0 ∈ X. Let (t j , x j ) ∈ X T be a sequence which converges to (0, x 0 ). We will show that lim sup j→+∞ φ(t j , x j ) ≤ φ 0 (x 0 ).

Since φ is bounded from above we can assume the functions φ t are negative. Let h t be a smooth local potential for ω t in an open neighborhood B of x 0 i.e. dd c h t = ω t . Up to replacing φ t by φ t + h t , we may assume that the functions φ t are psh and negative on B. Fix r so small that B(x, 2r) ⋐ B. For any δ ∈ [0, r), there exists j 0 such that x j ∈ B(x 0 , δ) for all j ≥ j 0 hence B(x 0 , r) ⊂ B(x j , r + δ). We have

φ(t j , x j ) ≤ 1 Vol(B(x j , r + δ)) ˆB(x j ,r+δ) φ(t j , x)dV ≤ 1 Vol(B(x j , r + δ)) ˆB(x 0 ,r) φ(t j , x)dV.
Since lim sup j φ t j (x) ≤ φ 0 (x) for all x ∈ X, Fatou's lemma implies that lim sup j→+∞ φ(t j , x j ) ≤ Vol(B(x 0 , r)) Vol(B(x 0 , r + δ)) 1 Vol(B(x 0 , r)) ˆB(x 0 ,r) φ 0 (x)dV(x).

Now we first let δ → 0 and then r → 0 to conclude the proof.

Definition 3.1.6. We say that φ ∈ P (X T , ω) has minimal singularities if φ t -V ω t is bounded for each t ∈ (0, T) fixed.

If

φ ∈ P (X T , ω) ∩ L ∞ loc (Ω T ) the product (ω t + dd c φ t ) n
is well defined as a positive measure in Ω as follows from the works of Bedford-Taylor [START_REF] Bedford | The Dirichlet problem for a complex Monge-Ampère equation[END_REF][START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF]. This Monge-Ampère measure extends trivially over X since Ω is a Zariski open subset in X. Since ω t ≤ Θ for 0 ≤ t ≤ T, the positive Borel measures (ω t + dd c φ t ) n have uniformly bounded masses on X :

ˆX(ω t + dd c φ t ) n ≤ ˆX(Θ + dd c φ t ) n ≤ ˆX Θ n . (3.1.4)
These can be considered as a family of currents of degree 2n in the real (2n + 1)-dimensional manifold X T = (0, T) × X. We have the following :

Lemma 3.1.7. Let φ ∈ P (X T , ω) ∩ L ∞ loc (Ω T )
and γ be a continuous test function in Ω T . Then t → ´Ω γ(t, •)(ω t + dd c φ t ) n is a Borel bounded measurable function in (0, T), and

sup 0<t<T ˆΩ γ(t, •)(ω t + dd c φ t ) n ≤ (max Ω T γ) ˆX Θ n .
Proof. For the first statement, the proof is identical to the corresponding one in the local context ; see [GLZ21b, Lemma 2.2]. The second one follows from the inequality (3.1.4) above.

This shows that dt ∧ (ω t + dd c φ t ) n is well-defined as a positive Borel measure in X T .

Definition 3.1.8. Fix φ ∈ P (X T , ω) ∩ L ∞ loc (Ω T ). The map γ → ˆXT γdt ∧ (ω t + dd c φ t ) n := ˆT 0 dt ˆΩ γ(t, •)(ω t + dd c φ t ) n
defines a positive (2n + 1)-current on Ω T , hence on X T , denoted by dt ∧ (ω t + dd c φ t ) n , which can be identified with a positive Radon measure on X T .

The following is a parabolic analogue of the convergence result of Bedford-Taylor [BT76, BT82]. Lemma 3.1.9. Assume that (φ j ) is a monotone sequence of functions in P (X T , ω) which converges almost everywhere to a function

φ ∈ P (X T , ω) ∩ L ∞ loc (Ω T ) on X T . Then dt ∧ (ω t + dd c φ j t ) n → dt ∧ (ω t + dd c φ t ) n in the sense of measures on Ω T .
Proof. The proof is similar to that of [GLZ20, Proposition 1.12] but, because of its crucial role in the sequel, we give the details here.

Let γ(t, x) be a continuous test function in Ω T . By definition we have, for any j, ˆΩT

γ(t, •)dt ∧ (ω t + dd c φ j t ) n = ˆT 0 dt ˆΩ γ(t, •)(ω t + dd c φ j t ) n .
We now apply Bedford-Taylor's convergence theorem (see e.g. [GZ17a, Theorem 3.23]) to infer that, for any t ∈ (0, T),

ˆΩ γ(t, •)(ω t + dd c φ j t ) n → ˆΩ γ(t, •)(ω t + dd c φ t ) n .
On the other hand, Lemma 3.1.7 yields, for all t ∈ (0, T),

ˆΩ γ(t, •)(ω t + dd c φ j t ) n ≤ max Ω |γ(t, •)| Vol(ω t ) ≤ C(γ) ˆX Θ n .
The result follows from Lebesgue dominated convergence theorem.

We say that φ : X T → R is locally uniformly semi-concave (resp. semi-convex) in Ω T = (0, T) × Ω if for any compact subset J × K ⊂ (0, T) × Ω there exists κ = κ(φ, J, K) > 0 (resp. κ < 0) such that for all x ∈ K, the function t → φ(t, x)κt 2 is concave (resp. convex) in t ∈ J. For any x ∈ Ω fixed, the left and right derivatives,

∂ + t φ(t, x) = lim s→0 + φ(t + s, x) -φ(t, x) s ,
and

∂ - t φ(t, x) = lim s→0 - φ(t + s, x) -φ(t, x) s
exist for all t ∈ (0, T), and they coincide when ∂ t φ(t, x) exists. Let ℓ denote the Lebesgue measure on R and µ denote a positive Borel measure on X. We have the following result whose proof is identical to that of [START_REF] Guedj | The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows[END_REF] The following convergence results play a key role in the sequel. We omit their proofs and refer the reader to [GLZ21a, Section 2].

≤ f ∈ L p (D) with p > 1. Let (v j ) be a sequence of Borel functions in J × D such that (e v j f ) is uniformly bounded in L 1 (J × D, dt ∧ dV).
Assume that for any x ∈ D, v j (., x) converges to a bounded Borel function v(., x) in the sense of distributions on J and for all η ∈ C ∞ 0 (J × D)

sup j∈N,x∈D
ˆJ η(t, x)v j (t, x)dt < +∞.

(3.1.5)

Then for any positive smooth test function η ∈ C ∞ 0 (J × D), lim inf j→+∞ ˆJ×D η(t, x)e v j (t,x) f (x)dt ∧ dV ≥ ˆJ×D η(t, x)e v(t,x) f (x)dt ∧ dV. Proof. See [GLZ21a, Proposition 2.6].
Proposition 3.1.12. Let ( f j ) be a sequence of positive functions converging to f in L

1 (X T , ℓ ⊗ µ). Let (φ j ) be a sequence of functions in P (X T , ω) ∩ L ∞ loc (Ω T ) which • converges ℓ ⊗ µ-almost everywhere in X T to a function φ ∈ P (X T , ω) ∩ L ∞ loc (Ω T ) ;
• is locally uniformly semi-concave in Ω T .

Then the limit lim j→+∞ φj (t, x) exists and is equal to φ(t, x) for ℓ ⊗ µ-almost every (t, x) ∈ Ω T , and

h( φj ) f j ℓ ⊗ µ → h( φ) f ℓ ⊗ µ,
in the weak sense of measures on Ω T , for all h ∈ C 0 (R, R).

Proof. We refer the reader to [GLZ21a, Proposition 2.9] for a proof (see also [GLZ20, Theorem 1.14]).

Pluripotential subsolutions/supersolutions

We assume that T < +∞. As explained in the introduction, we assume here that g(t)θ ≤ ω t ≤ Θ, where θ is a big (1, 1)-form, g is a smooth increasing positive function in t ∈ [0, T], and Θ is a Kähler form.

Let us emphasize here that by comparison with [START_REF] Guedj | The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows[END_REF][START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF], for an element φ ∈ P (X T , ω) the weak derivative ∂ τ φ(t, •) is merely locally bounded from above in Ω but φ is not locally uniformly Lipschitz in (0, T). This is natural as we are dealing with quasi-psh functions which are bounded from above but not from below.

Before defining pluripotential subsolutions (supersolutions), we need to make sense of the quantity φt = ∂ τ φ(t, •), in order to define the right-hand side of (CMAF). By the definition of P (X T , ω), for any compact subset K ⊂ Ω, J ⋐ (0, T), there exists a constant C = C K,J > 0 such that Ctφ(t, x) is increasing in t ∈ J for every x ∈ K. Thus, for every x ∈ K, ∂ τ φ t (x) is well defined for almost every t ∈ J (see e.g. [KK96, Lemma 1.2.8]). This implies that the right-hand side of (CMAF) is well-defined almost everywhere in Ω T (using Fubini's theorem). This analysis motivates the following : Definition 3.1.13. We say that a parabolic potential φ ∈ P (X T , ω) is a pluripotential subsolution to (CMAF) on X T if

• for each t ∈ (0, T) fixed, the ω t -psh function φ(t, •) is locally bounded in Ω • the inequality (ω t + dd c φ t ) n ∧ dt ≥ e φt +F(t,•,φ t ) f dV ∧ dt holds in the sense of measures in (0, T) × Ω.
Definition 3.1.14. We say that a parabolic potential φ ∈ P (X T , ω) is a pluripotential supersolution to (CMAF) on X T if • for each t ∈ (0, T) fixed, the ω t -psh function φ(t, •) is locally bounded in Ω,

• the inequality

(ω t + dd c φ t ) n ∧ dt ≤ e φt +F(t,•,φ t ) f dV ∧ dt holds in the sense of measures in (0, T) × Ω.
Remark 3.1.15. In these definitions the left-hand side is well-defined by using Bedford-Taylor's theory (see Definition 3.1.8).

Lemma 3.1.16. Let φ ∈ P (X T , ω) be a parabolic potential such that the restriction of φ to {t} × Ω is an ω t -psh function which is locally bounded on Ω. Then 1) φ is a pluripotential subsolution to (CMAF) if and only if for a.e. t ∈ (0, T),

(ω t + dd c φ t ) n ≥ e ∂ τ φ(t,•)+F(t,•,φ t ) f dV, (3.1.6)
in the sense of measures in Ω.

2) φ is a pluripotential supersolution to (CMAF) if and only if for a.e. t ∈ (0, T),

(ω t + dd c φ t ) n ≤ e ∂ τ φ(t,•)+F(t,•,.φ t ) f dV, (3.1.7)
in the sense of measures in Ω.

Proof. We shall prove the result for subsolutions. The proof for supersolutions is similar. We first assume that (3.1.6) holds for almost every t. Let η be a positive continuous test function in (0, T) × Ω. We thus obtain

ˆΩ η(t, •)(ω t + dd c φ t ) n ≥ ˆΩ η(t, •)e ∂ t φ t +F(t,•,φ t ) f dV.
Integrating with respect to t, we get

ˆT 0 ˆΩ η(ω t + dd c φ t ) n ∧ dt ≥ ˆT 0 ˆΩ ηe ∂ t φ t +F(t,•,φ t ) f dVdt,
hence φ is a pluripotential subsolution to (CMAF). Conversely, assume that φ is a pluripotential subsolution to (CMAF). We consider positive test functions that can be decomposed as

η(t, x) = λ(t)ξ j (x),
where (ξ j ) is a sequence of positive test functions on Ω which generates a dense subspace of the space C 0 c (Ω) in C 0 -topology. It follows from Fubini's theorem that

ˆT 0 ˆΩ ξ j (x)(ω t + dd c φ t ) n λ(t)dt ≥ ˆT 0 ˆΩ ξ j (x)e ∂ t φ t +F(t,.,φ t ) f dV λ(t)dt.
Hence for any j there exists a subset E j which has full measure in (0, T) so that for all t ∈ E j , ˆΩ ξ j (x)(ω t + dd c φ t ) n ≥ ˆΩ ξ j (x)e ∂ t φ t +F(t,•,φ t ) f (x)dV(x).

(3.1.8)

If we set E := ∩ j E j , then E has full measure in (0, T). Moreover, the inequality (3.1.8) holds for all t ∈ E and for all j. Let ξ be an arbitrary positive continuous function in Ω. We can approximate this function by finitely convex combinations of the ξ j , we infer that for all t ∈ E,

ˆΩ ξ(x)(ω t + dd c φ t ) n ≥ ˆΩ ξ(x)e ∂ t φ t +F(t,•,φ t ) f (x)dV(x),
from which (3.1.6) follows.

Lemma 3.1.17. Let φ, ψ ∈ P (X T , ω) be two pluripotential subsolutions to (CMAF). Then

1 {φ≥ψ} ∂ t max(φ, ψ) = 1 {φ≥ψ} ∂ t φ,
almost everywhere in Ω T , and

(ω + dd c max(φ, ψ)) n ∧ dt ≥ 1 {φ≥ψ} (ω + dd c φ) n ∧ dt in the sense of measures in Ω T .
Proof. Fix K ⋐ Ω and J ⋐ (0, T). Then there exists a constant C = C K,J > 0 such that, for any Lemma 3.1.18. For every λ ∈ [(1 + δ 0 g(0)) -1 , 1], the function λg(0)(ρ + χ)/2 is ω 0 -psh. In particular, there exists a uniform constant C 0 > 0 such that

x ∈ K fixed, Ct -φ(t,
λg(0) ρ + χ 2 -C 0 ≤ φ 0 .
Recall here that χ is a fixed θ-psh function with analytic singularities such that θ + dd c χ ≥ 2δ 0 Θ, for some δ 0 > 0.

Proof. By hypothesis (3.0.2), we first observe that

ω 0 + dd c λg(0) ρ + χ 2 = λ 2 (ω 0 + g(0)dd c ρ) + λ 2 (ω 0 + g(0)dd c χ) + (1 -λ)ω 0 ≥ λ 2 g(0)(θ + dd c ρ) + λ 2 g(0)(θ + dd c χ) + (1 -λ)ω 0 ≥ λg(0)δ 0 Θ -(1 -λ)Θ ≥ 0
where the last inequality follows from the choice of λ. Thus the function λg(0)(ρ + χ)/2 is ω 0psh. Since φ 0 is ω 0 -psh with minimal singularities, there exists a constant C 0 > 0 such that λg(0

)(ρ(x) + χ(x))/2 -C 0 ≤ φ 0 (x) for all x ∈ X.

The envelope of subsolutions

Definition

Definition 3.2.1. A Cauchy datum for (CMAF) is a ω 0 -psh function φ 0 : X → R with minimal singularities. We say φ ∈ P (X T , ω) is a subsolution to the Cauchy problem :

(ω t + dd c u t ) n = e ∂ t u t +F(t,•,u t ) f dV, u {0}×X = φ 0 if φ is a pluripotential subsolution to (CMAF) such that lim sup t→0 φ(t, x) ≤ φ 0 (x) for all x ∈ X.
We let S φ 0 , f ,F (X T ) denote the set of pluripotential subsolutions to the Cauchy problem above.

Lemma 3.2.2. The set S φ 0 , f ,F (X T ) is non-empty, uniformly bounded from above on X T , and stable under finite maxima.

Proof. Fix λ ∈ [(1 + δ 0 g(0)) -1 , 1]. Consider, for any (t, x) ∈ X T , u(t, x) := λg(t) ρ(x) + χ(x) 2 -C 1 (t + 1), (3.2.1)
where ρ and χ are defined in (3.1.2) and (3.1.1), the uniform constant C 1 > 0 will be chosen later. By hypothesis (3.0.2) on ω t we have

λ 2 (ω t + g(t)dd c χ) + (1 -λ)ω t ≥ λ 2 g(t)(θ + dd c χ) + (1 -λ)ω t ≥ λg(t)δ 0 Θ -(1 -λ)Θ = [λ(1 + δ 0 g(0)) -1]Θ ≥ 0.
since g(t) is increasing. Therefore, we obtain

(ω t + dd c u t ) n = λ 2 (ω t + g(t)dd c ρ) + λ 2 (ω t + g(t)dd c χ) + (1 -λ)ω t n ≥ λ 2 g(t)(θ + dd c ρ) n = (λg(t)) n e c 1 f dV. (3.2.2)
We set

C 1 = C 0 + M F + |n log(g(T))| + |c 1 |, (3.2.3) it thus follows from (3.2.2) that exp(∂ t u t + F(t, •, u t (•))) f dV = exp(λg ′ (t)(ρ + χ)/2 -C 1 + F(t, •, u t (•))) f dV ≤ exp(n log(λg(t)) + c 1 ) f dV ≤ (ω t + dd c u t ) n
using in the first inequality that g is increasing in t ∈ [0, T], sup X ρ = sup X χ = 0. It follows moreover from the choice of C 1 and Lemma 3.1.18 that u(0, •) ≤ φ 0 on X, hence u ∈ S φ 0 , f ,F (X T ).

Let now φ ∈ S φ 0 , f ,F (X T ) such that φ ≥ u. Set with µ := f dV. Consider the set

G := {x ∈ X : u(T, x) > -M},
for M > 0 so large that µ(G) > µ(X) 2 . We observe that for every t ∈ (0, T),

φ t (x) ≥ u(t, x) ≥ u(T, x) > -M, ∀ x ∈ G. Set -m F = inf [0,T)×X F(t, x, -M) > -∞. Since F(., ., r) is non-decreasing in r we obtain ˆG e φt -m F dµ ≤ ˆG e φt +F(t,•,φ t ) dµ ≤ ˆG(ω t + dd c φ t ) n ≤ ˆG(Θ + dd c φ t ) n ≤ ˆX Θ n .
On the other hand, it follows from Jensen's inequality that exp ˆG φt dµ

µ(G)
≤ ˆG e φt dµ µ(G) .

Combining these two estimates we get

ˆG φt dµ ≤ µ(G) log e m F ´X Θ n µ(G) ≤ µ(X) log 2e m F ´X Θ n µ(X) =: C.
We then infer that the function t → ´G φ t dµ -Ct is non-increasing in (0, T), hence

ˆG φ t dµ ≤ ˆG φ 0 dµ + Ct ≤ ˆG φ 0 dµ + CT. (3.2.4)
On the other hand, it follows from [GZ17a, Proposition 8.5] that there exists a uniform constant C ′ (only depending on µ) such that ˆX(ψsup X ψ)dµ ≥ -C ′ , for all ψ ∈ PSH(X, Θ).

Thus for each t ∈ (0, T),

-C ′ ≤ ˆX(φ t -sup X φ t )dµ ≤ ˆG(φ t -sup X φ t )dµ ≤ C ′′ -µ(G) sup X φ t .
We deduce that sup X φ t is uniformly bounded from above.

The stability under finite maxima follows immediately from Lemma 3.1.17.

From now on, we let M 0 > 0 denote a uniform upper bound of all pluripotential subsolutions φ to (CMAF) such that φ ≥ u on X T , and set

M F := sup X T F(•, •, M 0 ).
(3.2.5) Lemma 3.2.2 allows us to define the upper envelope of subsolutions :

Definition 3.2.3. We let

U = U φ 0 , f ,F,X T := sup{φ ∈ S φ 0 , f ,F (X T ) : u ≤ φ ≤ M 0 }
denote the upper envelope of all subsolutions.

Lemma 3.2.4. There exists φ ∈ S φ 0 , f ,F (X T ) such that for any x ∈ X,

lim t→0 φ(t, x) = φ 0 (x).
Proof. We set δ = δ 0 g(0). For any (t,

x) ∈ [0, δ) × X, consider φ(t, x) = (1 -α t )φ 0 (x) + α t g(0) ρ(x) + χ(x) 2 + nt(log(δ -1 0 t) -1) -Ct,
where α t = δ -1 t, the functions ρ, χ are defined in (3.1.2), (3.1.1), and

C := M F + δ -1 sup X g(0) ρ + χ 2 -φ 0 -min(c 1 , 0).
Lemma 3.1.18 with λ = 1 ensures that C < +∞. We compute

ω t + dd c φ t = (1 -α t )(ω 0 + dd c φ 0 ) + α t 2 (ω 0 + g(0)dd c ρ) + α t 2 (ω 0 + g(0)dd c χ) + ω t -ω 0 .
Since ω t + tΘ is increasing, we have ω tω 0 ≥ -tΘ, hence

α t 2 (ω 0 + g(0)dd c χ) + ω t -ω 0 ≥ α t g(0) 2 (θ + dd c χ) + ω t -ω 0 ≥ α t g(0)δ 0 Θ -tΘ = 0.
Computing the time derivative we obtain

∂ t φ(t, •) = (δ 0 g(0)) -1 g(0) ρ + χ 2 -φ 0 + n log(δ -1 0 t) -C.
Hence, for all t ∈ (0, δ) we have, by the choice of the constant C,

(ω t + dd c φ t ) n ≥ α t g(0) 2 (θ + dd c ρ) n = (δ -1 0 t) n e c 1 f dV ≥ e ∂ t φ t +F(t,•,φ t ) f dV.
We divide [0, T] into N small intervals of the same length [T k , T k+1 ], k = 0, ..., N -1 so that |T k+1 -

T k | ≤ δ := δ 0 g(0), T 0 = 0 and T N = T. For t ∈ [T k , T k+1 ] we define φ (k) (t, •) : = (1 -α (k) t )φ T k + α (k) t g(T k ) ρ + χ 2 -C (k) (t -T k ) + n(t -T k )(log(δ -1 0 (t -T k )) -1),
where α

(k) t = δ -1 (t -T k ), and
C (k) = M F + δ -1 sup X g(T k ) ρ + χ 2 -φ T k -min(c 1 , 0).
The subsolution constructed in the proof of Lemma 3.2.2 (see (3.2.1)) ensures that C (k) < +∞ is a uniform positive constant. The same arguments as above ensure that φ (k) is a pluripotential subsolution to (CMAF) in [T k , T k+1 ] × X. Gluing these functions, we get our desired pluripotential subsolution defined on [0, T) × X. It is also clear from the definition that φ(t, .) converges to φ 0 in L 1 (X, dV) as t → 0 + .

Lipschitz regularity in time

In this section, we study the regularity in time t of the Perron upper envelope by adapting some arguments in [GLZ21a, Section 4]. Proposition 3.2.5. For all 0 < S < T we have U φ 0 , f ,F,X S = U φ 0 , f ,F,X T in X S .

Proof. Set U T := U φ 0 , f ,F,X T and U S := U φ 0 , f ,F,X S . We can assume that |T -S| ≤ δ 0 g(0) 2 , since if we can show that U T = U S for such S we can restart the process to prove that U S = U S ′ for

S - δ 0 g(0) 2 < S ′ < S.
It suffices to show that U S ≤ U T because the reverse inequality is clear. Fix φ ∈ S φ 0 , f ,F (X S ). Fix 0 < t 0 < S such that Tt 0 < δ 0 g(0). Set, for (t, x) ∈ (t 0 , T) × X,

ψ(t, x) = (1 -α t )φ(t 0 , x) + α t g(t 0 ) ρ(x) + χ(x) 2 -C(t -t 0 ) + n(t -t 0 )(log[δ -1 0 (t -t 0 )] -1),
where α t = (δ 0 g(t 0 )) -1 (tt 0 ) < 1, the functions ρ, χ are defined in (3.1.2), (3.1.1), and

C := M F + (δ 0 g(t 0 )) -1 sup X g(t 0 ) ρ + χ 2 -φ t 0 + |c 1 |.
From (3.2.1), with λ = 1 and t = t 0 , we see that C < +∞ is a uniform constant. From (3.2.1) again we see that ∂ t ψ(t, x) satisfies (3.1.5). We compute

ω t + dd c ψ t = (1 -α t )(ω t 0 + dd c φ t 0 ) + α t 2 (ω t 0 + g(t 0 )dd c ρ) + α t 2 (ω t 0 + g(t 0 )dd c χ) + ω t -ω t 0 .
Since ω t + tΘ is increasing, we have ω tω t 0 ≥ -(tt 0 )Θ for all t ≥ t 0 . It thus follows that

α t 2 (ω t 0 + g(t 0 )dd c χ) + ω t -ω t 0 ≥ α t g(t 0 ) 2 (θ + dd c χ) + ω t -ω t 0 ≥ α t g(t 0 )δ 0 Θ -(t -t 0 )Θ = 0.
Hence, for all t ∈ [t 0 , T),

(ω t + dd c ψ t ) n ≥ α t g(t 0 ) 2 (θ + dd c ρ) n = (δ -1 0 (t -t 0 )) n e c 1 f dV ≥ e ∂ t ψ t +F(t,•,ψ t ) f dV,
by the choice of the constant C. Therefore the function

(t, x) → u(t, x) := φ(t, x), if t ∈ [0, t 0 ] ψ(t, x), if t ∈ [t 0 , T)
is a pluripotential subsolution to (CMAF) in [0, T) × X by using Lemma 3.1.16. We thus have

u ∈ S φ 0 , f ,F (X T ) since u(0, •) = φ 0 . This yields u ≤ U T in [0, T) × X. In particular φ ≤ U T in [0, t 0 ] × X,
and it follows that U S ≤ U T on [0, t 0 ] × X by taking supremum over all subsolutions. We now let t 0 → S to obtain U s ≤ U T in X S .

Next we introduce the mixed type inequality :

Lemma 3.2.6. Let θ 1 , θ 2 be two closed smooth (1, 1)-forms on X with big cohomology classes. Let φ 1 (φ 2 resp.) be a bounded θ 1 -psh (θ 2 -psh resp.) function such that

(θ 1 + dd c φ 1 ) n ≥ e f 1 µ and (θ 2 + dd c φ 2 ) n ≥ e f 2 µ
where f 1 , f 2 are bounded measurable functions and µ is a positive Radon measure with L 1 density with respect to Lebesgue measure. Then, for any λ ∈ (0, 1),

(λ(θ 1 + dd c φ 1 ) + (1 -λ)(θ 2 + dd c φ 2 )) n ≥ e λ f 1 +(1-λ) f 2 µ.
Proof. The proof is the same as that of [GLZ21a, Lemma 2.10] using the convexity of the exponential together with the mixed Monge-Ampère inequalities ; see e.g. [START_REF] Dinew | An inequality for mixed Monge-Ampère measures[END_REF].

Theorem 3.2.7. There exists a uniform constant L U > 0 such that for all (t, x) ∈ X T ,

t|∂ t U(t, x)| ≤ L U -L U (ρ(x) + χ(x)). (3.2.6)
Proof. Let φ ∈ S φ 0 , f ,F (X T ) such that φ ≥ u on X T , where u is defined in (3.2.1). Fix 0 < T ′ < T and ε 0 > 0 so small that (1

+ ε 0 )T ′ < T. Set, for all (t, x) ∈ X T ′ , s ∈ (1 -ε 0 , 1 + ε 0 ), u s (t, x) := α s s φ(st, x) + (1 -α s )g(t) ρ(x) + χ(x) 2 -C|s -1|(t + 1),
where ρ, χ are defined in (3.1.2), (3.1.1), α s = 1 -A|s -1|, and

C = C 0 (A + 2) + κ F T + AM F + (A + 2)C 1 (T + 1) + (A + 2)M 0 .
The constant C 1 is defined in (3.2.3), and the constant A will be chosen later that depends only on T. We will show that u s ∈ S φ 0 , f ,F (X T ). We compute

ω t + dd c u s (t, •) = α s s (ω st + dd c φ st ) + α s ω t - α s s ω st + (1 -α s ) ω t + g(t)dd c ρ + χ 2 .
Since φ is a subsolution to (CMAF), we have for almost every t ∈ (0, T ′ ),

(s -1 (ω st + dd c φ st )) n ≥ e -n log s+∂ t φ(st,•)+F(t,•,φ(st,•)) f dV.
Recalling the definition of ρ and ω t ≥ g(t)θ we also have

1 2 (ω t + g(t)dd c ρ) n ≥ g(t) 2 (θ + dd c ρ) n = e n log g(t)+c 1 f dV. (3.2.7)
On the other hand, since ωt ≥ -Θ, we have

α s ω t - α s s ω st = α s s (ω t -ω st ) + α s (1 -s -1 )ω t ≥ -α s t s -1 -1 Θ -α s s -1 -1 Θ ≥ -(t + 1)s -1 |s -1|Θ ≥ -(2T + 2)|s -1|Θ,
where the last line follows from s ≥ 1/2. Recall that θ + dd c χ ≥ 2δ 0 Θ for some δ 0 > 0. If we choose A ≥ 2(T + 1)(δ 0 g(0)) -1 , then

α s ω t - α s s ω st + (1 -α s ) ω t + g(t)dd c χ 2 ≥ -(2T + 2)|s -1|Θ + A|s -1|g(t)δ 0 Θ ≥ 0,
since g is increasing in t. Combining these estimates with the mixed Monge-Ampère inequality (Lemma 3.2.6) we obtain

(ω t + dd c u s (t, •)) n ≥ (α s (s -1 ω st + dd c φ st ) + (1 -α s )(2 -1 (ω t + dd c ρ))) n ≥ exp(α s ∂ t φ(st, •) + α s F(st, •, φ(st, •)) -α s n log s + (1 -α s )(n log g(t) + c 1 )) f dV ≥ exp(∂ t u s (t, •) + F(t, •, u s (t, •)) f dV. (3.2.8)
where the last line follows from the choice of C as we now explain. Indeed, observe that

α s ∂ t φ(st, •) = ∂ t u s (t, •) + C|s -1| -(1 -α s )g ′ (t) ρ + χ 2 ≥ ∂ t u s (t, •) + C|s -1|.
(3.2.9) Since g is non-decreasing, we also have g(ts)g(t) ≤ κ g t|s -1| ≤ κ g Tε 0 g(0) -1 g(t).

It thus follows that g(ts) ≤ γg(t) where γ = 1 + ε 0 κ g Tg(0) -1 . Choosing ε 0 small enough at the beginning we can ensure that γ(1

+ δ 0 g(0)) -1 < 1 . Up to increasing A so large that A A+2 ≥ γ(1 + δ 0 g(0)) -1 , hence 1 - α s s φ st = 1 - α s s (φ st -M 0 ) + 1 - α s s M 0 ≥ (A + 2)|s -1|(φ ts -M 0 ) ≥ (A + 2)|s -1| A (A + 2)γ g(ts) ρ + χ 2 -C 1 (T + 1) -M 0 ≥ A|s -1|g(t) ρ + χ 2 -((A + 2)C 1 (T + 1) + (A + 2)M 0 )|s -1|,
where the first inequality follows from the elementary one 1 -α s s ≤ (A + 2)|s -1|, while the second inequality follows from φ ≥ u. We infer

φ st ≥ α s s φ st + (1 -α s )g(t) ρ + χ 2 -C|s -1| ≥ u s (t, •).
Using this and the assumption that F is non-decreasing in r and uniformly Lipschitz in t, we get

α s F(st, •, φ(st, •)) = F(st, •, φ(st, •)) -(1 -α s )F(st, •, φ(st, •)) ≥ F(t, •, φ ts (•)) -κ F t|s -1| -|s -1|AM F ≥ F(t, •, u s (t, •)) -(κ F T + AM F )|s -1|.
(3.2.10) Combining (3.2.9), (3.2.10), and the definition of C, we obtain the last inequality in (3.2.8). Hence u s is a pluripotential subsolution to (CMAF) by Lemma 3.1.16. We now take care of the initial values. For any x ∈ X we have

u s (0, x) = φ 0 (x) -C|s -1| + (1 -α s )g(0) ρ(x) + χ(x) 2 -1 - α s s φ 0 (x) ≤ φ 0 (x) -C|s -1| + A|s -1|g(0) ρ(x) + χ(x) 2 -(A + 2)|s -1|φ 0 (x) ≤ φ 0 (x) -C|s -1| + (A + 2)|s -1| A A + 2 g(0) ρ(x) + χ(x) 2 -φ 0 (x) ≤ φ 0 (x)
where the last line follows again from the choice of C. Thus, for any x ∈ X we also get lim sup t→0 u s (t, x) ≤ φ 0 (x).

Therefore u s ∈ S φ 0 , f ,F (X T ), so u s ≤ U in X T ′ . We thus obtain

α s s φ(st, x) + (1 -α s )g(t) ρ(x) + χ(x) 2 -C|s -1|(t + 1) ≤ U(t, x).
We now take the supremum over all subsolutions φ ∈ S φ 0 , f ,F (X T ) to get

α s s U(st, x) + A|s -1|g(t)(ρ(x) + χ(x)) -C|s -1|(t + 1) ≤ U(t, x), ∀(t, x) ∈ X T ′ .
Letting s → 1, we infer, for all (t, x) ∈ X T ′ that

t|∂ t U(t, x)| ≤ C(T + 1) + AM 0 -Ag(T)(ρ(x) + χ(x)).
We can now define L U := Ag(T) + C(T + 1) + AM 0 . Finally, letting T ′ → T and applying Proposition 3.2.5 we finish the proof.

Convergence at initial time

We define the upper semi-continuous (u.s.c) regularization U * of U by the formula

U * (t, x) = lim sup X T ∋(s,y)→(t,x)
U(s, y), (t, x) ∈ X T .

We then prove that the upper envelope has the right initial values :

Theorem 3.2.8. The upper semi-continuous regularisation of the upper envelope U := U φ 0 , f ,F,X T satisfies, for all x ∈ Ω, lim

Ω T ∋(t,y)→(0,x) U * (t, y) = φ 0 (x).
Proof. Thanks to Lemma 3.2.4, it suffices to show that for all x ∈ Ω, lim sup

Ω T ∋(t,y)→(0,x) U * (t, y) ≤ φ 0 (x).
Theorem 3.2.7 ensures that for y ∈ Ω fixed, the upper envelope U(•, y) is locally Lipschitz in (0, T). Arguing exactly as in the proof of [GLZ21a, Lemma 1.7] we can show that U * (t, •) = (U t ) * for all t ∈ (0, T), where (U t ) * denotes the u.s.c regularization of u t (t fixed) in the x-variable only.

It thus remains to prove that, for all x ∈ Ω,

lim sup t→0 U * t (x) ≤ φ 0 (x).
Fixing M > 0, we set G := {u T > -M}, where u is defined as in (3.2.1). We claim that there exists a constant C > 0 (also depending on M) such that, for all t ∈ (0, T),

ˆG U * t f dV ≤ ˆG φ 0 f dV + Ct. (3.2.11)
Fix t 0 ∈ (0, T). By Choquet's lemma, there exists a sequence {φ j } in S φ 0 , f ,F (X T ) such that

U * t 0 = lim j→+∞ φ j t 0 * in X.
Since the set S φ 0 , f ,F (X T ) is stable under finite maximum, we can moreover assume that the sequence {φ j } is increasing with

φ j ≤ M 0 on X. It follows from (3.2.4) that ˆG φ j t f dV ≤ ˆG φ 0 f dV + Ct, ∀ t ∈ (0, T),
for a constant C = C(M) > 0 independent of the sequence {φ j }. For t = t 0 , letting j → +∞, we obtain ˆG U *

t 0 f dV ≤ ˆG φ 0 f dV + Ct 0 ,
thanks to a classical theorem of Lelong (see e.g. [GZ17a, Proposition 1.40]). Note that the sequence {φ j t } depends on t 0 , but the constant C does not. Therefore the claim (3.2.11) follows.

Let now u 0 ∈ PSH(X, ω 0 ) be any cluster point of U * t as t → 0. We can assume that U * t converges to u 0 in L q (X, dV) for any q > 1. Then U * t f converges to u 0 f in L 1 (X). Thus, the claim above ensures that ˆG u 0 f dV ≤ ˆG φ 0 f dV.

We infer that u 0 ≤ φ 0 almost everywhere on G with respect to f dV, hence everywhere on G by the assumption on f . Letting M → +∞, we thus conclude that lim sup t→0 U * t = φ 0 on Ω.

The envelope is a subsolution

We now consider the set of subsolutions which are locally uniformly Lipschitz.

Definition 3.2.9. Let κ be a fixed positive constant. We let S κ φ 0 , f ,F (X T ) denote the set of all functions φ ∈ S φ 0 , f ,F (X T ) such that, for all t ∈ (0, T), x ∈ Ω,

t∂ t φ(t, x) ≤ κ -κ(ρ(x) + χ(x)). Set U κ := U κ φ 0 , f ,F,X T := sup{φ : φ ∈ S κ φ 0 , f ,F (X T )}. Proposition 3.2.10. For all 0 < S < T we have U κ φ 0 , f ,F,X S = U κ φ 0 , f ,F,X T in X S .
Proof. The proof is the same as that of Proposition 3.2.5.

Theorem 3.2.11. We have, for all κ > 0 and (t, x) ∈ X T ,

t∂ t U κ (t, x) ≤ L U -L U (ρ(x) + χ(x)),
where L U is the constant defined in Theorem 3.2.7.

Proof. The proof is the same as that of Theorem 3.2.7. In fact, if φ ∈ S κ φ 0 , f ,F (X T ) then the function u s in the proof of Theorem 3.2.7 satisfies

t∂ t u s ≤ α s (κ -κ(ρ + χ)) ≤ κ -κ(ρ + χ) because 0 < α s ≤ 1. It follows that u s ∈ S κ φ 0 , f ,F (X T
) and we argue as in the proof of Theorem 3.2.7 to conclude. Theorem 3.2.12. The upper envelope U is a pluripotential subsolution to (CMAF) in X T .

Proof. We will first show that U κ = (U κ ) * is a subsolution to (CMAF). Indeed, Choquet's lemma implies that there exists a sequence {φ j } in S κ φ 0 , f ,F (X T ) such that

(U κ ) * = sup j∈N φ j * in X T .
Since S κ is stable under finite maximum, we can assume that the sequence {φ j } is non-decreasing. We now claim that

dt ∧ (ω t + dd c φ j t ) n → dt ∧ (ω t + dd c (U κ ) * t ) n
in the sense of measures in (0, T) × Ω.

Let K be a relatively compact open subset of Ω and J be a compact interval of (0, T). Then there exists a constant C = C(J, K) > 0 such that for all j ∈ N, φ j (t, x) -Ct is decreasing in t ∈ J, for any x ∈ K. Moreover, the sequence of functions φ j increases towards u, so for any x ∈ K, u(t, x) -Ct is decreasing in t. Thus for each x ∈ K, there exists a countable subset E x ⊂ J such that u(•, x) is continuous on J\E x . Now set

E := {(t, x) ∈ J × K : t ∈ E x }.
Note that E has zero (2n + 1)-dimensional Lebesgue measure by using Fubini's theorem. Let N be the set of t ∈ J such that E t = {x ∈ K : (t, x) ∈ E} has positive Lebesgue measure. We must have that N has zero Lebesgue measure. Thus for any t ∈ J ′ := J\N, the set E t has zero Lebesgue measure, and lim s→t u(s, x) = u(t, x) for all x ∈ K\E t . Fixing (t, x) ∈ J ′ × K, we want to show that lim sup

(s,y)→(t,x) u(s, y) ≤ (U κ t ) * (x), (3.2.12)
where the upper semicontinuous regularization on the RHS is in the x-variable only. Since the problem is local we may assume that the functions u s are psh and negative in a neighborhood B(x, 2r) ⊂ K. Fix δ ∈ (0, r). For y so close to x that B(x, r) ⊂ B(y, r + δ) we have

φ j (s, y) ≤ 1 Vol(B(y, r + δ)) ˆB(y,r+δ) φ j (s, z)dV(z) ≤ 1 Vol(B(y, r + δ)) ˆB(y,r+δ) u(s, z)dV(z).
Letting j → +∞ we get

u(s, y) ≤ 1 Vol(B(y, r + δ)) ˆB(y,r+δ) u(s, z)dV(z) ≤ Vol(B(x, r)) Vol(B(y, r + δ)) 1 Vol(B(x, r)) ˆB(x,r)
u(s, z)dV(z).

Since lim s→t u(s, z) = u(t, z) for almost every z ∈ B(x, r) ⊂ K, Fatou's lemma yields lim sup

(s,y)→(t,x) u(s, y) ≤ Vol(B(x, r)) Vol(B(x, r + δ)) 1 Vol(B(x, r)) ˆB(x,r) u(t, z)dV(z).
Now, we first let δ → 0 and then r → 0 to obtain the desired inequality (3.2.12) by definition of U κ . The reverse inequality is clear, hence we get the equality. Therefore, for each t ∈ J ′ we have that φ j t increase almost everywhere towards (U κ t ) * = (U κ ) * t on K, so Bedford-Taylor's convergence theorem yields

(ω t + dd c φ j t ) n → (ω t + dd c (U κ ) * t ) n
in the weak sense of measures in K. Thus the claim follows directly from Fubini's theorem. On the other hand, for each x ∈ K fixed, the sequence {∂ t φ j (t, x) + F(t, x, φ j (t, x))} converges to ∂ t (U κ ) * (t, x) + F(t, x, (U κ ) * (t, x)) in the sense of distributions in J, with the later being bounded in J × K. Applying Proposition 3.1.11 we obtain lim j→+∞ e ∂ t φ j +F(t,•,φ j ) f dt ∧ dV ≥ e ∂ t (U κ ) * +F(t,•,(U κ ) * ) f dt ∧ dV in the weak sense of measures in J × K. It thus follows that (U κ ) * is a pluripotential subsolution to (CMAF) in Ω T , and hence (U κ ) * ∈ S κ φ 0 , f ,F (X T ). We thus deduce that U κ = (U κ ) * . We have shown that, for some κ 0 > 0, U κ = U κ 0 for all κ > κ 0 (by Theorem 3.2.11). It thus remains to prove that U = U κ 0 in X T . We first assume that φ 0 = P ω 0 h := sup{ψ ∈ PSH(X, ω 0 ) : ψ ≤ h} for some continuous function h. Fix 0 < S < T, s > 0 sufficiently small, and φ ∈ S φ 0 , f ,F (X T ). For (t, x) ∈ [0, S] × X, we define

u s (t, x) := α s φ(t + s, x) + (1 -α s )g(t + s) ρ(x) + χ(x) 2 -Cs(t + 1) -η(s),
where α s = 1 -(δ 0 g(0)) -1 s, η(s) := sup X (α s φ sh) and

C = (δ 0 g(0)) -1 C 1 (T + 1) + 2δ -1 0 M F + n| log(g(T))| + |c 1 |,
with C 1 > 0 defined in (3.2.3). We compute

ω t + dd c u s (t, •) =α s (ω t+s + dd c φ t+s ) + 1 -α s 2 (ω t+s + g(t + s)dd c ρ) + 1 -α s 2 (ω t+s + g(t + s)dd c χ) + ω t -ω t+s .
It follows from the assumption (3.0.2) that ω tω t+s ≥ -sΘ.

Since θ + dd c χ ≥ 2δ 0 Θ we thus obtain

1 -α s 2 (ω t+s + dd c χ) + ω t -ω t+s ≥ (δ 0 g(0)) -1 2 sg(t + s)(θ + dd c χ) -sΘ ≥ 0.
We thus get

(ω t + dd c u s (t, •)) n ≥ (α s (ω t+s + dd c φ t+s ) + (1 -α s )g(t + s)(θ + dd c ρ)/2) n
≥ e α s (∂ t φ t+s +F(t+s,•,φ t+s ))+(1-α s )(n log g(t+s)+c 1 ) f dV where we apply Lemma 3.2.6 in the last line. Since the function (t, r) → F(t, •, r) is uniformly Lipschitz, it follows that

α s F(t + s, •, φ t+s ) = F(t + s, •, φ t+s ) -(1 -α s )F(t + s, •, φ t+s ) ≥ F(t, •, φ t+s ) -κ F s -(δ 0 g(0)) -1 sM F . Since φ ≥ u on X T we have (1 -α s )φ t+s ≥ (1 -α s )g(t + s) ρ + χ 2 -(δ 0 g(0)) -1 sC 1 (T + 1).
Consequently, it follows from the choice of C that φ t+s ≥ u s (t, •) for all t ∈ [0, S]. Therefore,

α s F(t + s, •, φ t+s ) ≥ F(t, •, u s (t, •)) -s(κ F + (δ 0 g(0)) -1 M F ), since the function r → F(•, •, r) is increasing. Observe now that α s ∂ t φ t+s = α s ∂ t u s + Cs -α s g ′ (t + s) ρ + χ 2 .
It thus follows from the choice of C and the estimates above that

(ω t + dd c u s t ) n ≥ e ∂ t u s t +F(t,•,u s (t,•)) f dV,
which means that u s is a pluripotential subsolution to (CMAF). By definition of u s we have u s (0, •) ≤ h on X since sup X ρ = sup X χ = 0. Since u s (0, •) is ω 0 -psh, we infer u s (0, •) ≤ φ 0 = P ω 0 h on X. It follows that u s ∈ S φ 0 , f ,F (X S ), and hence u s ∈ S κ φ 0 , f ,F (X S ) for some κ > 0 large enough. Therefore, u s ≤ U κ = U κ 0 in X S by Proposition 3.2.10. On the other hand it follows from Hartogs' Lemma that lim s→0 η(s) ≤ 0. Letting s → 0 we get φ ≤ U κ 0 in X S . Finally, letting S → T to obtain φ ≤ U κ 0 , so U ≤ U κ 0 on X T (see Proposition 3.2.5). Therefore U = U κ 0 is the maximal subsolution to (CMAF) with initial data φ 0 .

We now remove the extra assumption on φ 0 . Let {h j } be a sequence of continuous functions decreasing to φ 0 . Then φ j 0 = P ω 0 (h j ) is a decreasing sequence of ω 0 -psh functions converging to φ 0 . We thus obtain that the upper envelope U j := U φ j 0 , f ,F,X T is also a subsolution to (CMAF) by the previous arguments. We also provide a uniform Lipschitz constant for U j . Since φ j 0 decreases to φ 0 , we have that U j decreases to some V ∈ P (X T ) which is a subsolution to (CMAF) and U ≤ V. On the other hand we see that V {0}×X ≤ φ 0 . Hence V = U.

The envelope is locally uniformly semi-concave in time

Theorem 3.2.13. There exists a uniform constant C U > 0 such that

t 2 ∂ 2 t U(t, x) ≤ C U -C U (ρ(x) + χ(x)), (3.2.13)
in the sense of distributions in X T .

Proof. Fix 0 < T ′ < T and ε 0 > 0 small enough such that (1

+ ε 0 )T ′ < T, s ∈ [1 -ε 0 , 1 + ε 0 ]. Set, for any (t, x) ∈ X T ′ , u s (t, x) := α s s -1 U(st, x) + sU(s -1 t, x) 2 + (1 -α s )g(t) ρ(x) + χ(x) 2 -C|s -1| 2 (t + 1),
where α s = 1 -A(s -1) 2 for A > 0 a uniform constant to be chosen later, and

C := (A + 1)C 0 + κ F T + An| log(g(T)) + c 1 |.
We are going to prove that u s is a subsolution to (CMAF). We compute

ω t + dd c u s (t, •) = α s 2 1 s (ω st + dd c U st ) + s(ω s -1 t + dd c U s -1 t ) + (1 -α s ) ω t + g(t)dd c ρ 2 + α s ω t - α s 2 (s -1 ω st + sω s -1 t ) + (1 -α s ) ω t + g(t)dd c χ 2 . Consider, for s ∈ [1 -ε 0 , 1 + ε 0 ], h(s) := ω t - 1 2 (ω st + ω s -1 t ).
We have h(1) = h ′ (1) = 0, and |h ′′ (s

)| ≤ (T + 2) 2 Θ on [1 -ε 0 , 1 + ε 0 ]. Hence α s ω t - α s 2 (s -1 ω st + sω s -1 t ) ≥ -(T + 2) 2 Θ.
Recall that χ is a θ-psh function on X such that θ + dd c χ ≥ 2δ 0 Θ. If we take A > 0 such that A ≥ (T + 2) 2 (δ 0 g(0)) -1 then

α s ω t - α s 2 (s -1 ω st + sω s -1 t ) + (1 -α s ) ω t + dd c χ 2 ≥ 0, hence, ω t + dd c u s (t, •) ≥ α s 2 1 s (ω st + dd c U st ) + s(ω s -1 t + dd c U s -1 t ) + (1 -α s ) ω t + g(t)dd c ρ 2 .
It follows from Theorem 3.2.12 that U is a pluripotential subsolution to (CMAF). We then have for almost every t ∈ (0, T ′ ) (s -1 (ω st + dd c U st )) n ≥ e ∂ τ U st +F(st,•,U st )-n log s f dV,

and

(s(ω s -1 t + dd c U st )) n ≥ e ∂ τ U s -1 t +F(s -1 t,•,U s -1 t )+n log s f dV.

Combining these together with Lemma 3.2.6, we obtain

(ω t + dd c u s (t, •)) n ≥ e α s (a(s)+a(s -1 ))+(1-α s )(n log g(t)+c 1 ) f dV, where
a(s) = α s 2 (∂ τ U st + F(st, •, U st )).
Since F is a convex function in r we get

1 2 F(st, •, U st ) + 1 2 F(s -1 t, •, U s -1 t ) ≥ F (s + s -1 )t 2 , •, U(st, •) + U(s -1 t, •) 2 . (3.2.14)
Now we use the same arguments as in Theorem 3.2.7 to show that for each t ∈ [0, T),

U(st, •) + U(s -1 t, •) 2 ≥ u s (t, •). (3.2.15) It is equivalent to show that 1 2 (1 -s -1 α s )U st + (1 -sα s )U s -1 t ≥ (1 -α s )g(t) ρ + χ 2 -C(t + 1)(s -1) 2 . (3.2.16)
The left-hand side can be rewritten as

1 2 (1 -s -1 (1 -A(s -1) 2 ))U st + (1 -s(1 -A(s -1) 2 ))U s -1 t = 1 2 (s -1)(U st -U s -1 t ) + (A -1)s -1 (s -1) 2 U st + As(s -1) 2 U s -1 t .
By Theorem 3.2.7, we have

(s -1)(U st -U s -1 t ) ≥ 2(s -1) 2 (L U (ρ + χ) -L U ).
The same arguments as in the proof of Theorem 3.2.7 give

(A -1)s -1 U st ≥ AU st ≥ (A -L U /g(0))g(t) ρ + χ 2 -C 1 (T + 1), AsU s -1 t ≥ (A + 1)U s -1 t ≥ (A -L U /g(0))g(t) ρ + χ 2 -C 1 (T + 1).
where A is large enough so that (A -L U /g(0))/(A + 1) ≥ (1 + δ 0 g(0)) -1 γ (γ = 1 + ε 0 κ g Tg(0) -1 ).

Combining these estimates, it follows from the choice of C that (3.2.16) holds. Since F is nondecreasing in r and uniformly Lipschitz in t, it follows from (3.2.14) and (3.2.15) that

1 2 F(st, •, U st ) + 1 2 F(s -1 t, •, U s -1 t ) ≥ F t, •, U(st, •) + U(s -1 t, •) 2 -κ F t s + s -1 2 -1 ≥ F(t, •, u s (t, •)) -κ F T(s -1) 2 , hence α s 2 F(st, •, U st ) + F(s -1 t, •, U s -1 t ) ≥ F(t, •, u s (t, •)) -κ F T(s -1) 2 -AM F (s -1) 2 .
Therefore, we obtain

a(s) + a(s -1 ) + (1 -α s )(n log g(t) + c 1 ) ≥ ∂ τ u s (t, •) + F(t, •, u s (t, •)).
On the other hand, the choice of C ensures, for any (t, x) ∈ X T ′ , that

u s (0, x) ≤ φ 0 (x) -C(s -1) 2 + (1 -α s )g(0) ρ(x) + χ(x) 2 -1 - s + s -1 2 α s φ 0 (x) ≤ φ 0 (x) -C(s -1) 2 + A(s -1) 2 g(0) ρ(x) + χ(x) 2 -(A + 1)(s -1) 2 φ 0 (x) ≤ φ 0 (x) -C(s -1) 2 + (A + 1)(s -1) 2 A A + 1 g(0) ρ(x) + χ(x) 2 -φ 0 (x) ≤ φ 0 (x) -C(s -1) 2 + (A + 1)C 0 (s -1) 2 ≤ φ 0 (x).
Therefore, we conclude that u s ∈ S φ 0 , f ,F (X T ′ ), so we obtain for any (t, x) ∈ X T ′ that

α s s -1 U(st, x) + sU(s -1 t, x) 2 -U(t, x) + A/2(s -1) 2 (ρ(x) + χ(x)) ≤ C(T + 1)(s -1) 2 , hence s -1 U(st, x) + sU(s -1 t, x) 2 -U(t, x) + A(s -1) 2 (ρ(x) + χ(x)) ≤ (C(T + 1) + 2AM 0 )(s -1) 2 .
From this, we obtain for all (t, x) ∈ X T ′ ,

U(st, x) + U(s -1 t, x) 2 -U(t, x) + A(s -1) 2 (ρ(x) + χ(x)) ≤ (C(T + 1) + (2A + 1)M 0 + 2L U -L U (ρ(x) + χ(x)))(s -1) 2 . Letting s → 1 yields for all (t, x) ∈ X T ′ t 2 ∂ 2 t U(t, x) ≤ (C(T + 1) + (2A + 1)M 0 + 5L U ) -(A + L U )(ρ(x) + χ(x)).
We finally let T ′ → T and apply Proposition 3.2.5 to complete the proof.

Existence and Uniqueness

Existence of solutions

We shall prove in this section that U φ 0 , f ,F,X T is the unique pluripotential solution to the Cauchy problem (see Definition 3.2.1). Theorem 3.3.1. The upper envelope U := U φ 0 , f ,F,X T is a pluripotential solution to the Cauchy problem for the parabolic complex Monge-Ampère equation (CMAF) in X T . Moreover, U is locally uniformly semiconcave in (0, T) × Ω.

Proof. We have shown in Theorem 3.2.13 that U is locally uniformly semi-concave in t ∈ (0, T), and U ∈ S φ 0 , f ,F (X T ) and it satisfies the initial condition. It remains to show that U solves the parabolic equation (CMAF). We apply a local balayage process to modify the function U on a given "small ball" B ⋐ Ω by constructing a new ω t -psh function U B so that it satisfies the local Monge-Ampère flow

(ω t + dd c U B ) n = e ∂ t U B (t,•)+F(t,•,U B (t,•)) f dV on B T = (0, T) × B, U B ≥ U on B T and U B = U on X T \ B T .
Indeed, we choose complex coordinates z = (z 1 , • • • , z n ) identifying B with the complex unit ball B ⊂ C n . We can write ω t = dd c g t in a local holomorphic coordinate chart B ⊂ X, for some smooth local potential g t . Set f = f • z -1 ∈ L p (B) and d Ṽ is the restriction of the volume dV to B. We consider the following complex Monge-Ampère flow

dt ∧ (dd c u t ) n = e ∂ t u(t,•)+ F(t,•,u(t,•)) f d Ṽ ∧ dt (3.3.1)
in B T := (0, T) × B with the Cauchy-Dirichlet boundary data h being the restriction of U defined on the parabolic boundary of B T denoted by ∂ P B T := ([0, T) × ∂B) ∪ ({0} × B). Here F(t, x, r) = F(t, x, rg t (x)) -∂ t g t satisfies the same assumptions as F. We have shown that h is locally uniformly Lipschitz (see Theorem 3.2.7) and locally uniformly semi-concave (see Theorem 3.2.13) i.e. for all 0 < T ′ < T, and for all (t, z) ∈ (0, T ′ ) × ∂B, there exist constants L = L(T ′ ), and C = C(T ′ ) such that

t|∂ t h(t, z)| ≤ L, t 2 |∂ 2 t h(t, z)| ≤ C. (3.3.2)
Using mollifiers we can find a sequence h j of continuous functions on [0, T) × ∂B such that h j decreases pointwise to h. The function h j is thus the Cauchy-Dirichlet boundary data satisfying the same assumption (3.3.2) as h.

Then it follows from [GLZ21a, Theorem 6.4] that there exists a sequence of functions u j solving (3.3.1) with the boundary data h j . Moreover, u j is locally uniformly semi-concave in t ∈ [0, T). Since h j decreases to U • z -1 on ∂ P B T , so U • z -1 ≤ u j and the sequence u j decreases to some function v. The function v solves (3.3.1) by using Proposition 3.1.12, and lim sup t→0 v(t, z)

≤ U 0 • z -1 in B. But the comparison principle (see [GLZ21a, Theorem 6.5]) ensures that U • z -1 ≤ v in B T . Hence lim t→0 v(t, z) = U 0 • z -1 . We then define U B (t, x) = v(t, z(x)) in B T U(t, x) in X T \ B T .
as required. We infer that U B belongs to the set S φ 0 , f ,F (X T ), the maximal property ensures that U B ≤ U, hence equality. Since B is an arbitrary ball in Ω, this shows that U solves (CMAF) on Ω T , hence on X T . Moreover, by Theorem 3.2.7 and Theorem 3.2.13, U is locally uniformly uniformly Lipschitz and semi-concave in t.

The comparison principle

We first establish a version of the comparison principle which requires relatively strong regularity assumptions : Proposition 3.3.2. Let φ (resp. ψ) be a subsolution (resp. supersolution) to (CMAF) with initial value φ 0 (resp. ψ 0 ). We assume that a) φ is C 1 in t and continuous on (0, T) × Ω, b) ψ is locally uniformly semi-concave in t c) φ t → φ 0 and ψ t → ψ 0 in L 1 (X), as t → 0, d) for any t ∈ [0, T), ψ t has minimal singularities, e) the function (t, x) → ψ(t, x) is continuous on [0, T) × Ω.

Then

φ 0 ≤ ψ 0 ⇒ φ ≤ ψ in X T .
Proof. Fix 0 < T ′ < T, in particular T ′ < +∞. We shall prove that φ ≤ ψ on [0, T ′ ] × X. The result thus follows by letting T ′ → T. We fix λ, ε > 0 sufficiently small. Set for (t, x) ∈ [0, T ′ ] × X,

φ λ (t, x) := (1 -λ)φ(t, x) + λg(t) ρ(x) + χ(x) 2 ,
where ρ, χ are θ-psh functions defined in (3.1.2), (3.1.1). One can moreover impose χ < 0 to be smooth in the ample locus Ω = Amp{θ}, with analytic singularities, and such that χ(x) → -∞ as x → ∂Ω. We will show that φ λ ≤ ψ and we then let λ → 0 to conclude the proof. Set

w(t, x) := φ λ (t, x) + λg(0)δ 0 χ(x) -ψ(t, x) -3εt.
Observe that by Lemma 3.1.5, this function is upper semi-continuous on [0, T ′ ] × Ω. By the assumption d) we have φ λ (t, •) ≤ ψ(t, •) + O(1) for each t. Since χ is continuous in Ω and tends to -∞ on ∂Ω, we have that w tends to -∞ on ∂Ω. Hence w attains its maximum at some point

(t 0 , x 0 ) ∈ [0, T ′ ] × Ω.
We want to show that w(t 0 , x 0 ) ≤ 0. Assume by contradiction that it is not the case i.e w(t 0 , x 0 ) > 0, with t 0 > 0. The set

K := {x ∈ Ω : w(t 0 , x) = w(t 0 , x 0 )} is a compact subset of Ω since w(t 0 , x) tends to -∞ as x → ∂Ω. The classical maximum principle ensures for all x ∈ K that (1 -λ)∂ t φ(t 0 , x) ≥ ∂ - t ψ(t 0 , x) + 3ε, since g ′ (t) ≥ 0 for all t.
The partial derivative ∂ t φ(t, x) is continuous in Ω by assumption. Since the function t → ψ(t, x) is locally uniformly semi-concave, for any t ∈ (0, T), the left derivative ∂ - t ψ(t, •) is upper semi-continuous in Ω (see Proposition 3.1.10). We can thus find η > 0 small enough that, by introducing the open set containing K, D := {x ∈ Ω : w(t 0 , x) > w(t 0 , x 0 ) -η} ⋐ Ω.

We have for all x ∈ D,

(1λ)∂ t φ(t 0 , x) > ∂ - t ψ(t 0 , x) + 2ε.

(3.3.3)

Set u := φ λ (t 0 , •) + λg(0)δ 0 χ and v = ψ(t 0 , •). We observe that

λ ω t + dd c g(t)χ 2 + λg(0)dd c χ ≥ λg(0)δ 0 Θ + λg(0)δ 0 dd c χ ≥ 0.
Since φ is a pluripotential subsolution to (CMAF), we infer by using Lemma 3.2.6 that

(ω t 0 + dd c u) n ≥ ((1 -λ)(ω t 0 + dd c φ t 0 ) + λ(g(t 0 )θ + g(t 0 )dd c ρ)/2) n ≥ e (1-λ)(∂ t φ(t 0 ,•)+F(t 0 ,•,φ(t 0 ,•)))+λ(n log g(t)+c 1 ) f dV ≥ e (1-λ)∂ t φ(t 0 ,•)+F(t 0 ,•,φ(t 0 ,•))-λ(M F +|n log g(t)+c 1 |) f dV
in the weak sense of measures in D. Choosing λ so small that

λ < min [0,T] {(M F + |n log g(t) + c 1 |) -1 ε}.
It thus follows from (3.3.3) and the increasing property of F that

(ω t 0 + dd c u) n ≥ e ∂ - t ψ(t 0 ,•)+F(t 0 ,•,u)+ε f dV
in the weak sense of measures in D. On the other hand, ψ is a pluripotential supersolution to (CMAF), thus

(ω t 0 + dd c v) n ≤ e ∂ - t ψ(t 0 ,•)+F(t 0 ,•,ψ(t 0 ,•)) f dV
in the weak sense of measures in D. The last two inequalities yield

(ω t 0 + dd c u) n ≥ e F(t 0 ,•,u)-F(t 0 ,•,v)+ε (ω t 0 + dd c v) n .
Shrinking D if necessary, we can assume that u(x) > v(x) for any x ∈ D. We thus get

(ω t 0 + dd c u) n ≥ e ε (ω t 0 + dd c v) n
in the sense of measures in D.

Consider now ũ := u + min ∂D (vu). We observe that v ≥ ũ on ∂D, hence the elliptic comparison principle (see Proposition 3.3.3) yields ˆ{v< ũ}∩D e ε (ω

t 0 + dd c v) n ≤ ˆ{v< ũ}∩D (ω t 0 + dd c u) n ≤ ˆ{v< ũ}∩D (ω t 0 + dd c v) n .
Thus ũ ≤ v almost everywhere in D with respect to the measure (ω t 0 + dd c v) n . It thus follows from the domination principle (Proposition 3.3.3) that ũ ≤ v everywhere in D. In particular,

u(x 0 ) -v(x 0 ) + min ∂D (v -u) ≤ ũ(x) -v(x) ≤ 0. (3.3.4)
On the other hand, since K ∩ ∂D = ∅, we get w(t 0 , x) ≤ w(t 0 , x 0 ), for all x ∈ ∂D, hence

u(x) -v(x) < u(x 0 ) -v(x 0 ), contradicting (3.3.4
). Therefore, we must have t 0 = 0, hence Assume that v has minimal singularities. Then

(1 -λ)φ + λg(t) ρ + χ 2 + λg(0)χ -ψ -3εt ≤ λ sup X g(0) ρ + χ 2 -φ 0 , in [
ˆ{u<v}∩D MA θ (v) ≤ ˆ{u<v}∩D MA θ (u). Moreover, if MA θ (u)({u < v} ∩ D) = 0 then u ≥ v in D.
Proof. Fix ε > 0. For each C > 0 we set u C := max(u, V θ -C). Then the function max(u C , vε) is θ-psh with minimal singularities and coincides with u C in a neighborhood of ∂D. The boundary condition means that for any ε > 0 the subset {u

C < v -ε} is compact in D. Let {u C < v -ε} ⋐ D ′ ⋐ D. We claim that ˆD′ (θ + dd c u C ) n = ˆD′ (θ + dd c max(u C , v -ε)) n . (3.3.5)
Indeed, set w := max(u C , vε), using local regularization of plurisubharmonic functions, we observe that (θ + dd c w) n -(θ + dd c u C ) n = dd c S in the sense of currents on D, where S := (w -

u C )((θ + dd c w) n-1 + • • • + (θ + dd c u C ) n-1
) is a well-defined current with compact support in D. Pick any test function γ which is identically 1 in a neighborhood of the support of S. Then

ˆD′ dd c S = ˆD′ γdd c S = ˆD′ S ∧ dd c γ = 0,
where we have known that dd c γ = 0 on the support of S. This implies (3.3.5).

On the other hand, we apply [GZ17a, Theorem 3.27] to get ˆ{u C <v-ε}

(θ + dd c max(u C , v -ε)) n = ˆ{u C <v-ε} (θ + dd c v) n .
Combining this together with the equality (3.3.5), we obtain

ˆ{u C <v-ε}∩D ′ MA θ (v) = ˆD′ MA θ (max(u C , v -ε)) - ˆ{u C ≥v-ε} MA θ (max(u C , v -ε)) ≤ ˆD′ MA θ (u C ) - ˆ{u C >v-ε} MA θ (max(u C , v -ε)) ≤ ˆD′ MA θ (u C ) - ˆ{u C >v-ε} MA θ (u C ), which implies ˆ{u C <v-ε}∩D MA θ (v) ≤ ˆ{u C ≤v-ε}∩D MA θ (u C )
since D ′ was taken arbitrarily. Letting ε → 0 and then C → +∞ we obtain the required inequality. Arguing as in [BEGZ10, Corollary 2.5], we complete the last statement. Let χ be a θ-psh function defined in (3.1.1). Since v has minimal singularities, we may also assume that χ ≤ v. For any ε > 0 small enough, consider

v ε := (1 -ε)v + εχ, hence lim sup D∋x→∂D (u(x) -v ε (x)) ≥ 0. Then ε n ˆ{u<v ε } MA θ (χ) ≤ ˆ{u<v ε } MA θ (v ε ) ≤ ˆ{u<v ε } MA θ (u) = 0, since χ ≤ v implies that {u < v ε } ⊂ {u < v}.
On the other hand, MA θ (χ) dominates Lebesgue measure. We deduce that u ≥ v ε almost everywhere with respect to Lebesgue measure in the open set D, hence everywhere in D. The result follows by letting ε → 0.

We now slightly relax the hypothesis e) in Proposition 3.3.2.

Proposition 3.3.4. Let φ (resp. ψ) be a pluripotential subsolution (resp. supersolution) to (CMAF) with initial value φ 0 (resp. ψ 0 ). We assume that a) φ is C 1 in t and continuous on (0, T) × Ω, b) ψ is locally uniformly semi-concave in t, c) φ t → φ 0 and ψ t → ψ 0 , as t → 0, d) for any t ∈ (0, T), ψ t has minimal singularities, e') the function (t, x) → ψ(t, x) is continuous on (0, T) × Ω.

Then φ 0 ≤ ψ 0 ⇒ φ ≤ ψ in X T .
Proof. We proceed as in the proof of Theorem 3.2.7. We fix s > 0 sufficiently small and consider

v s (t, x) = ψ(t + s, x) + Cs(t + 1) -Cs log δ -1 0 s, and
u s (t, x) := α s φ(t, x) + (1 -α s )g(t) ρ(x) + χ(x) 2 -Cs(t + 1).
Here α s = 1 -As ∈ (0, 1), A > 0 is determined in Lemma 3.3.5, the functions ρ, χ are defined in (3.1.2), (3.1.1) and C is a positive constant which will be chosen later. We want to show that for C > 0 large enough, u s is a subsolution while v s is a supersolution to (CMAF) and u s (0, •) ≤ v s (0, •). We can then apply Proposition 3.3.2 and let s → 0 to complete the proof. We first observe that

ω t+s + dd c u s = α s (ω t + dd c φ t ) + 1 -α s 2 (ω t + g(t)dd c ρ) + 1 -α s 2 (ω t + g(t)dd c χ) + ω t+s -ω t .
By assumptions, we see that for s > 0 ω t+sω t ≥ -sΘ. Since θ + dd c χ ≥ 2δ 0 Θ we thus obtain

1 -α s 2 (ω t + g(t)dd c χ) + ω t+s -ω t ≥ Asg(t)δ 0 Θ -sΘ ≥ 0.
Hence,

(ω t+s + dd c u s ) n ≥ (α s (ω t + dd c φ t ) + (1 -α s )g(t)(θ + dd c ρ)/2) n ≥ e α s (∂ t φ t +F(t,•,φ t ))+(1-α s )(n log g(t)
(ω t+s + dd c u s ) n ≥ e ∂ t u s +F(t,•,u s ) f dV.
On the other hand, since ψ is a supersolution, we have

(ω t+s + dd c v s ) n ≤ e ∂ t v s -Cs+F(t+s,•,ψ(t+s,•)) f dV ≤ e ∂ t v s -Cs+F(t,•,v s (t,•))+κ F s f dV ≤ e ∂ t v s +F(t,•,v s (t,•)) f dV,
where the second line follows from the Lipschitz condition and the increasing monotonicity of F, the last line follows from the choice of C. Up to increasing C > 0 it follows from Lemma 3.3.5 that for any x ∈ X,

u s (0, x) ≤ (1 -As)φ(0, x) + Ag(0)s(ρ + χ)/2 ≤ ψ s (x) + Cs -Cs log Ag(0)s = v s (0, x).
It then follows from Proposition 3.3.2 that u s (t, x) ≤ v s (t, x) for all (t, x) ∈ X T . Letting s → 0, we finish the proof.

Lemma 3.3.5. With the same assumptions of ψ as in Proposition 3.3.4. Then there exist uniform constants A > 0, C > 0, and t 0 > 0 small enough such that for all (t, x) ∈ (0, t 0 ) × X,

ψ(t, x) ≥ (1 -At)ψ 0 (x) + C(t log(Ag(0)t) -t) + Ag(0)t(ρ(x) + χ(x))/2.
Proof. The proof is similar to that of [GLZ20, Lemma 3.14]. We recall that the function F satisfies the Lipschitz condition i.e., there exists a constant κ F > 0 such that, for all t, t ′ ∈ [0,

T/2], x ∈ X, r ∈ R, |F(t, x, r) -F(t ′ , x, r)| ≤ κ F |t -t ′ |.
Set t 0 = min(1, A, T/4) with A > 0 under control. Fix s > 0 sufficiently small and consider for (t, x) ∈ (0, t 0 ) × X,

u s (t, x) := (1 -At)ψ s (x) + Atg(s)(ρ(x) + χ(x))/2 + n(t log Ag(0)t -t) -Ct, v s (t, x) := ψ(t + s, x) + 2κ F ts,
where ρ, χ are θ-psh functions on X defined in (3.1.2),(3.1.1), and C is a positive constant to be chosen later. We see that u s is of class C 1 in t, and for any t ∈ (0, t 0 ) fixed, u s (t, •) is continuous in Ω since ρ is continuous in Ω (see e.g. [GZ17a, Theorem 12.23]). Now let A ≥ (δ 0 g(0)) -1 . One has, for any t ∈ (0, t 0 ),

ω t+s + dd c u s (t, •) = (1 -At)(ω s + dd c ψ s ) + At(ω s + g(s)dd c ρ)/2 + At(ω s + g(s)dd c χ)/2 + ω t+s -ω s .
By hypothesis (3.0.2), we have ω t+sω s ≥ -tΘ. Since θ + dd c χ ≥ 2δ 0 Θ, we thus obtain

At(ω s + g(s)dd c χ)/2 + ω t+s -ω s ≥ Atg(s)δ 0 Θ -tΘ ≥ 0, hence (ω t+s + dd c u s ) n ≥ (At(ω s + g(s)dd c ρ)/2) n ≥ (Ag(0)t) n e c 1 f dV, since g(s) ≥ g(0). We now choose C = A sup X (ρ + χ -2φ 0 )/2 + M F -min(c 1 , 0), hence (ω t+s + dd c u s ) n ≥ e ∂ t u s (t,•)+F(t,•,u s (t,•)) f dV.
It follows from the definition that u s (t, •) converges in L 1 (X, dV) to u s (0, •) = ψ s . On the other hand, since ψ is a supersolution to (CMAF), we have

(ω t+s + dd c v s ) n ≤ e ∂ t ψ t+s +F(t+s,•,ψ(t+s,•)) f dV = e ∂ t v s -2κs+F(t+s,•,ψ(t+s,•)) f dV.
Since the function F is Lipschitz in t and is increasing in r, for all t, s ∈ (0, t 0 ), x ∈ X,

F(t + s, x, ψ(t + s, x)) ≤ F(t, x, ψ(t + s, x)) + κ F s ≤ F(t, x, v s (t, x)) + κ F s,
this yields (ω t+s + dd c v s ) n ≤ e ∂ t v s (t,•)+F(t,•,v s (t,•)) f dV.

Since for each s that ψ s is continuous on Ω hence v s is continuous on [0, t 0 ) × Ω and it is clear that v s (t, •) converges to v s (0, •) = ψ s in L 1 (X, dV) as t → 0. We moreover see that for each t, v s (t, •) has minimal singularities because ψ t has. We can now apply Proposition 3.3.2 and get u s ≤ v s on (0, t 0 ) × X. Letting s → 0 we have for all (t, x) ∈ (0, t 0 ) × X,

(1 -At)ψ 0 (x) + Ag(0)t(ρ(x) + χ(x))/2 + C(t log(Ag(0)t) -t) ≤ ψ(t, x),
as desired.

Space regularity

In this section, we shall use the extra assumption

ωt ≤ Aω t , ∀ t ∈ [0, T), (3.3.6)
for some constant A > 1.

Theorem 3.3.6. Under the extra assumption (3.3.6), the envelope U has minimal singularities and U t is moreover continuous in Ω, for each t ∈ (0, T).

Proof. We first show that for each t, U t has minimal singularities. Observe that

η t := e -At ω t
is decreasing in t. By [BEGZ10, Theorem 6.1], for each t ∈ [0, T) there exists a unique η t -psh function ϕ t with full Monge-Ampère mass such that

(η t + dd c ϕ t ) n = e ϕ t +c 1 f dV,
for c 1 > 0 so that sup X ϕ 0 = 0. For 0 < s ≤ t we have

(η s + dd c ϕ t ) n ≥ (η t + dd c ϕ t ) = e ϕ t +c 1 f dV.
It follows that ϕ t is a subsolution to (η s + dd c ϕ s ) n = e ϕ s +c 1 f dV, so a classical comparison principle (see e.g. [BEGZ10, Proposition 6.3]) ensures that ϕ t ≤ ϕ s for s ≤ t. Therefore the function t → ϕ t (x) is decreasing for all x ∈ X. Since t → η t is decreasing in t, we may assume that η t ≥ θ ′ for some big (1, 1)-form θ ′ . As explained in Section 3.1.3, φt = ∂ t ϕ t is well-defined almost everywhere and negative. Set for any (t, x) ∈ X T ,

u(t, x) := e At ϕ t (x) -C 2 (t + 1)
where C 2 > 0 is a constant to be chosen later. Since sup X ϕ 0 = 0 and ϕ t is decreasing in t, we have ϕ t ≤ 0 for all t ∈ [0, T). We infer for almost every t ∈ [0, T),

u(t, •) = e At φt + Ae At ϕ t -C 2 ≤ ϕ t -C 2 , hence (ω t + dd c u t ) n = e nAt+ϕ t +c 1 f dV ≥ e ut +F(t,•,u(t,•)) f dV,
where for the last inequality we have chosen C 2 > 0 so big that C 2 > nAT + c 1 + M F . Hence u t is a subsolution to (CMAF). Moreover, we can choose C 2 > 0 so big that u(0, •) = ϕ 0 -C 2 ≤ φ 0 since φ 0 is a η 0 -psh function with minimal singularities. Therefore, u is a subsolution to the Cauchy problem, i.e. u ∈ S φ 0 , f ,F (X T ). By [BEGZ10, Theorem 6.1] we have

ϕ t ≥ V η t -C(t) for some time-dependent constant C(t). Since V ω t = e At V η t , we thus infer u t ≥ V ω t -C ′ (t), hence U t ≥ V ω t -C ′ (t) for all t ∈ [0, T).
It remains to show that for each t ∈ (0, T), U t is continuous in Ω. We have that for each t > 0, ∂ t U + F(t, •, U t ) ≤ κκ(ρ + χ) + M F on X, for some constant κ > 0. The continuity of U t in Ω thus follows from [Dan21, Theorem 3.2].

We now show that the solution constructed in Theorem 3.3.1 is unique : Theorem 3.3.7. Let Φ be a pluripotential solution to the Cauchy problem for (CMAF) with initial data φ 0 . Assume that (3.3.6) holds and • Φ is locally uniformly semi-concave in (0, T) ;

• for each t, Φ t has minimal singularities ;

• Φ t is continuous in Ω. Then Φ = U.
Proof. Since Φ is locally uniformly Lipschitz in t we infer that Φ is continuous on (0, T) × Ω. We would like to apply Proposition 3.3.4 but U is not C 1 in t. We are going to regularize it by taking convolution in t as in [GLZ20, Proposition 3.16]. Fix 0 < T ′ < T, s > 0 near 1. Set, for any

(t, x) ∈ X T ′ , V s (t, x) := α s s U(st, x) + (1 -α s )g(t) ρ(x) + χ(x) 2 -C|s -1|(t + 1),
where α s , C are defined as in the proof of Theorem 3.2.7 so that V s ∈ S φ 0 , f ,F (X T ′ ). Let η be a smooth function with compact support in [-1, 1] such that ´η(t)dt = 1. Set, for ε > 0 small, η ε (t) = ε -1 η(t/ε), and we define for any (t,

x) ∈ X T ′ u ε (t, x) := ˆR V s (t, x)η ε (s -1)ds -Bε(t + 1).
When K X is not nef, the flow has a finite time singularity at T (< +∞). The limiting class at T of the flow

α T := lim t→T {ϑ t } = e -T {ω 0 } + (1 -e -T )c 1 (K X )
is big and nef. In [CT15, Theorem 1.5], Collins and Tosatti showed that the flow ϑ t exists on the maximal time interval [0, T) and develops singularities precisely on the Zariski closed set X\Amp(α T ) as t → T -. For t > T, the cohomology class {ϑ t } is still big, but no longer nef, we can not continue the flow in the classical sense.

In [FIK03, Section 10], Feldman, Ilmanen and Knopf have asked the question : can one define and construct weak solutions of the Kähler-Ricci flow beyond the singular time ? In [BT12, Theorem 4], Boucksom and Tsuji have constructed the normalized Kähler-Ricci flow on smooth projective varieties with pseudoeffective canonical class for all times. They used the discretization of the Kähler-Ricci flow and some algebro-geometric tools. In the end, they have conjectured the same result for the case of general Kähler manifolds (see [BT12, Conjecture 1]). Tô [START_REF] Tô | Convergence of the Weak Kähler-Ricci Flow on Manifolds of General Type[END_REF] used the viscosity theory to show that the weak Kähler-Ricci flow exists for all time in the viscosity sense and converges to the unique singular Kähler-Einstein metric in the class c 1 (K X ) constructed in [START_REF] Eyssidieux | Singular Kähler-Einstein metrics[END_REF][START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF].

In this section we show that the normalized Kähler-Ricci flow can be extended through a finite time singularity and understood in the weak sense of parabolic pluripotential theory developed in Section 3.3. A compact Kähler manifold of general type turns out to be projective by a classical result of Moishezon. Our result thus gives an alternative approach to the existence of weak Kähler-Ricci flows previously obtained by Boucksom-Tsuji and Tô. The main point in our proof is that the flow survives through a finite time singularity provided that the limiting class is big. The argument is thus working also on a general Kähler manifold with pseudoeffective canonical class as will be shown later in Section 3.4.2. Now let θ be a smooth closed (1, 1)-form representing c 1 (K X ). Set ω t := e -t ω 0 + (1e -t )θ. Since dV X is a smooth volume form on X, -Ric(dV) ∈ c 1 (K X ), and so there is a smooth function f such that θ = -Ric(dV X ) + dd c f . We then define µ = e f dV which is a smooth positive volume with θ = Ric(µ). Thus the normalized Kähler-Ricci flow (3.4.1) can be written as the complex Monge-Ampère flow (ω t + dd c φ t ) n = e φt +φ t dµ.

(3.4.2)

It has been shown in [BEGZ10, Theorem 6.1] that there exists a unique θ-psh function φ KE with minimal singularities such that

(θ + dd c φ KE ) n = e φ KE µ. (3.4.3)
The current ω KE := θ + dd c φ KE is called the singular Kähler-Einstein metric. It has bounded poten- tials and is smooth in Ω := Amp(K X ), where it satisfies

Ric(ω KE ) = -ω KE .
We are going to show that the (normalized) pluripotential Kähler-Ricci flow (3.4.2) exists for all time and continuously deforms any initial Kähler form ω 0 towards ω KE on Amp(K X ), as t → +∞.

The result even holds for an initial datum S 0 which is a positive current with bounded potentials :

Theorem 3.4.1. Let φ 0 be a bounded ω 0 -psh function. Then there exists a unique pluripotential solution φ to (3.4.2) with initial data φ 0 for t > 0. Furthermore, φ t converges exponentially fast towards φ KE on Amp(K X ), as t → +∞.

Proof. Recall that θ is a smooth representative of c 1 (K X ). Since ω 0 is a Kähler form, there exists a small constant c > 0 such that ω 0 ≥ cθ. Hence ω t = e -t ω 0 + (1e -t )θ ≥ g(t)θ, where g(t) = ce -t + 1e -t is a smooth (strictly) positive function with g ′ (t) = e -t (1c) > 0 for c > 0 small enough. The existence of the unique pluripotential solution follows from the results in Section 3.3. It thus remains to study its long-term behavior. We first establish a lower bound for the solution φ by constructing a subsolution to the Cauchy Problem. Set, for any (t, x) ∈ (0, T) × X, u(t, x) := e -t φ 0 (x) + (1e -t )φ KE (x) + h(t), for a C 1 function h to be chosen later so that u is a subsolution to the Cauchy Problem. We observe that for all t > 0,

ω t + dd c u t = e -t (ω 0 + dd c φ 0 ) + (1 -e -t )(θ + dd c φ KE ) ≥ 0
in the weak sense of currents, so u t is ω t -psh and

(ω t + dd c u t ) n ≥ (1 -e -t ) n (θ + dd c φ KE ) n = e n log(1-e -t ) e φ KE f dV.
On the other hand

∂ t u t + u t = φ KE + h ′ (t) + h(t) hence u is a subsolution if n log(1 -e -t ) ≥ h(t) + h ′ (t).
We thus choose h to be the unique solution of the ODE :

h(t) + h ′ (t) = n log(1 -e -t ), h(0) = 0. We compute (e t h(t)) ′ = e t (h(t) + h ′ (t)) = ne t log(1 -e -t ), hence h(t) = ne -t ˆet log(1 -e -t )dt = ne -t (e t -1) log(e t -1) -te t + C .
for some constant C. Since h(0) = 0 the constant C must be zero, hence h(t) = ne -t (e t -1) log(e t -1)te t = O(te -t ) as t → ∞.

It follows from the comparison principle that u ≤ φ hence

φ KE (x) + e -t (φ 0 (x) -φ KE (x)) + h(t) ≤ φ(t, x).
(3.4.4)

For the upper bound, we argue as in the proof of [START_REF] Tô | Convergence of the Weak Kähler-Ricci Flow on Manifolds of General Type[END_REF]Theorem 4.4]. Since the cohomology class of θ is big, we can find a θ-psh function χ 0 with analytic singularities such that θ + dd c χ 0 ≥ εω 0 for some small constant ε > 0. We can assume that ε ≤ 1. Replacing χ 0 by χ 0sup X χ 0 , we can always assume that χ 0 ≤ 0 hence χ 0 ≤ V θ . We then have

ω t + dd c φ t = e -t (ω 0 -ε -1 dd c χ 0 ) + (1 -e -t )θ + dd c (φ t + ε -1 e -t χ 0 ) ≤ [e -t ε -1 + (1 -e -t )]θ + dd c (φ t + ε -1 e -t χ 0 ). (3.4.5)
Set g(t) = 1 + (ε -1 -1)e -t and u(t, x) = φ t (x) + e -t (ε -1 χ 0 (x) -C) for a constant C > 0 to be chosen later. It follows from (3.4.5) that (g(t)θ + dd c u t ) n ≥ (ω t + dd c φ t ) n ≥ e φt +φ t µ = e ut +u t µ, Let ϕ 0 be a ε -1 θ-psh function with minimal singularities. We can find a constant C > 0 such that ϕ 0ε -1 χ 0 ≥ φ 0 -C. Therefore u is a subsolution of the following Cauchy problem

(g(t)θ + dd c ϕ t ) n = e ∂ t ϕ t +ϕ t µ ϕ(0, •) = ϕ 0 , (3.4.6)
where ϕ denotes the pluripotential solution to (3.4.6). Thus the comparison principle (Proposition 3.3.4) yields u ≤ ϕ on [0, ∞) × Amp(K X ), i.e.

φ(t, x) + e -t (ε -1 χ 0 (x) -C) ≤ ϕ(t, x).
On the other hand, the function ϕ t converges to φ KE on Amp(K X ) as t → +∞ by Lemma 3.4.2 below. Combining this with (3.4.4), we infer that φ t converges to φ KE on Amp(K X ) as t → +∞.

Lemma 3.4.2. The solution ϕ t of (3.4.6) converges locally exponentially fast to φ KE on Amp(K X ) as t → +∞.

Proof. Set φt := g(t) -1 (ϕ t -a(t)),
where g(t) = 1 + (ε -1 -1)e -t , and a is the unique solution of the ODE : a(t) + a ′ (t) = n log g(t), with a(0) = 0. An easy computation shows that a(t) = O(te -t ). Now the flow (3.4.6) becomes

(θ + dd c φt ) n = e g(t)∂ t φt + φt µ φ(0, •) = εϕ 0 . (3.4.7)
We now normalize in time ψ(t, •) = φ(s(t), •), where s(t) is the unique solution of the ODE s ′ (t) = g(s(t)) with s(0) = 0. Then the flow (3.4.6) can be written as

(θ + dd c ψ t ) n = e ∂ t ψ t +ψ t µ ψ(0, •) = εϕ 0 . (3.4.8)
We set for any (t, x) ∈ (0, +∞) × X, u(t, x) := e -t ψ 0 (x) + (1e -t )φ KE (x) + h(t), where h is the unique solution to the ODE h ′ (t) + h(t) = n log(1e -t ), with h(0) = 0. As in the proof of Theorem 3.4.1 we can check that u is a subsolution to the Cauchy problem for the flow (3.4.8).

On the other hand, since φ KE is a θ-psh function with minimal singularities we can choose a constant C > 0 such that φ KE + C ≥ ψ 0 . Set for (t, x) ∈ (0, +∞) × X, v(t, x) := φ KE (x) + Ce -t . One can check that v is supersolution to (3.4.8). Therefore, the comparison principle yields

e -t ψ 0 (x) + (1 -e -t )φ KE (x) + O(te -t ) ≤ ψ(t, x) ≤ φ KE (x) + Ce -t ,
which implies ψ t → φ KE on Amp(K X ) as t → +∞. So does the flow ϕ t since s(t) → +∞, g(t) → 1 and a(t) → 0 as t → +∞.

Extending the Kähler-Ricci flow through finite time singularities

In this subsection, we apply our results to prove the existence of the (pluripotential) Kähler-Ricci flow on manifolds through a finite time singularities. In particular, the answer of Feldman-Ilmanen-Knopf's question is affirmative also in this case.

Let (X, ω 0 ) be a compact Kähler manifold of dimension n and consider the Kähler-Ricci flow with initial data ω 0 , ∂θ ∂t = -Ric(θ), θ| t=0 = ω 0 .

(3.4.9)

The maximal existence time T of the flow is defined by

T := sup{t > 0 : {ω 0 } + tc 1 (K X ) is Kähler}.
Suppose that T < ∞ (K X is not nef). Then the limiting class {θ T } := {ω 0 } + Tc 1 (K X ) is nef, but not Kähler. If we assume moreover that ´X θ n T > 0, then the class {θ T } is big by a fundamental theorem of Demailly and Paun [DP04, Theorem 2.12]. Since the set of big cohomology classes is open, there is a constant ε > 0 so small that the class {θ t } is big for t ∈ [0, T + ε). We can prove the existence of a pluripotential solution of the flow on [0, T + ε). Theorem 3.4.4. Let (X, ω 0 ) be a compact Kähler manifold. Assume that the solution θ(t) of the Kähler-Ricci flow (3.4.9) starting at ω 0 exists on the maximal time interval [0, T) with T < ∞, and that the limiting class {ω 0 } + Tc 1 (K X ) is big. Then the pluripotential Kähler-Ricci flow starting with ω 0 exists for t ∈ [0, T + ε) for some small ε > 0.

Proof. Let η be a smooth representative of the class {θ T+ε }, and set

χ = 1 T + ε (η -ω 0 ) ∈ c 1 (K X )
;

ω t = ω 0 + tχ = 1 T + ε ((T + ε -t)ω 0 + tη) ∈ {ω 0 } + tc 1 (K X ).
Fix a volume form dV on X with dd c log V = χ. Then the Kähler-Ricci flow can be written as the complex Monge-Ampère flow

(ω t + dd c φ t ) n = e φt dV, φ(0) = 0.
Since ω 0 is a Kähler form, there exists a small constant c ∈ (0, 1) such that ω 0 ≥ cη. Hence

ω t ≥ g(t)η for t ∈ [0, T ′ ], where g(t) = (T + ε) -1 (c(T + ε) + t(1 -c)
) is a positive increasing function. Theorem 3.3.1 can be applied (with F(t, x, r) ≡ 0, f = 1) and guarantees the existence of a pluripotential solution to the Monge-Ampère flow on X T+ε .

Using the same argument as above the pluripotential Kähler-Ricci flow can be continued as long as the class {ω 0 } + tc 1 (K X ) is big. If X has nonnegative Kodaira dimension, then c 1 (K X ) is pseudoeffective, and hence the class {ω 0 } + tc 1 (K X ) is big for any t > 0. In particular, the flow is volume non-collapsing at a finite time singularity, as emphasized by Collins-Tosatti (see [CT15, Proposition 4.2]). We thus obtain a longtime pluripotential solution :

Theorem 3.4.5. Let (X, ω 0 ) be a compact Kähler manifold with nonnegative Kodaira dimension. Then the pluripotential Kähler-Ricci flow starting with ω 0 exists for t ∈ [0, ∞).

Proof. Fix T < +∞. Let η be a smooth representative of the class {θ T }, and set

χ = 1 T (η -ω 0 ) ∈ c 1 (K X ); ω t = ω 0 + tχ = 1 T ((T -t)ω 0 + tη) ∈ {ω 0 } + tc 1 (K X ).
Fix a volume form dV on X with dd c log V = χ. Then the Kähler-Ricci flow can be written as the complex Monge-Ampère flow

(ω t + dd c φ t ) n = e φt dV, φ(0) = 0.
Since ω 0 is a Kähler form, there exists a small constant c ∈ (0, 1) such that ω 0 ≥ cη. Hence ω t ≥ g(t)η for t ∈ [0, T], where g(t) = T -1 (cT + t(1c)) is a positive increasing function. Again, by Theorem 3.3.1 there exists a pluripotential solution U = U φ 0 , f ,F,X T with φ 0 = 0, f = 1, F = 0. We next claim that U t has minimal singularities for each t ∈ (0, T). The proof of the claim is very similar to that of Theorem 3.3.6, but for completeness we provide the details below. We first observe that t ωt ≤ ω t for all t > 0, yielding that η t := t -1 ω t is decreasing in t. By [BEGZ10, Theorem 6.1], for each t ∈ (0, T) there exists a unique η t -psh function ϕ t with full Monge-Ampère mass such that (η t + dd c ϕ t ) n = e ϕ t dV. For 0 < s ≤ t we have

(η s + dd c ϕ t ) n ≥ (η t + dd c ϕ t ) = e ϕ t dV.
It follows that ϕ t is a subsolution to (η s + dd c ϕ s ) n = e ϕ s f dV, so the comparison principle (see e.g. [BEGZ10, Proposition 6.3]) ensures that ϕ t ≤ ϕ s for s ≤ t. Therefore the function t → ϕ t (x) is decreasing for all x ∈ X. As explained in Section 3.1.3, φt = ∂ t ϕ t is well-defined almost everywhere on X T . Set u(t, x) := tϕ t (x) + n(t log tt). We infer, for almost every (t, x) ∈ (0, T) × X,

u(t, x) = t φt (x) + ϕ t (x) + n log t ≤ ϕ t + n log t, hence (ω t + dd c u t ) n = e n log t+ϕ t dV ≥ e ut dV.
Moreover, since u(0, •) = 0 = φ 0 , we have that u is a subsolution to the Cauchy problem, i.e. u ∈ S φ 0 , f ,F (X T ) with φ 0 = 0, F ≡ 0. By [BEGZ10, Theorem 6.1] we have

ϕ t ≥ V η t -C(t) for some time-dependent constant C(t). Since V ω t = tV η t , we thus infer u t ≥ V ω t -C ′ (t), hence U t ≥ V ω t -C ′ (t)
for all t ∈ (0, T). This completes the proof of the claim.

If T ′ > T, then by the above arguments there exist a pluripotential solution U ′ = U φ 0 , f ,F,X ′ T of the flow. Both U and U ′ satisfy the assumptions in Theorem 3.3.7, hence U = U ′ on X T . We can thus glue all these solutions to get a longtime solution of the flow, finishing the proof.

Stable varieties

Log canonical pairs.

A pair (X, D) is by definition a complex normal compact projective variety carrying a Weil Q-divisor D (not necessary effective). We will say that the pair (X, D) is a log canonical (lc) pair if K X + D is Q-Cartier, and if for some (or equivalently any) log resolution π : X ′ → X, we have

K X ′ = π * (K X + D) + ∑ a i E i
where E i are either exceptional divisors or components of the strict transform of D, and the coefficients a i satisfy the inequality a i ≥ -1.

When D ≡ 0, we say that X has log canonical singularities.

Semi-log canonical singularities.

We give here a short overview of the notion of semi-log canonical singularities and stable varieties. We refer to the survey [Kov13, §5, 6] and the references therein for more details.

In the sequel, X will be a reduced and equidimensional scheme of finite type over C unless stated otherwise, and we set n := dim C X. In order to study the normalized Kähler-Ricci flow, one needs a canonical sheaf (or a canonical divisor). Let us stress that the dualizing sheaf, even if it exists, is not necessarily a line bundle (or a divisor).

We say that the scheme (variety) X is Cohen-Macaulay if for every x ∈ X the depth of O X,x , denoted by depth(O X,x ), is equal to its Krull dimension. If X is Cohen-Macaulay, then X admits a dualizing sheaf ω X .

We say that X is Gorenstein if X is Cohen-Macaulay (X admits a dualizing sheaf ω X ) and ω X is a line bundle. A scheme (variety) X is called G 1 if it is Gorenstein in codimension 1, which means that there exists an open subset U ⊂ X such that codim X (X\U) ≥ 2 and U is Gorenstein.

We say that X satisfies the S 2 condition of Serre if for all x ∈ X, we have depth(O X,x ) ≥ min{dim O X,x , 2}. This condition is equivalent to saying that for each closed subset ı : Z → X of codimension at least two, the natural map O X → ı * O X\Z is an isomorphism.

We now want to have an interpretation of ω X in terms of Weil divisor. If X satisfies the conditions G 1 and S 2 , and U is a Gorenstein open subset whose complement has codimension at least 2, we may define the "canonical=dualizing" sheaf ω U as the determinant of the cotangent bundle, i.e., the sheaf of top differential forms, ω U = det Ω U . One can then define the canonical sheaf ω X by ω X = ȷ * ω U where ȷ : U → X is the open embedding.

As U is non-singular, ω U is a line bundle, hence corresponds to a Cartier divisor. Let K U := ∑ a i K i be a Weil divisor associated to this Cartier divisor such that for all i, K i does not contain any component of X sing of codimension 1. Let Ki denote the closure of K i and

K X := ∑ a i Ki .
Since codim X (U) ≥ 2, this is the unique Weil divisor for which

K X | U = K U . We see that the divisorial sheaf O X (K X ) := { f ∈ K(X) : K X + div( f ) ≥ 0}
is reflexive, and coincides with ω U = ω X | U , hence the S 2 condition implies that

ω X ≃ O X (K X ).
Remark 3.4.6. The condition G 1 guarantees the existence of the canonical sheaf ω X , and the condition S 2 ensures its uniqueness. When X is projective, we know that it admits a dualizing sheaf, as it is reflexive, it coincides with ω X by the S 2 condition.

We let ω [m]

X denote the m-th reflexive power of the canonical sheaf ω X (defined by ω

[m]

X := (ω ⊗m X ) * * ). The same arguments above yield ω

[m] X ≃ O X (mK X ). Thus the Weil divisor K X is Q- Cartier if and only if ω X is a Q-line bundle, i.e., ω [m]
X is a line bundle for some m > 0. From now on we work with the canonical divisor K X instead of its associated canonical sheaf ω X .

We say that a closed point x ∈ X is double crossing if it is locally analytically isomorphic to the singularity {0 ∈ (z 0 z 1 = 0) ⊂ C n+1 }.

A scheme X is called demi-normal if it satisfies the S 2 condition and has only double crossing singularities in codimension 1. We now give the definition of semi-log canonical models : Definition 3.4.7. We say that X has semi-log canonical (slc) singularities if K X is Q-Cartier and there exist two Zariski open sets U, V such that

• X = U ∪ V,
• U is a normal variety with log canonical singularities,

• V has only double crossing points.

We mention that semi-log canonical models may not be normal varieties. Let µ : X n → X be a normalization of X. We emphasize again that X is not irreducible in general, so its normalization is defined to be the disjoint union of the normalization of its irreducible components. The conductor ideal sheaf

I C X := Ann O X (µ * O X n /O X )
is defined to be the largest ideal sheaf on X that is also an ideal sheaf on X n . If we consider the affine case where A n is the integral closure of some integral ring A, then one can see that the annihilator Ann A (A n /A) := {a ∈ A : aA n ⊂ A} is the largest ideal in A that is also an ideal in B.

For the case of schemes (varieties), we let I C X n denote the corresponding conductor ideal sheaf on X n , and we define the conductor subscheme as C X := Spec X (O X /I C X ) on X and C X n := Spec X n (O X n /I C X n ) on X n . If X is seminormal (i.e. every finite morphism X ′ → X, with X ′ is reduced, that is a bijection on points is an isomorphism) and S 2 , then one can show that these subschemes have pure codimension 1 hence they define Weil divisors which are moreover reduced (cf. [START_REF] Kovács | The canonical sheaf of Du Bois singularities[END_REF]4.5]).

If X is demi-normal and K X is Q-Cartier, then we have the following relation

µ * K X = K X n + C C n . (3.4.10)
Under the previous seminormality and S 2 assumptions, the G 1 condition is equivalent to the demi-normality. In other words, we may alternatively define slc models as follows :

Definition/Proposition 3.4.8. A scheme X has semi-log canonical singularities if and only if

• X is G 1 and S 2 , • K X is Q-Cartier (of index m), • The pair (X n , C X n ) is log-canonical.
Note that there are many schemes satisfying the S 2 condition and the seminormality but not demi-normality. For instance, a reduced scheme consisting of the three axes in A 3 does not have double crossings in codimension 1, but is both S 2 and seminormal.

We can finally give the definition of stable variety :

Definition 3.4.9. A projective variety X is called stable if • X has semi-log canonical singularities, • K X is an ample Q-Cartier divisor.
From Definition 3.4.7, we can see that X is a stable variety if K X is ample and X = U ∪ V, where U, V are Zariski open sets, U is a normal variety with log canonical singularities, and V has only double crossing singularities.

Convergence of NKRF on stable varieties

Let X be a complex projective variety with semi-log canonical singularities such that K X is ample (stable variety). We now consider the normalized Kähler-Ricci flow starting at any Kähler metric ω 0 on X, this is the evolution following equation :

∂θ t ∂t = -Ric(θ t ) -θ t , θ| t=0 = ω 0 . (3.4.11)
After passing to a suitable resolution of singularities, we may as well assume that X is smooth if we study the setting of log pairs (X, D), where D = ∑ N i=1 a i D i is the Q-divisor on X with simple normal crossing (snc), where the role of the canonical line bundle is played by the log canonical line bundle K X + D (which occurs as the pull-pack to the resolution of the original canonical line bundle). In this setting the original variety has semi-log canonical singularities precisely when the log pair (X, D) is log canonical (lc) in the usual sense of Minimal Model Program (MMP), i.e. the coefficients of D are at most equal to one (but negative coefficients allowed). Let us mention that even if the original canonical line bundle is ample, the corresponding log canonical line bundle is merely semi-ample and big on the resolution, since it is trivial along the exceptional divisors of the corresponding resolution. The initial data ω 0 may now assume to be a smooth semi-positive (1,1) form with big cohomology class.

Let X be a compact Kähler manifold and (X, D) be a log canonical pair such that K X + D is semi-ample and big (i.e., (K X + D) n > 0). We fix θ a smooth representative of the class c 1 (K X + D). It has been shown in [BG14, Theorem C] that there exists a unique closed positive current

ω KE = θ + dd c ψ KE in c 1 (K X + D) which is smooth on a Zariski open set U of X and satisfies Ric(ω KE ) = -ω KE + [D]
in the sense of currents on X. The current ω KE is called the singular Kähler-Einstein metric.

Our aim is to prove the existence and the convergence of the pluripotential solutions of the normalized Kähler-Ricci flow like the previous one, this is the content of the following theorem : Theorem 3.4.10. Let S 0 be a positive closed current with bounded potentials. Then the normalized Kähler-Ricci flow starting with S 0 admits a unique pluripotential solution defined on [0, +∞) × X. Furthermore, the pluripotential normalized Kähler-Ricci flow converges towards ω KE on Amp(K X + D), as t → +∞.

Observe that Theorem 3.4.10 implies Theorem D in the introduction : indeed, if Y is a projective variety with semi-log canonical singularities such that K Y is ample (stable variety) and π : (X, D) → Y is a log resolution of the normalization (endowed with its conductor), then the exceptional locus of π is contained in the complement of the ample locus of K X + D.

Remark 3.4.11. A similar result has been obtained in [CGLS19, Theorem 1.3] with a very different approach. These authors generalize a priori estimates of Song-Tian [START_REF] Song | The Kähler-Ricci flow through singularities[END_REF] to the case of Q-factorial projective varieties with log canonical singularities. They also show that if X is stable, then the normalized Kähler-Ricci flow (3.4.11) has a unique maximal weak solution on [0, +∞) which is smooth in (0, +∞) × X reg and converges to the singular Kähler-Einstein metric ω KE both in the sense of currents and in the C ∞ loc (X reg )-topology as t tends to infinity. Our approach allows one to treat more general equations, avoiding any projectivity assumption on the variety nor any integrality on the initial cohomology class, and applies to big classes for which no smooth deformation is available.

Proof of Theorem 3.4.10. By definition, D = ∑ N i=1 a i D i is a simple normal crossings R-divisor with a i ∈ (-∞, 1] and defining section s i . The normalized Kähler-Ricci flow (3.4.11) can be written as the following complex Monge-Ampère flow

(ω t + dd c φ t ) n = e φt +φ t dµ, (3.4.12)
where ω t := e -t ω 0 + (1e -t )θ, and dµ is a measure on X which is of the form

dµ = dV X ∏ N i=1 |s i | 2a i = f dV X
where s i are non-zero sections of O X (D i ), | • | i are smooth hermitian metrics on O X (D i ), and dV X is a smooth volume form on X. We let D lc := ∪ a k =1 D k denote the "non-klt" locus.

Step 1 : constructing a subsolution. We let Ω denote the ample locus of the class {θ}. Since the latter is big, there exists a θ-psh function χ 0 such that θ := θ + dd c χ 0 ≥ δω X on Ω for some δ > 0 and χ 0 → -∞ near ∂Ω.

Up to multiplying by a positive constant we can assume that |s i | 2 ≤ 1/e so thatlog(|s i | 2 ) ≥ 1 out of D i . Note also that dd c (log(|s i | 2 )) extends as a smooth real (1, 1)-form on X whose cohomology class is 2πc 1 (D). We compute

-dd c (log(λ -log(|s i | 2 ))) = - dd c (λ -log(|s i | 2 )) λ -log(|s i | 2 ) + ds i ∧ d c s i |s i | 2 (λ -log(|s i | 2 )) 2 .
The second term is a semipositive (1, 1)-form. Sincelog(|s i | 2 ) goes to ∞ near D j , we infer that θdd c log(λlog |s i | 2 ) is positive on Ω\D j when λ is big enough. Replacing θ by 1 N θ and increasing

λ if necessary one has 1 N θ -dd c log(λ -log |s i | 2 ) > 0, hence ∑( 1 N θ -dd c log(λ -|s i | 2
)) defines a Kähler form on Ω\D. Then for suitable positive constants λ, A the function

v := -2 N ∑ i=1 log(λ -log |s i | 2 ) + χ 0 -A (3.4.13)
is a subsolution of the complex Monge-Ampère equation :

(θ + dd c ψ KE ) n = e ψ KE dV X ∏ i |s i | 2a i . (3.4.14)
By the arguments above, we get the lower bound (see also [START_REF] Berman | Kähler-Einstein metrics on stable varieties and log canonical pairs[END_REF]5.5.2]) :

ψ KE ≥ χ 0 -∑ a k =1 log(-log |s k | 2 ) -A
for some uniform constant A > 0. Here the hermitian metrics | • | k are chosen conveniently. Since θ is semi-positive we see that for all t, ω t ≥ cθ for some c > 0 small enough. For simplicity, we may assume that c = 1. We can check that

u(t, x) := ψ KE (x) -C 0 e -t (3.4.15)
is a subsolution to (3.4.12), for a constant C 0 > 0 so large that u(0, •) ≤ φ 0 .

Step 2 : the approximating flows. We now establish the existence of the flow (3.4.12) by an approximation argument using ideas from [DL15, Theorem 4.5]. Fix T < +∞. The difficulty is that the density f = Π i |s i | -2a i is not in L p , p > 1, (not even in L 1 ) since some of the coefficients a j might be equal to 1. For each j ∈ N, Theorems A and B provide a unique φ t,j ∈ P (X T , ω) such that (ω t + dd c φ t,j ) n = e φt,j +φ t,j min( f , j)dV X , φ 0,j = φ 0 .

(3.4.16)

Since ω t is the pull-back of a smooth family of Kähler forms, we have

-Aω t ≤ ωt ≤ Aω t ,
for a uniform constant A > 0. We can proceed as in the proof of Theorem 3.2.7 and Theorem 3.2.13 to establish the following uniform bounds : for each T ∈ (0, +∞) and any compact K ⊂ Ω\D lc , there is a constant C(T, K) such that

t|∂ t φ t,j (x)| ≤ C(T, K), and t 2 ∂ 2 t φ t,j (x) ≤ C(T, K), ∀ (t, x) ∈ (0, T) × K.
Indeed, on (0, T) the forms ω t satisfy ω t ≥ g(t)θ, where g(t) = c 0 > 0 is a constant. The function F in our case is defined by r → F(t, x, r) ≡ r which satisfies the assumptions in the introduction. More precisely, we have the following : Proposition 3.4.12. Let J = [a, b] be a compact interval of (0, T). There exist uniform constants C 0 , C 1 , C 2 > 0 such that for all j ∈ N, t ∈ J,

(1) C 0 ≥ φ t,j (x) ≥ ψ KE (x) -C 0 e -t , (2) |∂ t φ t,j | ≤ C 1 + ∑ a k =1 log(-log |s k | 2 ) -χ 0 , (3) ∂ 2 t φ t,j ≤ C 2 + ∑ a k =1 log(-log |s k | 2 ) -χ 0 .
Proof. We first prove (1). For the lower bound, we can check that the function u in (3.4.15) is also a subsolution to (3.4.16). For the upper bound, we pick C > 0 so big that ω n t ≤ e C f dV for all t ∈ [0, T]. The domination principle (see e.g. [BEGZ10, Corollary 2.5]) yields φ t,j ≤ C holds everywhere for all t, j.

We next prove (2).

Fix ε 0 > 0 such that (1 + ε 0 )b < T. For all t ∈ J and s ∈ (1 -ε 0 , 1 + ε 0 ) there exists a constant A 1 > 0 such that ω t ≥ (1 -A 1 |s -1|)ω ts .
(3.4.17)

For s small enough we set

λ s := |1 -s| s , α s := s(1 -λ s )(1 -A 1 |s -1|) ∈ (0, 1), (3.4.18) hence γ s := λ s /(1 -α s ) ≥ ε 1 > 0.
Shrinking ε 1 we may assume that γ s ω t ≥ ε 1 θ. Let v 1 be a solution to the following equation

(ε 1 θ + dd c v 1 ) n = e v 1 f dV X . (3.4.19)
The same argument in the Step 1 yields

v 1 ≥ ε 1 χ 0 -ε 1 ∑ a k =1 log(-log |s k | 2 ) -A,
for some uniform constant A > 0. For any (t, x) ∈ J × X we set

u s (t, x) := α s s φ j (ts, x) + (1 -α s )v 1 (x) -C|s -1|e -t ,
for C > 0 to be chosen later. We have

(ω t + dd c u s (t, •)) n = (1 -λ s )ω t + α s s dd c φ ts + (1 -λ s )ω t + (1 -α s )dd c v 1 n ≥ [α s (ω ts + dd c φ ts,j ) + (1 -α s )(γ s ω t + dd c v 1 )] n ≥ e α s (∂ t φ j (ts,•)+φ j (ts,•))+(1-α s )v 1 min( f , j)dV = e ∂ t u s (t,•)+u s (t,•) min( f , j)dV
where we use (3.4.17) in the second line and Lemma 3.2.6 in the third one. Therefore u s is a subsolution to (3.4.16). Since φ 0 is bounded we can choose C > so large that u s (0, •) ≤ φ 0 on X.

Hence the comparison principle (Proposition 3.3.4) ensures that for any j, u s ≤ φ j in J × X, i.e.,

α s s φ j (ts, x) + (1 -α s )v 1 -C|s -1|e -t ≤ φ j (t, x), ∀ (t, x) ∈ J × X.
Letting s → 1 we infer for all (t, x) ∈ J × X,

|∂ t φ j (t, x)| ≤ C 1 -C 1 v 1 (x),
for a uniform constant C 1 > 0.

To prove (3) we argue as above. Set for any (t, x)

∈ J × X, v s (t, x) := α s s -1 φ j (ts, x) + sφ j (ts -1 , x) 2 + (1 -α s )v 1 (x) -C|s -1|e -t ,
for a constant C > 0 so large that v s (0, •) ≤ φ 0 . We can check as above that v s is a subsolution to (3.4.16). By the same arguments we can obtain the estimate (3).

We now finish the proof of Step 2. For t ∈ (0, T) fixed, φ t,j is decreasing as j → ∞ by the comparison principle (Proposition 3.3.4). It follows from Proposition 3.4.12 that

φ t,j (x) ≥ ψ KE (x) -C 0 , ∀ (t, x) ∈ [0, T) × X, for a large constant C 0 > 0. It has been shown in [BEGZ10, Theorem 4.2] that ψ KE ∈ E (X, ω t ) for each t since 0 ≤ θ ≤ ω t , hence φ t,j ∈ E (X, ω t ).
We want to prove that lim j φ t,j = φ t is a solution to the flow (3.4.12).

Fix a compact sub-interval J ⋐ (0, T), a compact subset K ⊂ (Ω \ D lc ). Proposition 3.4.12 implies that there exists a constant C = C J > 0 such that the function t → φ j (t, x) -Ct 2 is concave in J, for all x ∈ K. Moreover, the function x → φ j (t, x) is ω t -psh and uniformly bounded on K for all j. We obtain the same properties for the limiting function φ(t, x) by letting j → +∞. It follows from Proposition 3.1.10 that φj , φ are well-defined and lim j φj (t,

•) = φ(t, •). Consider G := {x ∈ X : f (x) > M} ∪ {x ∈ X : -v 1 (x) > M},
where v 1 is a solution to (3.4.19). Since -v 1 is locally bounded outside a divisor, we can choose M > 0 so large that G has small Monge-Ampère capacity Cap Θ (G) < ε for some Kähler form Θ and for any ε > 0. Hence for all t ∈ J, j ∈ N, we have that φt,j is uniformly bounded from above on X\G. Therefore Lebesgue dominated convergence theorem ensures that lim j→+∞ ˆJ ˆX\G e φt,j +φ t,j min( f , j)dVdt = ˆJ ˆX\G e φt +φ t f dVdt.

Using the notations from [DL15, Section 2], it follows from [DL15, Theorem 2.9] that, for all t ∈ (0, T), j ∈ N,

ˆG(ω t + dd c φ t,j ) n ≤ Cap ψ KE -C 0 ,0 (G) ≤ h(ε), for some continuous function h : [0, +∞) → [0, +∞) with h(0) = 0. Hence ˆJ ˆX e φt +φ t f dVdt ≥ ˆJ ˆX\G e φt +φ t f dVdt = lim j→+∞ ˆJ ˆX\G (ω t + dd c φ t,j ) n = lim j→∞ ˆT 0 ˆX(ω t + dd c φ t,j ) n -lim j→+∞ ˆJ ˆG(ω t + dd c φ t,j ) n ≥ ˆJ ˆX ω n t -Th(ε).
Letting ε → 0 we obtain ´J ´X e φt +φ t f dVdt ≥ ´J ´X ω n t . On the other hand, since (ω t + dd c φ t,j ) n converges to (ω t + dd c φ t ) n , Fatou's lemma yields dt ∧ (ω t + dd c φ t ) n ≥ e φt +φ t f dVdt in the sense of measures in (0, T) × X, whence equality. This implies that φ is a solution to (3.4.12). Proposition 3.4.13. For each t, the solution φ t of (3.4.12) is continuous on Ω \ D lc .

Proof. It follows from Proposition 3.4.12 that

e φt +φ t f ≤ exp C + ∑ a k =1 log(-log |s k | 2 ) -χ 0 -∑ i log(|s i | 2 ) .
The proof thus follows from [Dan21, Theorem 3.2].

Step 3 : convergence at time zero. Using similar arguments as in the proof of Theorem 3.2.8, we are going to check that the solution φ t of the equation (3.4.12) converges pointwise towards φ 0 as t → 0 + . Arguing as at the beginning of the proof of Theorem 3.4.1, we can check that

φ(t, x) ≥ u(t, x) := e -t φ 0 (x) + (1 -e -t )ψ KE (x) + h(t), ∀ (t, x) ∈ (0 + ∞) × X,
where ψ KE is the solution of (3.4.14) and h(t) = ne -t (e t -1) log(e t -1)te t . It thus remains to show that for all x ∈ X, lim t→0 φ t (x) ≤ φ 0 (x). Fix T < +∞ and consider

G := {x ∈ X : u(T, x) > -M}, where M > 0 is a constant such that µ(G) > µ(X)/2 (recall that ψ K E is smooth outside a divisor). Observe that φ(t, x) ≥ u(t, x) ≥ u(T, x) > -M for all x ∈ G, t ∈ (0, T).
Following the proof of Theorem 3.2.8, we obtain as in (3.2.11) that

ˆG φ t dµ ≤ ˆG φ 0 dµ + Ct, (3.4.20)
for a constant C > 0 depending on G.

Let now u 0 ∈ PSH(X, ω 0 ) be any cluster point of φ t as t → 0. We can assume that φ t converges to u 0 in L q (X, dV) for any q > 1. On the other hand, dµ = ∏ i |s i | -2a i dV X has density f = ∏ i |s i | -2a i ∈ L p loc (X\D) for any p > 1. Hence, φ t f converges to u 0 f in L 1 (K) for any compact subset K of X\D. Thus, the claim above ensures that ˆG u 0 f dV ≤ ˆG φ 0 f dV.

We infer that u 0 ≤ φ 0 almost everywhere on G with respect to f dV, hence everywhere on G.

Letting M → +∞, we conclude that lim sup t→0 φ t = φ 0 on X\D, hence on the whole X.

Step 4 : uniqueness of the flow. By the previous steps, we have shown that there exists a solution φ to (3.4.12) with initial data φ 0 . This function satisfies the following properties :

• φ is locally uniformly semi-concave in t,

• (t, x) → φ(t, x) is continuous on (0, +∞) × U, where U := Ω \ D lc ,
• φ t → φ 0 pointwise as t → 0 + .

We are going to show that such a solution is unique. Let Φ be a solution to (3.4.12) with the same properties as above. We shall prove that φ ≤ Φ on [0, +∞) × X, whence equality. The proof follows step by step from the uniqueness result obtained in Section 3.3.2.

Step 4.1. Assume moreover that :

1. φ is C 1 in t, 2. Φ is continuous on [0, +∞) × U.
Since θ is semi-positive we fix c > 0 such that ω t ≥ cθ for all t. For simplicity we again assume that c = 1. Let χ be a θ/2-psh function with analytic singularities such that χ is smooth in U, χ = -∞ on ∂U, and sup X χ = 0. We will use this function in order to apply the classical maximum principle in U. The standard strategy is to replace φ by (1λ)φ + λχ. Nevertheless, the time derivative φt may blow up as t → 0 so we need another auxiliary function. Let ρ ∈ PSH(X, θ/2) be the unique solution to

(θ/2 + dd c ρ) n = e ρ dµ, (3.4.21)
normalized by sup X ρ = 0, where dµ = ∏ i | f i | -2a i dV. It follows from [Dan21, Corollary 3.5] that ρ is continuous in U. Fix 0 < T < +∞. For ε, λ > 0 small enough we set

w(t, x) := (1 -λ)φ(t, x) + λ(ρ(x) + χ(x)) -Φ(t, x) -3εt, ∀ (t, x) ∈ (0, T) × X.
By Lemma 3.1.5, this function is upper semi-continuous on [0, T] × U. Since ρ + χ is a θ-psh function which is continuous in U and tends to -∞ on ∂U, the function w attains its maximum at some point (t 0 , x 0 ) ∈ [0, T] × U.

We want to show that w(t 0 , x 0 ) ≤ 0. Assume by contradiction that it is not the case i.e., w(t 0 , x 0 ) > 0 with t 0 > 0. The set

K := {x ∈ U : w(t 0 , x) = w(t 0 , x 0 )} is a compact subset of U since w(t 0 , x) tends to -∞ as x → ∂U. The classical maximum principle ensures that for all x ∈ K, (1 -λ)∂ t φ(t 0 , x) ≥ ∂ - t Φ(t 0 , x) + 3ε. The partial derivative ∂ t φ(t, x
) is continuous on U by assumption. Since the function t → Φ(t, x) is locally uniformly semi-concave, for any t ∈ (0, T), the left derivative ∂ - t Φ(t, •) is upper semicontinuous in Ω (see Proposition 3.1.10). We can thus find η > 0 small enough that, by introducing the open set containing K,

D := {x ∈ U : w(t 0 , x) > w(t 0 , x 0 ) -η} ⋐ U.
We have for all

x ∈ D (1 -λ)∂ t φ(t 0 , x) > ∂ - t Φ(t 0 , x) + 2ε. (3.4.22)
Proof. Fix ε > 0, and consider

w ε (t, x) = e -t Φ ε + (1 -e -t )ψ K E + h(t).
A direct computation shows that

(ω t+ε + dd c w ε ) n = e -t (ω ε + dd c Φ ε ) + (1 -e -t )(θ + dd c ψ KE ) n ≥ e log(1-e -t )+ψ KE µ.
where we have used

ω ε + dd c Φ ε ≥ 0. Since h ′ (t) + h(t) = n log(1 -e -t ) we have (ω t+ε + dd c w ε ) n ≥ e ∂ t w ε +w ε µ.
It is also clear from the definition that w ε (t, •) converges in L 1 (X) to w ε (0, •) = Φ ε as t → 0 + . On the other hand, w ε is C 1 in t and Φ t+ε is continuous on [0, +∞) × U. We can thus apply Step 4.1 to obtain w ε (t, x) ≤ Φ(t + ε, x). The proof follows by letting ε → 0.

Step 4.3. We are now ready to treat the general case by removing the extra assumption on φ.

For s > 0 near 1 we set, for any (t,

x) ∈ (0, T) × X V s (t, x) := α s s φ(ts, x) + (1 -α s )v 1 (x) -C|s -1|e -t ,
where α s is defined as in (3.4.18), and v 1 is a solution to (3.4.19). For C > 0 large enough, the proof of Proposition 3.4.12 ensures that V s is a subsolution to (3.4.12) that satisfies V s (0, •) ≤ φ 0 on X.

Let {η ε } ε>0 be a family of smoothing kernels in R approximating the Dirac mass δ 0 . For ε > 0 small enough we define

φ ε (t, x) := ˆR V s (t, x)η ε (s -1)ds
We proceed as in the proof of Theorem 3.3.7 to show that φ ε -O(ε) is again a subsolution and apply the previous step to conclude.

Step 5 : the long-term behavior of the flow. It remains to establish the convergence at t = +∞.

We have seen that u(t, x) := e -t φ 0 + (1

-e -t )ψ KE (x) + h(t)
is a subsolution to (3.4.12). The comparison principle (see Step 4) yields for any t > 0, x ∈ X,

ψ KE (x) -C(t + 1)e -t ≤ u(t, x) ≤ φ(t, x)
for some uniform constant C > 0.

For the upper bound, since θ = θ + dd c χ 0 is a Kähler current we can fix a constant A > 0 such that

ω 0 ≤ (1 + A) θ on Ω, thus ω t ≤ (1 + Ae -t ) θ for all t. Set v(t, x) := (1 + Ae -t )ψ KE (x) + Be -t
where B is chosen so that v 0 ≥ φ 0 . Thus the function v is a supersolution to the Cauchy problem for the parabolic equation

((1 + Ae -t ) θ + dd c v t ) n ≤ e vt +v t +nAe -t
with initial data φ 0 , while w(t, x) := φ(t, x) -nAe -t is a subsolution to this equation since

((1 + Ae -t ) θ + dd c w t ) n ≥ (ω t + dd c φ t ) n = e φt +φ t f dV = e ẇt +w t +nAe -t f dV.
The comparison principle thus yields

φ(t, x) ≤ (1 + Ae -t )ψ KE (x) + C ′ e -t ,
as desired.

Introduction

The Chern-Ricci flow is an evolution equation of Hermitian metrics by their Chern-Ricci forms. It was first investigated by M. Gill [START_REF] Gill | Convergence of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds[END_REF] in the setting of complex manifolds with vanishing first Bott-Chern class. In [TW13, TW15, TWY15, Zhe17] the Chern-Ricci flow was studied on more general complex manifolds and a number of further results were established, several of which are analogous to those for the Kähler-Ricci flow. These results provide affirmative evidence that the Chern-Ricci flow is a natural geometric flow on complex manifolds and that its behavior reflects the underlying geometry. Alternative flows of Hermitian metrics have been previously studied by Streets-Tian [ST10, ST11a, ST13], and also Liu-Yang [START_REF] Liu | Geometry of Hermitian manifolds[END_REF], motivated in part by open classification problems. Following Song-Tian's program [START_REF] Song | The Kähler-Ricci flow through singularities[END_REF], our aim in this chapter is to establish the existence of the (weak) Chern-Ricci flow on mildly singular varieties.

Let (X, ω 0 ) be a compact n-dimensional Hermitian manifold. The Chern-Ricci flow ω = ω(t) starting at ω 0 is an evolution equation of metrics

∂ ∂t ω = -Ric(ω), ω| t=0 = ω 0 , (4.0.1)
where Ric(ω) is the Chern-Ricci form of ω associated to the Hermitian metric g = (g i j), which in local coordinates is given by Ric(ω) = -dd c log det(g). In the Kähler setting, Ric(ω) = iR j kdz j ∧ d zk , where R j k is the Ricci curvature of ω. Thus if ω 0 is Kähler, i.e., dω 0 = 0, (4.0.1) coincides with the Kähler-Ricci flow. Tosatti and Weinkove [TW15, Theorem 1.3] showed that there exists a unique maximal solution to (4.0.1) on [0, T) for a number T ∈ (0, ∞] determined by ω 0 .

Solving the Chern-Ricci flow boils down to solving a parabolic scalar equation modeled on

∂φ ∂t = log det(g i j + ∂ i ∂ j φ) det(g i j) , g i j + ∂ i ∂ j φ > 0 with initial data φ(0, x) = 0.

Parabolic pluripotential method

Recently, Guedj-Lu-Zeriahi [START_REF] Guedj | The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows[END_REF][START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF] have developed the first steps of a parabolic pluripotential approach both in the local and Kähler cases. This enabled them to study the behavior of the Kähler-Ricci flow on compact Kähler varieties with Kawamata log terminal (klt) singularities, extending works by Cao [START_REF] Cao | Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds[END_REF], Tsuji [START_REF] Tsuji | Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type[END_REF], Tian-Zhang [START_REF] Tian | On the Kähler-Ricci flow on projective manifolds of general type[END_REF], Song-Tian [START_REF] Song | The Kähler-Ricci flow through singularities[END_REF], and Eyssidieux-Guedj-Zeriahi [START_REF] Eyssidieux | Weak solutions to degenerate complex Monge-Ampère flows II[END_REF][START_REF] Eyssidieux | Convergence of weak Kähler-Ricci flows on minimal models of positive Kodaira dimension[END_REF].

We remark that any compact complex manifold admits a Hermitian metric but there are many of them which are not Kähler. It is thus desirable to extend the previous works [START_REF] Guedj | The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows[END_REF][START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF][START_REF] Dang | Pluripotential Monge-Ampère flows in big cohomology classes[END_REF] to the Hermitian setting. It is expected that Hermitian analogues of the Kähler-Ricci flow will play an important role in understanding the geometry of compact complex manifolds.

More precisely, we consider the following parabolic complex Monge-Ampère type equation

dt ∧ (ω t + dd c φ t ) n = e ∂ t φ t +F(t,x,φ t ) f (x)dV(x) ∧ dt (CMAF)
in X T := (0, T) × X, where dV is a fixed normalized volume form on X and

• T ∈ (0, ∞), X denotes a compact Hermitian manifold ;

• (ω t ) t∈[0,T] is a smooth family of semi-positive forms such that

-Aω t ≤ ωt ≤ Aω t and ωt ≤ Aω t , ∀ t ∈ [0, T], (4.0.2)
for some fixed constant A > 0 ;

• for all t ∈ [0, T], θ ≤ ω t for some semi-positive and big (1, 1)-form θ, i.e., there exists a quasi-plurisubharmonic function ρ with analytic singularities such that θ + dd c ρ dominates a Hermitian form ;

• (t, x, r) → F(t, x, r) is continuous on [0, T] × X × R, quasi-increasing in r, uniformly Lipschitz in (t, r), and uniformly convex in (t, r) ;

• 0 ≤ f ∈ L p (X) for some p > 1, and f > 0 almost everywhere ;

• φ : [0, T] × X → R is the unknown function, with φ t := φ(t, •).

Here d = ∂ + ∂ and d c = i( ∂ -∂)/2 are both real operators, so that dd c = i∂ ∂.

Our plan is to extend the pluripotential approach developed by Guedj-Lu-Zeriahi [START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF] to the Hermitian context. We are going to introduce a notion of pluripotential solutions to (CMAF), a parabolic analogue of the theory developed by Bedford and Taylor [START_REF] Bedford | The Dirichlet problem for a complex Monge-Ampère equation[END_REF][START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF]. The local side of this theory has been developed in [START_REF] Guedj | The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows[END_REF] by a direct approach, taking advantage of the euclidean structure of C n .

The above parabolic equation can be interpreted as a second-order PDE on the (2n + 1)dimensional manifold X T :

• the left-hand side dt ∧ (ω t + dd c φ t ) n is a well-defined positive Radon measure for the path t → φ t of bounded ω t -plurisubharmonic functions ;

• the right-hand side e φt +F(t,x,φ) f (x)dV(x) ∧ dt is a well-defined positive Radon measure if t → φ t (x) is (locally) uniformly Lipschitz.

It is also useful in practice to allow the Lipschitz constant to blow up as t approaches zero, so we introduce the corresponding class P (X T , ω) of parabolic potentials (see Definition 4.1.6)

We approximate the equation (CMAF) by smooth parabolic complex Monge-Ampère ones and establish various a priori estimates to prove our first main result : Theorem A. Let φ 0 be a bounded ω 0 -psh function. Then there exists a parabolic potential φ ∈ P (X T , ω) to (CMAF) such that

• (t, x) → φ(t, x) is locally bounded in [0, T) × X, • (t, x) → φ(t, x) is continuous in (0, T) × (X \ {ρ = -∞}),
• t → φ t is locally uniformly semi-concave in (0, T) × X,

• φ t → φ 0 as t → 0 + in L 1 (X) and pointwise.

It turns out that t → φ t (x)n(t log tt) + Ct is increasing for some fixed C > 0. The convergence at time zero is therefore rather strong. For instance, it holds in the sense of capacity, and is even uniform if φ 0 is continuous.

The semi-concavity property of the solution φ constructed in Theorem A is a key ingredient for the approximation process (see [GLZ20, Theorem 1.14]). We prove that it is the unique pluripotential solution with such regularity by establishing the following comparison principle : Theorem B. Let φ ∈ P (X T , ω) (resp. ψ) be a bounded pluripotential subsolution (resp. supersolution) to (CMAF) with initial data φ 0 (resp. ψ 0 ). Assume that φ is continuous in (0, T) × (X \ {ρ = -∞}) and ψ is locally uniformly semi-concave in t. Then

φ 0 ≤ ψ 0 =⇒ φ ≤ ψ.
In particular, there is a unique pluripotential solution to (CMAF) which is continuous in (0, T) × (X \ {ρ = -∞}) and locally uniformly semi-concave in t.

We let Φ(ω t , F, f , φ 0 ) denote the unique solution to (CMAF) with given data (ω t , F, f , φ 0 ) as in Theorem B. This comparison principle also allows us to establish the following stability result :

Theorem C. Assume that (ω t , F, f , φ 0 ) and (ω t,j , F j , f j , φ 0,j ) satisfy the assumptions above with uniform constants independently of j, and

• ω t,j are smooth Hermitian forms converging uniformly to ω t ,

• F j uniformly converge to F with uniform constants,

• f j are densities which converge in L p (X) to f ,

• (φ 0,j ) is a sequence of (smooth) bounded ω 0 -psh functions converging in L 1 (X) towards φ 0 ∈ PSH(X, ω 0 ) ∩ L ∞ (X).

Then Φ(ω t,j , F j , f j , φ 0,j ) locally uniformly converge to Φ(ω t , F, f , φ 0 ).

We then move on to study higher regularity properties of such solutions under some extra assumptions. In this context we prove the following : Theorem D. Given the initial data ω t , F, f , φ 0 as above, assume moreover F to be smooth in [0, T] × X × R. Assume also that f = e ψ + -ψ -with • ψ ± are quasi-psh functions on X,

• sup X ψ ± ≤ C, and ∥e -ψ -∥ L p ≤ C for some constant C > 0,

• ψ ± are (smooth) locally bounded in a Zariski open subset U of Ω.

The unique pluripotential solution Φ(ω t , F, f , φ 0 ) to (CMAF) is smooth on (0, T) × U.

This result can be seen as a generalization of the main result of [START_REF] Tosatti | On the evolution of a Hermitian metric by its Chern-Ricci form[END_REF][START_REF] Tô | Regularizing properties of complex Monge-Ampère flows II : Hermitian manifolds[END_REF]. It encompasses the case of smooth parabolic Monge-Ampère equations on mildly singular compact hermitian varieties, as well as more degenerate settings, hermitian analogues of the main results of [START_REF] Song | The Kähler-Ricci flow through singularities[END_REF][START_REF] Boucksom | Regularizing properties of the Kähler-Ricci flow[END_REF].

We finally apply our results to study the Chern-Ricci flow on mildly singular varieties. We can, in particular, define a good notion of weak Chern-Ricci flow on varieties with log terminal singularities (and more generally on klt pairs) ; see Section 4.5.1 for a precise definition. Our main results extend to this context as follows :

Theorem E. Let Y be a compact complex variety with log terminal singularities. Assume that θ 0 is a Hermitian metric such that

T max := sup{t > 0 : ∃ χ ∈ C ∞ (Y) such that θ 0 -tRic(θ 0 ) + dd c χ > 0} > 0.
If S 0 = θ 0 + dd c ϕ 0 is a positive (1,1)-current with bounded potential ϕ 0 , then there exists a unique solution ω of the weak Chern-Ricci flow (4.0.1) starting with S 0 for t ∈ [0, T max ).

The definition of the weak Chern-Ricci flow is given in Section 4.5.1. Theorem E shows that we can start the Chern-Ricci flow from a positive current with bounded potentials. The weak Chern-Ricci flow smoothes out the initial current in the sense that the flow becomes smooth on the nonsingular part of Y once t > 0 and the evolving metrics always admit bounded local potentials for any t ∈ [0, T max ). In particular, the smoothing property of the Chern-Ricci flow holds when Y is a compact complex manifold (see Section 4.4 or [START_REF] Tosatti | On the evolution of a Hermitian metric by its Chern-Ricci form[END_REF][START_REF] Tô | Regularizing properties of complex Monge-Ampère flows II : Hermitian manifolds[END_REF]).

Organization of the chapter

In Section 4.1.2 we construct the class of parabolic potentials and define parabolic complex Monge-Ampère operators. We establish a priori estimates in Section 4.2, which will be used to prove Theorem A in Section 4.3.1. While, Theorem B and Theorem C will be proved in Section 4.3.3, by establishing uniqueness and stability of pluripotential solutions with time regularity. In Section 4.4, we establish the smoothness of the pluripotential solutions, constructed in the previous sections, outside an analytic subvariety under some extra assumptions, proving Theorem D. In Section 4.5 we apply these tools to prove the existence and uniqueness for the weak Chern-Ricci flow on compact complex varieties with log terminal singularities, proving Theorem E.

Preliminaries

Throughout this article, we let X denote a compact complex manifold of complex dimension n ≥ 1. We always denote by ω X a Hermitian metric on X.

Recap on elliptic pluripotential theory

Big forms

We fix θ a smooth semi-positive (1,1) form on X. Recall that a function is said to be quasiplurisubharmonic function (quasi-psh for short) if it is locally given as the sum of a smooth and a plurisubharmonic function. 2 ( ∂ -∂) are both real operators, so that dd c = i∂ ∂. We let PSH(X, θ) denote the set of all θ-psh functions which are not identically -∞.

In this chapter , we consider the equivalence relation of real (1,1) forms (or currents) on X :

θ ∼ θ ′ ⇐⇒ θ = θ ′ + dd c χ for some function χ ∈ C ∞ (X).
We denote by {θ} the equivalence class of θ. Definition 4.1.2. We say that a smooth real (1,1) form θ is big if its equivalence class contains a positive Hermitian current ; i.e., there exists a positive (1,1) current T ∈ {θ} such that T ≥ δω X for some small constant δ > 0.

It follows from an approximation result of Demailly [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF] that one can weakly approximate a Hermitian current by the ones with analytic singularities.

A basic example the we are going to consider in Section 4.5 is the following : if Y is a compact complex space endowed with a Hermitian form ω Y , and π : X → Y is a resolution of singularities, then θ = π * ω Y is big as follows from classical arguments ; see e.g. [FT09, Proposition 3.2]. Moreover, we can find a θ-psh function ρ with analytic singularities such that θ + dd c ρ ≥ δω X , and {ρ > -∞} = X \ Exc(π) = π -1 (Y reg ) ≃ Y reg .

Monge-Ampère operators.

Throughout the article, we let θ denote a smooth real semi-positive and big (1,1) form. An adaptation of [START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF] to the Hermitian context allows us to define the complex Monge-Ampère operator (θ + dd c u) n for any θ-psh function u which is bounded. We refer the reader to [DK12, KN15, GL21] for more details.

The mixed Monge-Ampère operators (θ 1 + dd c u 1 ) j ∧ (θ 2 + dd c u 2 ) n-j are also well-defined for any 0 ≤ j ≤ n, and any bounded θ i -psh functions u i , for i = 1, 2. We recall the following mixed type inequality which will be used in the sequel.

Lemma 4.1.3. Let θ 1 , θ 2 be a semi-positive (1,1)-forms. Let u 1 ∈ PSH(X, θ 1 ) ∩ L ∞ (X) and u 2 ∈ PSH(X, θ 2 ) ∩ L ∞ (X) be such that (θ 1 + dd c u 1 ) n ≥ e f 1 µ, (θ 2 + dd c u 2 ) n ≥ e f 2 µ,
where f , f 2 are bounded measurable functions and µ is a positive Radon measure with L 1 density with respect to Lebesgue measure. Then for any δ ∈ (0, 1),

(δ(θ 1 + dd c u 1 ) + (1 -δ)(θ 2 + dd c u 2 )) n ≥ e δ f 1 +(1-δ) f 2 µ.
Proof. See [Ngu16, Lemma 1.9].

Minimum principle

We will also need the following version of the minimum principle inspired by the one in the local setting ; see [START_REF] Bedford | The Dirichlet problem for a complex Monge-Ampère equation[END_REF]Theorem A]. This somehow generalizes the one established by Kolodziej-Nguyen [KN19, Proposition 2.5], assuming that the form is merely big.

Since θ is big we can find a θ-psh function ρ with analytic singularities such that θ + dd c ρ ≥ δω X for some small δ > 0. We set Ω := {ρ > -∞}.

u ∈ PSH(X, θ) ∩ L ∞ loc (Ω) satisfying θ + dd c u ≥ ε 0 ω X for some ε 0 > 0. Let v ∈ PSH(X, θ) ∩ L ∞ (X) ∩ C 0 (Ω) be such that (θ + dd c v) n ≤ c(θ + dd c u) n on D, (4.1.1) for some c ∈ [0, 1). Then min D (v -u) = min ∂D (v -u).
The proof is the same as that of [KN19, Propostion 2.5], but we shall apply the "comparison principle" established in [START_REF] Guedj | Quasi-plurisubharmonic envelopes 3 : Solving Monge-Ampère equations on hermitian manifolds[END_REF]Theorem 1.11].

Proof. Without loss of generality we assume that min ∂D (vu) = 0, i.e., u ≤ v on ∂D.

We need to show that min D (vu) = 0. Assume by contradiction that m := min D (vu) < 0. We set

ϕ = v, on D, max(u, v), on X \ D.
Observe that inf X (ϕu) ≤ m < 0, hence the set

U(s) := {ϕ < u + inf X (ϕ -u) + s} ⊂ {ϕ < u} ⊂⊂ D is a nonempty open set for s > 0 small enough. It follows from [GL21, Theorem 1.11] that 0 < (1 -Bs) n ˆU(s) (θ + dd c u) n ≤ ˆU(s) (θ + dd c v) n ≤ c ˆU(s) (θ + dd c u) n ,
for a uniform constant B = B(ε 0 , ω X ) > 0, where we have used the facts that (θ + dd c u) n ≥ ε n 0 ω n X and (θ + dd c ϕ) n = (θ + dd c v) n on the open set U(s) ⊂ D. Therefore, for every s > 0 small enough,

(1 -Bs) n < c which is impossible. This completes the proof.

Stability

We now establish the following L 1 -L ∞ -stability estimate which allows us to show uniform convergence of solutions as long as they converge in L 1 .

Theorem 4.1.5. Fix f 1 , f 2 ∈ L p (X) with p > 1 and B -1 ≤ ´X f 1 n i dV X n ≤ ´X f p i dV X 1 p ≤ B, for some constant B > 1. Assume φ 1 , φ 2 ∈ PSH(X, θ) ∩ L ∞ (X) satisfy (θ + dd c φ i ) n = f i dV X .
Then there exist α = α(p) > 0 and a constant C > 0 depending on n, p, and B such that

∥φ 1 -φ 2 ∥ L ∞ (X) ≤ C ∥φ 1 -φ 2 ∥ α 1 + ∥ f 1 -f 2 ∥ p 1/n .
Proof. Without loss of generality we may assume that φ 1 , φ 2 ≤ 0. Set r = 2p p+1 , hence 1 < r < p. Since φ 1 , φ 2 are uniformly bounded we have e -φ i f i ∈ L r (X) for i = 1, 2. Therefore, there exists a constant A > 0 depending on B such that

A -1 ≤ ˆX e -φ i n f 1 n i dV X n ≤ ˆX e -rφ i f r i dV X 1 r ≤ A
It follows from [GL21, Theorem 3.5 (1)] that there is a constant T > 0 depending on n, r, A such that

∥φ 1 -φ 2 ∥ n L ∞ (X) ≤ T∥e -φ 1 f 1 -e -φ 2 f 2 ∥ r ≤ ∥e -φ 1 f 1 -e -φ 2 f 1 ∥ r + ∥e -φ 2 f 1 -e -φ 2 f 2 ∥ r .
By Hölder inequality, the latter term is bounded from above by

C(p)∥ f 1 -f 2 ∥ p . Now, we have ˆX | f 1 | r |e -rφ 1 -e -rφ 2 |dV X ≤ ˆX | f 1 | r r|φ 1 -φ 2 |e -r(φ 1 +φ 2 ) dV X ≤ ˆX r p | f 1 | p dV X r p ˆX |φ 1 -φ 2 | p p-r e - pr p-r (φ 1 +φ 2 ) dV X p-r p ≤ C(p)∥ f 1 ∥ r ∥φ 1 -φ 2 ∥ p-r r 1 ,
where we have used Hölder inequality again, and the elementary inequality The conclusion thus follows.

Parabolic potentials

Parabolic potentials form the basic objects of our study. They can be seen as weakly regular family of plurisubharmonic functions. In this section we define them and recall their first properties. We refer the reader to [START_REF] Guedj | The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows[END_REF][START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF] for more details.

Families of quasi-plurisubharmonic functions.

We start with some basic definitions which will be used throughout the paper. Definition 4.1.6. The set of parabolic potentials P (X T , ω) is the set of functions φ : (0,

T) × X → [-∞, +∞) satisfying • x → φ(t, x) is ω t -psh for all t ∈ (0, T),
• φ is locally uniformly Lipschitz in (0, T).

The last condition means that for any compact sub-interval J ⊂ (0, T) there exists κ = κ J (φ) > 0 such that for every x ∈ X,

φ(t, x) ≤ φ(s, x) + κ|t -s|, ∀ s, t ∈ J.
A parabolic potential φ ∈ P (X T , ω) can be extended as a upper semi-continuous function on [0, T) × X with ω t -psh slices. Proposition 4.1.7. Let φ 0 be a ω 0 -psh function. Assume φ ∈ P (X T , ω) satisfying

φ t → φ 0 in L 1 (X) as t → 0. Then the extension φ : [0, T) × X → [-∞, +∞) is upper semi-continuous.
Proof. The proof is almost identical to that of [GLZ20, Proposition 1.2]. The problem is local, we can thus assume here that X = B ⊂ C n is a neighborhood of x 0 . Let Θ be a Kähler form in B satisfying Θ = dd c h ≥ ω t for all t, where h is a local potential on B. Changing φ t by φ t + h we can assume that φ t are psh and negative in B. We can proceed exactly the same as in [GLZ20, Proposition 2.1] to conclude.

We also have a compactness result for this class of functions. Proposition 4.1.8. Let (φ j ) ⊂ P (X T , ω) be a sequence which

• is locally uniformly bounded from above in X T ,

• is locally uniformly Lipschitz in (0, T),

• does not converge locally uniformly to -∞ in X T .

Then (φ j ) is bounded in L 1 loc (X T ) and there exists a subsequence which converges to some parabolic potential φ in the L 1 loc (X T )-topology.

Proof. The proof of this result is local and quite close to the classical proof of the analogous result for quasi-psh functions. We refer the reader to [GLZ21a, Proposition 1.14] for more details.

Fix µ a Borel measure on X, and let ℓ denote the Lebesgue measure on R.

Lemma 4.1.9. Fix φ ∈ P (X T , ω). Then ∂ t φ(t, x) exists for all (t, x) ∈ X T \ E where E ⊂ X T is a ℓ ⊗ µ-negligible set.

In particular, ∂ t φ ∈ L ∞ loc (X T ) and h(∂ t φ)ℓ ⊗ µ is a well defined Borel measure on X T for any continuous function h : R → R.

Proof. The problem is local. We refer the reader to [GLZ21a, Lemma 1.13] for a proof.

Parabolic complex Monge-Ampère operators

We assume here that φ ∈ P (X T , ω) ∩ L ∞ (X T ). For each t ∈ (0, T), the function x → φ t (x) = φ(t, x) is ω t -psh and bounded, hence the wedge product

(ω t + dd c φ t ) n := (ω t + dd c φ t ) ∧ • • • ∧ (ω t + dd c φ t )
is well-defined as a positive Borel measure on X, following the works of Bedford-Taylor [START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF] (see e.g. [START_REF] Dinew | Pluripotential estimates on compact Hermitian manifolds[END_REF][START_REF] Kołodziej | Weak solutions to the complex Monge-Ampère equation on Hermitian manifolds[END_REF]). Lemma 4.1.10. Let φ ∈ P (X T ) ∩ L ∞ (X T ) and χ a continuous test function on X T . Then the function

Γ χ : t -→ ˆX χ(t, •)(ω t + dd c φ t ) n is continuous in (0, T).
Proof. We refer to [GLZ20, Lemma 1.9] for more details.

This lemma allows us to define the (2n + 1)-current dt ∧ (ω t + dd c φ T ) n on X T .

Definition 4.1.11. Let φ ∈ P (X T , ω) ∩ L ∞ loc (X T ). The map χ → ˆXT χdt ∧ (ω t + dd c φ t ) n := ˆT 0 dt ˆX χ(t, •)(ω t + dd c φ t ) n
defines a positive (2n + 1)-current on X T , denoted by dt ∧ (ω t + dd c φ t ) n , which can be identified with a positive Borel measure on X T .

The operator can also be defined by approximation in the spirit of Bedford and Taylor convergence results [START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF] : Proposition 4.1.12. Let (φ j ) be a monotone sequence of functions in P (X T , ω) converging to a function φ ∈ P (X T , ω) ∩ L ∞ loc (X T ) almost everywhere on X T . Then

dt ∧ (ω t + dd c φ j t ) n → dt ∧ (ω t + dd c φ t ) n
in the sense of measures on X T .

Proof. See [GLZ20, Proposition 1.11].

We have the following stronger version of the Chern-Levine-Nirenberg inequalities :

Proposition 4.1.13. Assume φ 1 , • • • , φ n ∈ P (X T , ω) ∩ L ∞ (X T ).
Then there exists a constant C > 0 such that for any non-positive parabolic potential ψ ∈ P (X T , ω),

ˆXT -ψ(ω t + dd c φ 1 t ) ∧ • • • ∧ (ω t + dd c φ n t ) ≤ C(∥ψ∥ L 1 (X T ) + 1) n ∏ j=1 (∥φ j ∥ L ∞ (X T ) + 1).
Proof. In each local chart of triple covers U i ⋐ V i ⋐ W i one can find a Kähler form Ω = dd c ρ i > ω t for all t ∈ [0, T). Then we have

ˆ(U i ) T -ψ(ω t + dd c φ 1 t ) ∧ • • • ∧ (ω t + dd c φ n t ) ≤ ˆ(U i ) T -ψ(dd c (ρ i + φ 1 t )) ∧ • • • ∧ (dd c (ρ i + φ n t )).
It follows from the local version of the C-L-N inequalities that the latter integral is bounded by

C ˆT 0 ∥ψ t ∥ L 1 (V i ) + 1 n ∏ j=1 ∥ρ i + φ j t ∥ L ∞ (V i ) + 1 dt < +∞
and the proof is completed since ρ i is uniformly under control in V i .

Assumptions on the densities

We assume throughout the article that

• dV is a fixed volume form on X ;

• 0 ≤ f ∈ L p (X, dV) for some p > 1, and Vol({ f = 0}) = 0 ;

• (t, x, r) → F(t, x, r) is a continuous function on [0, T) × X × R ;

• r → F(•, •, r) is uniformly quasi-increasing, i.e., there exists a constant λ F ≥ 0 such that for every (t, x) ∈ [0, T) × X, the function

r → F(t, x, r) + λ F r is increasing in R; (4.1.6) • (t, r) → F(t,
•, r) is locally uniformly Lipschitz, i.e., for any J ⋐ [0, T) × R there exists a constant κ J > 0 such that for every x ∈ X, (t, r), (t ′ , r ′ ) ∈ J, Our assumptions on the data F, f are mild enough so that the results of this article can be applied to the study of the Chern-Ricci flow on mildly singular Hermitian varieties. We refer to Sect. 4.5 for more details and geometric applications.

|F(t, x, r) -F(t ′ , x, r ′ )| ≤ κ J (|t -t ′ | + |r -r ′ |); (4.1.7) • (t, r) → F(t,

A priori estimates

In this section we assume that φ t = φ(t, •) is a smooth solution to (CMAF) with given smooth data (ω t , F, f , φ 0 ), i.e., t → ω t is a smooth family of Hermitian forms, F and f are smooth densities with f being strictly positive, and φ 0 is smooth and strictly ω 0 -psh.

Our aim is to establish various a priori estimates that will allow us to construct weak solutions to the corresponding degenerate equations. We will make several extra assumptions, depending on the a priori estimates in which we are interested.

Bounding the oscillation of φ t

Recall that ω t ≥ θ for all t ∈ [0, T], where θ is a semi-positive and big (1,1) form. For finite time we can assume without loss of generality that ω t ≤ ω X for t ∈ [0, T]. It follows from [GL21, Theorem 3.4] (respectively [KN15, Theorem 5.8]) that there exist a constant c -and a bounded θ-psh function ϕ (respectively a constant c + and a bounded ω X -psh function Φ) such that

(θ + dd c ϕ) n = e c -f dV, (ω X + dd c Φ) n = e c + f dV.
Up to adding a constant we may assume that sup

X ϕ = inf X Φ = 0.
Proposition 4.2.1. There exists a uniform constant C 0 > 0 only depending on sup X |φ 0 |, λ F , inf X ϕ, sup X Φ, and sup X T F(t, x, 0) such that

|φ t (x)| ≤ C 0 , ∀ (t, x) ∈ [0, T] × X.
Recall that λ F ≥ 0 is a constant such that, for all (t, x) ∈ X T , the function F(t, x, r) + λ F r is increasing on R.

Proof. Set, for any t ∈ R,

γ(t) := sup X |φ 0 |e λ F t + C e λ F t -1 λ F ,
for C > 0 to be chosen later. It is clear that γ(0) = sup X |φ 0 | and γ satisfies the following ordinary differential equation γ

′ (t) -λ F γ(t) = C.
We first get a lower bound for φ t . Set, for (t,

x) ∈ [0, T] × X, u(t, x) := ϕ(x) -γ(t).
We observe that u t is hence θ-psh such that u 0 ≤ φ 0 , and

(ω t + dd c u t ) n ≥ (θ + dd c u t ) n = e c -f dV.
On the other hand, since r → F(•, •, r) + λ F r is increasing and ϕ ≤ 0, we have for

(t, x) ∈ [0, T] × X, ut (x) + F(t, x, u t (x)) = -γ ′ (t) + F(t, x, u t (x)) ≤ -γ ′ (t) + F(t, x, 0) -λ F (ϕ(x) -γ(t)) ≤ λ F |ϕ(x)| -C + F(t, x, 0) ≤ c -,
choosing C > 0 large enough. Therefore (ω t + dd c u t ) n ≥ e ut +F(t,•,u t (•)) f dV.

Now the maximum principle ensures that φ ≥ u on [0, T] × X.

For a upper bound, set for any (t, x) ∈ [0, T] × X, v(t, x) := Φ(x) + γ(t), where γ(t) is the solution to ODE : γ ′ (t)λ F γ(t) = C with C = c +inf X T F(t, x, 0), and γ(0) = sup X φ 0 . We can check that e vt +F(t,x,v t ) f dV ≥ e c + f dV = (ω X + dd c v t ) n Since v 0 ≥ φ 0 it follows the maximum principle that v(t, x) ≥ φ(t, x) for any (t, x) ∈ X T which implies the upper bound for φ. More precisely, we have for any

(t, x) ∈ [0, T] × X, |φ t (x)| ≤ C 0 := sup X |φ 0 |e λ F t + C e λ F t -1 λ F
where C is the following uniform constant

C = sup X T |F(t, x, 0)| + (λ F + 1) sup X (|ϕ| + |Φ|) + max(-c -, c + ).
Let α > 0 be such that αT < 1. The following barrier constructions will be useful in showing that the pluripotential solution to (CMAF) has the right value at t = 0 : Proposition 4.2.2. There exists a uniform constant C > 0 such that for all (t, x) ∈ [0, T) × X,

φ t (x) ≥ (1 -αt)e -At φ 0 (x) + αtϕ + n(t log αt -t) -C e λ F t -1 λ F .
Proof. Recall that ωt ≥ -Aω t for some constant A > 0. In particular ω t ≥ e -At ω 0 and ω t ≥ θ. Consider for any (t,

x) ∈ [0, T) × X, u(t, x) := (1 -αt)e -At φ 0 + αtϕ + n(t log αt -t) -C e λ F t -1 λ F
for C > 0 a uniform constant. We proceed the same as in [GLZ20, Proposition 2.2] to show that (ω t + dd c u t ) n ≥ e ut +F(t,•,u t ) f dV, and moreover u 0 ≤ φ 0 . Therefore the maximum principle yields the desired estimate.

Lipschitz control in time

Our goal in this section is to establish an a priori bound which allows us to show that the solutions φ to degenerate complex Monge-Ampère flows (CMAF) are locally uniformly Lipschitz in time, away from zero.

We recall here that there exists a constant A > 0 such that

-Aω t ≤ ωt ≤ Aω t ∀ t ∈ [0, T]. (4.2.1)
Recall also that F(t, x, r) is quasi-increasing in r, i.e., there exists a constant λ F > 0 such that for every (t, x) ∈ [0, T) × X, the function r → F(t, x, r) + λ F r is increasing in R.

Proposition 4.2.3.

There exists a uniform constant C > 0 such that for all (t, x) ∈ X T ,

n log t -C ≤ φt (x) ≤ C t .
Proof. The proof is identical to that of [GLZ20, Theorem 2.

Semi-concavity in time

We now establish that ω t -psh solutions φ t to (CMAF) are semi-concave in time away from zero.

We assume in this subsection that there exists a constant A > 0 such that, for all t ∈ [0, T],

-Aω t ≤ ωt ≤ Aω t , and ωt ≤ Aω t . (4.2.2)

We also assume that (t, r) → F(t, x, r) is uniformly semi-convex, i.e., there exists a constant C F > 0 such that for every x, the function Proof. The proof is almost identical to that of [GLZ20, Theorem 2.7, 2.9]. We mention here that Lemma 2.8 in [START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF] still holds when ω is Hermitian and η is no longer closed.

(t, x) → F(t, x, r) + C F (t 2 + r 2 ) is convex on [0, T] × [-C 0 , C 0 ].
Theorem 4.2.5. Let ( f j ) j∈N be a sequence of L p (X)-densities converging towards f in L 1 (X). Let F j (t, x, r) be continuous densities which uniformly converge towards F. Let φ j (t, x) be a family of ω t -psh functions such that

• φ j are uniformly bounded,

• for any x ∈ X, φj (t, x) ≤ C/t 2 for some uniform constant C > 0.

Then there exists a bounded function φ ∈ P (X T , ω) such that, up to passing a subsequence, φ j → φ in L 1 loc (X T ), and e φj +F j (t,x,φ j (t,x)) f j (x)dV(x) ∧ dt → e φ+F(t,x,φ(t,x)) f (x)dV(x) ∧ dt in the weak sense of measures on X T .

Proof. We refer to [GLZ20, Theorem 2.11] for more details.

Stability estimates

Proposition 4.2.6. Let 0 < ε < T ′ < T. Assume that φ 1 , φ 2 ∈ P (X T , ω) ∩ C ∞ (X T ) solve dt ∧ (ω t + dd c φ i t ) n = e φi t +F i (t,•,φ i t ) f i dV ∧ dt, with assumptions that (F i , f i ) are smooth for i = 1, 2. Then for all (t, x) ∈ [ε, T ′ ] × X, |φ 1 (t, x) -φ 2 (t, x)| ≤ B ∥φ 1 -φ 2 ∥ γ L 1 (X T ) + ∥ f 1 -f 2 ∥ L p (X) 1 n ,
where 0 < γ = γ(n, p) while 0 < B depends on ε, T ′ , θ, ω X , and upper bounds for

∥ f i ∥ L 1 n , ∥ f i ∥ L p , ∥φ i ∥ L ∞ (X T ) ,
:= e C 1 /ε+M F (∥ f 1 ∥ L p + ∥ f 2 ∥ L p ) .
Similarly, the L 1 n -norms of the densities f i t are uniformly bounded from below by

A 2 := e -C 1 /ε-M F min ∥ f 1 ∥ L 1 n , ∥ f 2 ∥ L 1 n for t ∈ [ε, T ′ ]. Theorem 4.1.5 ensures that ∥φ 1 t -φ 2 t ∥ L ∞ (X) ≤ C ∥φ 1 t -φ 2 t ∥ γ L 1 (X) + ∥ f 1 t -f 2 t ∥ L p (X) 1 n , ∀t ∈ [ε, T ′ ],
where γ ∈ (0, 1) only depends on p, n, and C only depends on n, p, upper bounds for ∥φ

i ∥ L ∞ (X T ) , ∥ f i t ∥ L 1 n
and ∥ f i t ∥ L p . On the other hand, it follows from [GLZ20, Lemma 1.5] that

∥φ 1 t -φ 2 t ∥ L 1 (X) ≤ B max{∥φ 1 -φ 2 ∥ L 1 (X T ) , ∥φ 1 -φ 2 ∥ 1/2 L 1 (X T ) }, with B = 2 max{ √ κ, (T -T ′ ) -1 }, where κ is a uniformly Lipschitz constant of φ 1 -φ 2 in [ε, T ′ ].
The proof follows from the last two inequalities.

The previous result yields easily the following useful information : Corollary 4.2.7. Assume that ∥ f j ∥ L p , ∥F j ∥ L ∞ , ∥∂ t F j ∥ L ∞ and ∥∂ r F j ∥ L ∞ are uniformly bounded. If a sequence (φ j ) of solutions to (CMAF) f j ,F j converges in L 1 (X T ) to φ, then it converges uniformly on any compact subsets of (0, T) × X.

Pluripotential solutions

From now on we assume that the family (ω t ) and the density F and f satisfy the conditions given in the introduction.

For bounded parabolic potentials φ ∈ P (X T , ω) ∩ L ∞ (X T ), we can define the parabolic complex Monge-Ampère equation in the sense of measures on (0, T) × X :

(ω t + dd c φ t ) n ∧ dt = e φt +F(t,x,φ t ) f (x)dV(x) ∧ dt.

(CMAF) Indeed, it follows from Definition 4.1.11 that the left-hand side is a well-defined Radon positive measure, while Lemma 4.1.9 ensures that so is the right-hand side.

Existence results

Our goal in this section is to run the flow (CMAF) from a singular initial data φ 0 ∈ PSH(X, ω 0 ) having uniform bounds. The strategy is to approximate φ 0 by a decreasing sequence of smooth ω 0 -psh functions, f by a sequence of smooth functions f j in L p (dV), and F by a sequence of smooth functions F j , we then consider unique solutions φ j t to the flow (CMAF) with smooth data φ 0,j , F j , f j . In order to pass to the limit we shall use the a priori estimates established in the previous section.

Before proving Theorem A in the introduction, we give the following definition of pluripotential solutions to the Cauchy problem.

Definition 4.3.1. A parabolic potential φ ∈ P (X T , ω) is a pluripotential solution to (CMAF) with initial data φ 0 ∈ PSH(X, ω 0 ) ∩ L ∞ (X) if φ satisfies (CMAF)
in the sense of measures on X T and φ t → φ 0 in L 1 (X) as t → 0 + . Theorem 4.3.2. Let φ 0 be a bounded ω 0 -psh function on X. Assume that (ω, F, f ) is as in the introduction. Then there exists φ ∈ P (X T , ω) solving (CMAF) with initial data φ 0 such that for all 0 < T ′ < T,

• (t, x) → φ(t, x) is uniformly bounded in (0, T ′ ] × X,
• t → φ(t, •)n(t log tt) + Ct is increasing on (0, T ′ ),

• t → φ(t, •) -Ct 2 is concave on each compact subset of (0, T ′ ) for some C > 0,

• φ t → φ 0 as t → 0 + in L 1 (X).

Proof. We first approximate • F by smooth densities F j with uniform constants L F j , C F j , λ F j ;

• f by smooth densities f j > 0 in L p (X) ;

• φ 0 by smoothly decreasing ω 0 -psh functions φ 0,j thanks to [Dem92, BK07] ;

• ω t by smooth (in t) ω j t = ω t + 2 j ω X and ω j t satisfies the assumptions in the introduction.

It is well-known (see e.g. [TW15, Theorem 1.2]) that for every j, there exists a unique smooth function φ j ∈ P (X T , ω) to (CMAF) F j , f j , i.e., dt ∧ (ω j t + dd c φ j t ) n = e φj (t,x)+F j (t,x,φ j (t,x)) f j dt ∧ dV(x).

By the previous section, we see that the φ j 's are uniformly bounded and the derivatives φj are locally uniformly bounded from above in X T . Extracting and relabelling, it follows from Theorem 4.2.5 that there exists φ ∈ P (X T , ω) ∩ L ∞ loc (X T ) such that φ j → φ in L 1 loc (X T ) and e φj +F j (t,x,φ j (t,x)) f j (x)dV(x) ∧ dt → e φ+F(t,x,φ j (t,x)) f (x)dV(x) ∧ dt in the weak sense of measures on X T . We claim that φ j → φ locally uniformly in X T . Indeed, fix 0 < ε < T ′ < T. Since the densities f j have uniform L p norms, it follows from Proposition 4.2.3 that for each j, φj is uniformly bounded on [ε, T ′ ] × X. By Proposition 4.2.6, we have

|φ j (t, x) -φ k (t, x)| ≤ B ∥φ j -φ k ∥ α L 1 (X T ) + ∥ f j -f k ∥ p 1/n
, where B > 0 and α ∈ (0, 1) are uniform constants which do not depend on j, k, and t ∈ [ε, T ′ ]. The claim thus follows. Therefore

dt ∧ (ω t + dd c φ j t ) n → dt ∧ (ω t + dd c φ t ) n
in the sense of measures on X T , hence φ solves (CMAF).

We now show similarly that φ is locally uniformly semi-concave in (0, T). Since the densities f j are uniformly bounded in L p (X), Proposition 4.2.4 ensures that there is a uniform constant C > 0 such that for all j,

φj (t, x) ≤ C/t 2 , ∀(t, x) ∈ X T .
Hence, for each compact sub-interval J ⊂⊂ (0, T) there exists a constant C = C J > 0 such that the functions t → φ j (t, x) -Ct 2 are concave in J for all x ∈ X. The same properties hold for the limiting function φ(t, x) by letting j → +∞. Combining these two estimates we obtain

ˆX φt dµ ≤ C := mµ(X) + µ(X) log M -µ(X) log µ(X).
We then infer that the function t → ´X φ t dµ -Ct is decreasing in t ∈ (0, T), hence ˆX φ t dµ ≤ ˆX φ 0 dµ + C, as required.

Pluripotential subsolutions/supersolutions

Before establishing a pluripotential parabolic comparison principle, we introduce in this section some essential materials.

Regularization of subsolutions.

In this part, we regularize subsolutions of (CMAF) in time. This is analogous to that of [START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF]Sect. 3.4.3]. We always assume that for all t ∈ [0, T],

-Aω t ≤ ωt ≤ Aω t , (4.3.1) for some constant A > 0. We briefly recall a regularization process for subsolutions. Fix 0 < T ′ < T and ε 0 > 0 such that (1 + ε 0 )T ′ ≤ T. It follows from (4.3.1) that there exists A 1 > 0 such that for all t ∈ (0, T ′ ) and |s -1| < ε 0 ,

ω t ≥ (1 -A 1 |s -1|)ω ts .
For |s -1| < ε 0 we set

λ s := |s -1| s , α s = s(1 -λ s )(1 -A 1 |s -1|) ∈ (0, 1).
Up to shrinking ε 0 , we can also assume that

λ s 1 -α s ≥ ε 1 > 0.
It follows from [GL21, Theorem 3.4] that there exist an ε 1 θ-psh function ρ 1 and a constant c 1 ∈ R satisfying

(ε 1 θ + dd c ρ 1 ) n = e c 1 f dV.
This solution can be normalized by sup X ρ 1 = 0.

Lemma 4.3.7. Assume that φ ∈ P (X T ) is a bounded pluripotential subsolution to (CMAF). Then there exists a uniform constant C > 0 such that for every s ∈ (1

-ε 0 , 1 + ε 0 ), (t, x) → u s (t, x) := α s s φ(ts, x) + (1 -α s )ρ 1 (x) -C|s -1|t
is a pluripotential subsolution of (CMAF) in X T ′ .

Proof. The proof is similar to that of [GLZ20, Lemma 3.14]. Notice that we apply the mixed type inequality in the Hermitian setting ; see Lemma 4.1.3.

Let χ : R → [0, +∞) be a smooth function with compact support in [-1, 1] such that ´R χ(s)ds = 1. For any ε > 0, we set χ ε (s) := ε -1 χ(s/ε). Proposition 4.3.8. Assume that φ ∈ P (X T ) is a bounded pluripotential subsolution to (CMAF). Let u s be defined as in Lemma 4.3.7. Then there exists a constant B > 0 such that for all ε > 0 small enough, the function

Φ ε (t, x) := ˆR u s (t, x)χ ε (s -1)ds -Bε(t + 1)
is a pluripotential subsolution of (CMAF) which is C 1 in t and such that sup

X [Φ ε (0, •) -φ 0 ] ε→0 --→ 0.
Proof. We use Lemma 4.3.7 and proceed exactly the same as in [GLZ20, Proposition 3.16].

Uniqueness

We first establish the following comparison principle : Proposition 4.3.9. Let φ ∈ P (X T , ω) ∩ L ∞ (X T ) (resp. ψ) be a pluripotential subsolution (resp. supersolution) to (CMAF) with initial data φ 0 (resp. ψ 0 ). Assume that

• φ t → φ 0 and ψ t → ψ 0 in L 1 as t → 0, • x → φ(•, x) is continuous in Ω and |∂ t (t, x)| ≤ C/t for all (t, x) ∈ X T ;
• ψ is locally uniformly semi-concave in time t ∈ (0, T) ;

Then φ 0 ≤ ψ 0 ⇒ φ ≤ ψ on X T .
The proof below relies on the arguments in [START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF] (see also [START_REF] Guedj | Stability of solutions to complex Monge-Ampère flows[END_REF]).

Proof. Fix 0 < T ′ < T, we will prove that φ ≤ ψ on [0, T ′ ] × X. The proof then follows by letting T ′ → T. Using the invariance properties of the set of assumptions (see [GLZ20, Sect. 3.3]) we may assume that the function r → F(•, •, r) is increasing in the last variable. We divide the proof in several steps.

Step 1. We assume moreover that

• φ is C 1 in t ∈ (0, T) ; • the function (t, x) → ψ(t, x) is continuous on [0, T) × Ω.
The proof, in this step, relies on the arguments in [GLZ20, Proposition 4.2]. The only difference is that we will use the minimum principle established in the previous section.

We recall that ρ is a θ-psh function with analytic singularities such that θ + dd c ρ ≥ δω X for some δ > 0, and Ω = {ρ > -∞}. In particular, ρ is smooth in Ω and ρ = -∞ on ∂Ω. The standard strategy is to work with (1λ)φ t + λρ instead of φ t . However, the time derivative φt may blow up at time zero, so we really need another auxiliary function. It was shown in [GL21, Theorem 3.4] that there exist c = c(θ, f ) > 0 and ϕ ∈ PSH(X, θ) ∩ L ∞ (X) normalized by sup X ϕ = 0 such that (θ + dd c ϕ) n = c f dV X .

Fix λ, ε > 0 small enough and consider

w(t, x) := (1 -λ)φ(t, x) + λ ρ(x) + ϕ(x) 2 -ψ(t, x) -2εt, (t, x) ∈ [0, T ′ ] × X.
This function is upper semi-continuous on [0, T ′ ] × Ω, and goes to -∞ on ∂Ω. Hence w attains its maximum at some point (t 0 , x 0 ) ∈ [0, T ′ ] × Ω. We claim that w(t 0 , x 0 ) ≤ 0. Suppose by contradiction that w(t 0 , x 0 ) > 0, in particular t 0 > 0. Set K := {x ∈ X : w(t 0 , x) = w(t 0 , x 0 )}.

Since φ(•, x 0 ) is differentiable in (0, T), the classical maximum principle insures that for all x ∈ K,

(1 -λ)∂ t φ(t 0 , x) ≥ ∂ - t ψ(t 0 , x) + 2ε.
By assumption the partial derivative ∂ t φ(t 0 , x) is continuous on Ω. By local semi-concavity of t → ψ t it then follows that for every t ∈ (0, T), ∂ - t ψ(t, •) is upper semi-continuous in Ω. Moreover, by Proposition 4.3.6, ψ t is continuous on Ω for each t ∈ (0, T). We thus can find δ > 0 so small that, by introducing an open neighborhood of K

D := {x ∈ Ω : w(t 0 , x) > w(t 0 , x 0 ) -δ} ⊂⊂ Ω, one has (1 -λ)∂ t φ(t 0 , x) > ∂ - t ψ(t 0 , x) + ε, ∀ x ∈ D. (4.3.2) Set u := (1 -λ)φ(t 0 , •) + λ ρ+ϕ 2 and v = ψ(t 0 , •).
Since φ is a subsolution and ψ is a supersolution of (CMAF) we infer (ω t 0 + dd c u) n ≥ e F(t 0 ,x,u(x))-F(t,x,v(x))+ε (ω t 0 + dd c v) n in the weak sense of measures in D, where we have used (4.3.2). Recall that u(x) > v(x) + εt 0 for all x ∈ K. Shrinking D if necessary, we can assume that u(x) > v(x) for all x ∈ D. Since F is increasing in r we thus infer

(ω t 0 + dd c u) n ≥ e ε (ω t 0 + dd c v) n
∈ D : v(x) -u(x)} = min{x ∈ ∂D : v(x) -u(x)}.
In particular, for all x ∈ D,

u(x) -v(x) + min ∂D (v -u) ≤ 0. (4.3.3) Since K ∩ ∂D = ∅, we infer w(t 0 , x) < w(t 0 , x 0 ) for all x ∈ ∂D, hence u(x) -v(x) < u(x 0 ) -v(x 0 ), ∀ x ∈ ∂D which contradicts (4.3.3). Altogether this shows that t 0 = 0, therefore (1 -λ)φ(t, x) + λ ϕ(x) + ρ(x) 2 -ψ(t, x) -2εt ≤ λ sup X |φ 0 | in [0, T ′ ] × Ω. Letting λ → 0 and then ε → 0 we obtain φ ≤ ψ in [0, T ′ ] × Ω, hence in [0, T ′ ] × X.
Step 2. We finally get rid of the assumptions in Step 1. For the assumption that φ is C 1 in t, we use Proposition 4.3.8 and proceed the same way as in [GLZ20, Theorem 4.1].

To remove the assumption on ψ, the argument is the same as that of [GLZ20, Lemma 3.15, Proposition 4.4].

Moreover, we can get rid of the assumption φ 0 ≤ ψ 0 . Indeed, if M 0 := sup X (φ 0ψ 0 ) then φ 0 -M 0 is a subsolution of the same equations since F is increasing in the last variable. Hence

φ -M 0 ≤ ψ in X T , i.e., sup X T (φ -ψ) ≤ sup X (φ 0 -ψ 0 ) + .
The proof of Theorem B follows immediately from the comparison principle that we have just established.

Stability

We assume here that

• F, G : [0, T) × X × R → R are continuous ;
• F and G are increasing in the last variable ;

• F and G are uniformly Lipschitz in the last variable with Lipschitz constants L F and L G ,

• 0 ≤ f , g ∈ L p (X) for some p > 1,

• the family (ω t ) t∈[0,T) is assumed to be as in the introduction ; see (4.0.2).

The Lipschitz assumption on F means that for all (t, x) ∈ X T ,

|F(t, x, r) -F(t, x, r ′ )| ≤ L F |r -r ′ |, r, r ′ ∈ R.
Proposition 4.3.10. Let φ ∈ P (X T , ω) ∩ L ∞ (X T ) (resp. ψ) be a subsolution (resp. supersolution) to (CMAF) with data (F, f , ω t , φ 0 ) (resp. (G, g, ω t , ψ 0 )). Assume that ψ is locally semi-concave in time t ∈ (0, T) and is continuous on (0, T) × X. Assume also that for each t ∈ (0, T), φ t is continuous on X. Fix ε > 0. There exist constants α, A, B > 0 such that, for all (t,

x) ∈ [ε, T) × X, φ(t, x) -ψ(t, x) ≤ B∥φ ε -ψ ε ∥ α L 1 (X) + T sup X T ×R (G -F) + + A∥ f -g∥ 1/n p ,
where A, B > 0 depend on X, θ, n, p, a uniform bound for φ, ψ, φ, and ψ on the set [ε, T) × X, L G , and sup X T G(•, •, sup X T |φ|).

Proof. We are going to apply a perturbation argument as in [GLZ18, GLZ20] which was used in the work of Kolodziej [START_REF] Kołodziej | Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge-Ampère operator[END_REF]. If ∥ f -g∥ p = 0 then g ≤ f almost everywhere on X. In this case, we can easily check that the function (t, x) → φ(t, x) -Mt is a subsolution to (CMAF) with data (G, g, ω t , φ 0 ) and the proof thus follows from the comparison principle. Assume now ∥ f -g∥ p > 0. It was shown in [GL21, Theorem 3.4] that there exist a constant c h > 0 and a function ϕ ∈ PSH(X, θ) ∩ L ∞ (X), normalized by sup X ϕ = 0, such that

(θ + dd c ϕ) n = c h hdV = c h 1 + | f -g| ∥ f -g∥ p dV.
Since 1 ≤ ∥h∥ p ≤ 2, it follows from [GL21, Lemma 3.3] that c h ≥ c 1 > 0 for some uniform constant c 1 (this means that c h is bounded from below away from 0). Set

M 0 := sup X T φ, M 1 (ε) := sup [ε,T)×X φ, M G = sup X T G(t, x, M 0 ).
We also set M 2 (ε

) := M G + max(L G M 0 , M 1 (ε)) and δ := ∥ f -g∥ 1/n p e (M 2 (ε)-ln c 1 )/n .
We may assume that δ is small enough. The computations in [GLZ20, p. 1280-1281] shows that

φ δ (t, x) := (1 -δ)φ(t, x) + δϕ(x) + n log(1 -δ) -Bδt -Mt
(t, x) ∈ [ε, T) × X, φ(t, x) -ψ(t, x) ≤ max X (φ(ε, •) -ψ(ε, •)) + + TM + A(ε)∥ f -g∥ 1/n p ,
where A(ε) := (M 0 + ∥ϕ∥ L ∞ + 2n log 2 + BT)e (M 3 (ε)-ln c 1 )/n . Now we see that ψ ε is a supersolution to the degenerate elliptic equation

(θ + dd c ψ ε ) n ≤ e D(ε) f dV where D(ε) is a upper bound of ψ(t, x) + G(t, x, ψ(t, x)) on [ε, T) × X.
By Theorem 4.1.5, there exists α > 0 depending on n, p and a constant C(ε) > 0 depending on p, θ, D(ε), and ∥ f ∥ p such that max

X (φ ε -ψ ε ) + ≤ C(ε) ∥(φ ε -ψ ε )∥ α L 1 (X) + ∥ f -g∥ 1/n p .
This finishes the proof.

We now finish the proof of Theorem C. Set Φ j = Φ(ω t,j , F j , f j , φ 0,j ) and Φ = Φ(ω t , F, f , φ). By Proposition 4.3.10 above, it suffices to show that the norm ∥Φ j ε -Φ ε ∥ L 1 (X) goes to zero as j → +∞ for some small ε > 0. Now we observe that the norm ∥Φ j ε -Φ ε ∥ L 1 (X) is controlled by ∥Φ j -Φ∥ L 1 ([ε,T)×X) as follows from [GLZ21a, Lemma 1.8]. The proof thus follows from Theorem 4.2.5.

Smoothness of the pluripotential flow

In this section, we establish, under some extra assumptions, a partial regularity of the pluripotential solution to the complex Monge-Ampère flow (CMAF) constructed in previous sections. The assumptions on the density f will become more transparent in the context of the Chern-Ricci flow on complex varieties with log terminal singularities.

Setup

Recall that (X, ω X ) is a compact Hermitian manifold endowed with a reference hermitian form. Fixing T ∈ (0, +∞), let (ω t ) t∈[0,T] be a smooth family of semi-positive forms such that for all t ∈ [0, T], where θ ≤ ω t with θ a semi-positive and volume non-collapsing (1, 1)-form with big equivalence class, and -Aω t ≤ ωt ≤ Aω t and ωt ≤ Aω t , ∀ t ∈ [0, T]. for some fixed constant A > 0. Up to multiplying ω X with a large constant, we may assume that ω t ≤ ω X for all t ∈ [0, T]. We let Ω denote the "positive" locus of θ ; see Sect. 4.1.3. In this section, we assume that f is a positive measurable function on X which is of the form f = e ψ + -ψ - where • ψ ± are quasi-plurisubharmonic functions on X ;

• e -ψ -∈ L p for some p > 1 ;

• ψ ± are smooth in a given Zariski open subset U ⊂ Ω.

Up to multiplying ω X with a large positive constant, we may assume that ψ ± are both ω X -psh. Assume furthermore that the function F : [0, T] × X × R → R is smooth. Given a ω 0 -psh function φ 0 , our goal is to prove that there exists a unique bounded pluripotential solution φ ∈ P (X T , ω) ∩ L ∞ (X T ) with φ t → φ 0 in L 1 (X) as t → 0 + and such that on (0, T) × U, φ is smooth and satisfies (ω t + dd c φ t ) n = e ∂ t φ t +F(t,x,φ t ) f (x)dV(x).

A priori Laplacian estimates

We are now going to estimate ∆φ t , where ∆ denotes the Laplacian with respect to the reference metric ω X . The arguments follow the ones in [START_REF] Boucksom | Regularizing properties of the Kähler-Ricci flow[END_REF] but there are extra difficulties coming from the torsion terms.

Since U ⊂ Ω is a Zariski open subset, we may choose a θ-psh function ρ U with analytic singularities along ∂U such that θ + dd c ρ U ≥ 2δω X for some δ > 0 (see e.g. [BG13, Lemma 4.3.2]). In particular, ρ U → -∞ near ∂U. We may normalize ρ U so that sup X ρ U = 0. Theorem 4.4.1. With the previous setup, assume also that (ω t ) t∈[0,T] is a smooth path of Hermitian forms on X. Assume moreover that ψ ± are smooth ω X -psh functions on X and sup X ψ ± ≤ C for some constant C > 0. Let φ 0 be a smooth ω 0 -psh function on X. Suppose that φ ∈ C ∞ ([0, T] × X) satisfies the parabolic complex Monge-Ampère equation

    
(ω t + dd c φ t ) n = e φt +F(t,•,φ t )+ψ + -ψ - ω n X ω t + dd c φ t > 0 φ| {0}×X = φ 0 .

Then for each ε 0 > 0 and for each compact set K ⊂ U, there exists a constant B > 0 only depending on K, θ, ε 0 , T, C, sup Proof. Fix ε > 0 small. By previous estimates, we have sup

[0,T]×X |φ| ≤ C 0 -1, sup [ε,T]×X | φ| ≤ C ′ 0 ,
where C 0 explicitly depends on A, θ, ω X , T, ∥ f ∥ L p , sup X |φ 0 |, and sup X T F(t, x, 0) while C ′ 0 depends on C 0 , ε, ∥∂F/∂r∥ L ∞ .

We consider the function

H(t, •) := t log tr ω X ( ωt ) -γ(u t ) + 1 φ t+ε + C 0 ,
where ωt = ω t+ε + dd c φ t+ε , u t (x) = φ t+ε (x)ρ U (x)δψ -(x) + C 0 + δC ≥ 1, and γ : R → R is a smooth concave increasing function such that lim t→+∞ γ(t) = +∞. We are going to show that H is uniformly bounded from above for an appropriate choice of γ. The proof will thus follow since u is uniformly bounded on compact subsets of U. We let g denote the Hermitian metric associated to ω X and gt the one associated to ωt = ω t+ε + dd c φ t+ε . Since ρ U → -∞ near ∂U, the function H thus attains its maximum at some point (t 0 , x 0 ) ∈ [0, T] × U. If t 0 = 0 then sup [0,T]×U H ≤ -γ(1) + 1.

It thus suffices to treat the case t 0 > 0. We may assume tr ω X ( ωt ) ≥ 1 at (t 0 , x 0 ).

In order to apply the maximum principle, we need to perform a change of coordinates made possible by the following lemma from [GL10, Lemma 2.1] (see also [ST11a, Lemma 2.9] for a similar argument). Lemma 4.4.2. There exists a holomorphic coordinate system centered at x 0 such that for all i, j,

g i j = δ ij , ∂g i ī ∂z j = 0
and also that the matrix gi j(t 0 , x 0 ) is diagonal.

We let ∆ t = tr ωt dd c denote the Laplacian with respect to ωt . Following computations in [Tô18, Eq. (3.20)] we obtain ∆ t tr ω X ( ωt ) ≥ ∑ i,j,k gi ī g j j gj j gi jk gj īktr ω X Ric( ωt ) -C 1 tr ω X ( ωt ) tr ωt (ω X ).

(4.4.3)

On the other hand, we have Ric( ωt ) = Ric(ω X )dd c ( φt+ε + F(t + ε, •, φ t+ε ))dd c (ψ +ψ -) ≤ C 2 ω X + dd c ψ -dd c ( φt+ε + F(t + ε, •, φ t+ε ))

with C 2 > 0 under control, using dd c ψ + ≥ -Cω X and Ric(ω X ) ≤ Cω X . Plugging this into (4.4.3) we have

∆ t tr ω X ( ωt ) ≥ ∑ i,j
gi ī gj j gi jj gj ī jtr ω X (C 3 ω X + dd c ψ -) + ∆( φt+ε + F(t + ε, •, φ t+ε )) gi ī gj j gi jj gj ī j + C tr ωt (ω X ) (tr ω X ( ωt )) 2 + 2 (tr ω X ( ωt )) 2 Re ∑ i,j,k gi ī T ij j gk īk , (4.4.6) where T ij j := gj jigi jj is the torsion term corresponding to ω t+ε which is under control : |T ij j| ≤ C. Now at the point (t 0 , x 0 ), we have ∂ ī H = 0, hence

-
t ∑ k gk kī = tr ω X ( ωt ) γ ′ (u)u ī + φ ī (φ t+ε + C 0 ) 2 .
Cauchy-Schwarz inequality yields 2 (tr ω X ( ωt )) 2 Re ∑ i,j,k gi ī T ij j gk īk = 2

(tr ω X ( ωt )) 2 Re ∑ i,j,k gi ī T ij j gk kī + Re ∑ i,j,k using that tr ω X ( ωt ) ≥ 1 at (t 0 , x 0 ). We now take care of the term ∆ψ -. Observe that 0 ≤ ω X + dd c ψ -≤ tr ωt (ω X + dd c ψ -) ωt , and we take the trace with respect to ω X 0 ≤ n + ∆ψ - tr ω X ( ωt ) ≤ tr ωt (ω X + dd c ψ -). We write φ instead of φ t+ε . For the Laplacian of F, we note that

gi ī T ij j T ik k ≤ C γ ′ (u) 2 -tγ ′′ (u)
∆F(•, z, φ) = ∆F(•, z, •) + 2Re ∑ g i j ∂F ∂r i φ j + ∂F ∂r ∆φ + ∂ 2 F ∂r 2 |∂φ| 2 ω X .
On the right hand-side, the first term is uniformly bounded on X, the last one is non-negative by the convexity assumption of F. Next, there is a constant C > 0 depending on sup

-ρ U ) + 1 + 1 φ + C 0 .
Now, using the obvious inequality -bx + log x ≤ O(1) for x ∈ (0, +∞, ) we get a uniform upper bound for H at the point (t 0 , x 0 ). Applying this for ε = ε 0 /2 we obtain the theorem.

Higher order estimates

Thanks to Demailly's regularisation theorem [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF], there exists two sequences ψ ± j ∈ C ∞ (X) such that • ψ ± j decreases pointwise to ψ ± on X, the convergence is in C ∞ topology on U, sup X ψ ± j ≤ C, and • dd c ψ ± j ≥ -Bω for some B > 0 under control.

In fact, the constant B > 0 depends on the Lelong numbers of quasi-psh functions ψ ± according to Demailly's result. Nevertheless, these Lelong numbers can be uniformly bounded in terms of the lower bound -Cω for dd c ψ ± by standard arguments ; see e.g. [Bou02, Lemma 2.5]. Moreover, sup X ψ ± j is bounded independently of j, while we have for all j ∈ N, ∥e -ψ - j ∥ L p ≤ ∥e ψ -∥ L p . Using Demailly's regularization theorem again, we also get a decreasing sequence of smooth functions φ j 0 such that φ j 0 decreases pointwise to φ 0 as j → +∞, and ω 0 + dd c φ j 0 ≥ -ε j ω X with ε j → 0. We then set

ω j t := ω t + ε j ω X , f j = e ψ + j -ψ - j .
Since for each j, (ω j t ) is a smooth family of Hermitian forms, and f j is smooth. Moreover, φ j 0 is smooth and strictly ω j 0 -psh. It follows from [START_REF] Tosatti | On the evolution of a Hermitian metric by its Chern-Ricci form[END_REF][START_REF] Tô | Regularizing properties of complex Monge-Ampère flows II : Hermitian manifolds[END_REF] that there exists a unique function φ j ∈ C ∞ ([0, T] × X) such that (ω j t + dd c φ j t ) n = e φj +F(t,x,φ j ) f j dV

φ j | {0}×X = φ j 0 .
(4.4.16)

As in the proof of Theorem 4.3.2, up to extracting and relabelling, there exists a (unique) bounded function φ such that φ j → φ in L 1 loc (X T ) and

(ω t + dd c φ t ) n ∧ dt = e φt +F(t,x,φ) f (x)dV(X) ∧ dt in the sense of measures on X T . The convergence φ j → φ is moreover locally uniform in X T . We are going to prove that φ t is smooth in U for each t > 0.

Observe that f j = e ψ + j -ψ Then using the complex parabolic Evans-Krylov theory and Schauder's estimates, we obtain higher order estimates for φ Up to extracting a subsequence, we have φ j → φ in L 1 (X T ) such that on (0, T) × U, φ is smooth and satisfies (ω t + dd c φ t ) n = e φt +F(t,x,φ t ) f (x)dV(x)

with φ 0 = lim j φ j 0 .

Chern-Ricci flows on varieties with log terminal singularities

In this section we prove the existence and uniqueness of the weak Chern-Ricci flow on complex varieties with log terminal singularities. We prove that the hypotheses from previous sections are satisfied, allowing one run the Chern-Ricci flow from an arbitrary closed positive current with bounded potentials.

Notation

We refer the reader to [EGZ09, Sect. 5] for basic facts on pluripotential theory on compact complex variety with log terminal singularities. Roughly speaking one can consider local embedding j α : U α → C N and objects that are restrictions of global ones.

We assume here that Y is a Q-Gorenstein variety, i.e., Y is a normal complex space such that its canonical divisor K Y is Q-Cartier. We denote the singular set of Y by Y sing and let Y reg := Y \ Y sing . Given a log resolution of singularities π : X → Y (which may and will always be chosen to be an isomorphism over Y reg ), there exists a unique (exceptional) Q-divisor ∑ a i E i with simple normal crossings (snc for short) such that

K X = π * K Y + ∑ i a i E i ,
The coefficients a i ∈ Q are called discrepancies of Y along E i . Definition 4.5.1. We say that X has log terminal (lt for short) singularities if and only if a i > -1 for all i. This definition is independent of the choice of a log resolution. We refer the reader to [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF] for more details.

Let m be a positive integer such that mK Y is Cartier. The point of the definition is that the measure µ Y,h does not depend on the choice of σ, so is globally defined. The arguments above show that Y has lt singularities if and only if µ Y,h has finite total mass on Y, in which case we can consider it as a Radon measure on the whole of Y. Then χ = dd c log µ Y,h is well-defined smooth closed (1, 1)-form on Y.

where µ Y,h is the adapted measure on Y corresponding to h. Since θt is a Hermitian form, there is a small constant c > 0 such that θt ≥ cθ 0 for all t ∈ [0, T]. Now let π : X → Y be a log resolution of singularities as defined in the previous subsection. We have seen that the measure µ := f dV where f = ∏ i |s i | 2a i has poles (corresponding to a i < 0) or zeroes (corresponding to a i > 0) along the exceptional divisors E i = (s i = 0), dV is a smooth volume form. Passing to the resolution, the flow (4.5.3) becomes (ω t + dd c φ t ) n = e ∂ t φ µ (4.5.4)

where ω t := π * θt and φ := π * ϕ. Since ( θt ) t∈[0,T] is a smooth family of Hermitian forms, it follows that the family of semipositive forms [0, T] ∋ t → ω t satisfies all our requirements. We also have ω t ≥ θ := π * (cθ 0 ), the latter is smooth semipositive and big (see e.g. [FT09, Proposition 3.2]), but no longer hermitian. We fix a θ-psh function ρ with analytic singularities along a divisor E = π -1 (Y sing ) such that θ + dd c ρ ≥ 2δω X for some δ > 0.

If we set ψ + = ∑ a i >0 2a i log |s i |, ψ -= ∑ a i <0 -2a i log |s i |, we observe that ψ ± are quasipsh functions with logarithmic poles along the exceptional divisors, smooth on X \ Exc(π) = π -1 (Y reg ), and e -ψ -∈ L p (dV) for some p > 1. Finally, we have

{ ρ > -∞} = X \ Exc(π) = π -1 (Y reg ) ≃ Y reg .
Applying Theorem A and B, there exists a unique pluripotential solution to the Monge-Ampère flow on X T starting with φ 0 = π * ϕ 0 for any fixed T ∈ (0, T max ), denoted by φ t . The higher regularity properties of φ follows from Theorem D. Pushing this solution down to Y, we obtain a weak solution to the Chern-Ricci flow starting with S 0 , denoted by ϕ t . Now assume that ψ t is another solution to the weak Chern-Ricci flow on Y. Then π * ψ is a weak solution to the flow (4.5.4) on π -1 (Y reg ), it can be extended trivially on the whole X. The uniqueness result thus yields that φ t = π * ψ t on X, hence ϕ t = ψ t on Y. flow (5.2.1). Moreover, the evolved currents ω t are smooth outside an analytic subset. This reinforces the analogy between the weak Chern-Ricci flow and the Demailly's equisingular regularization procedure for currents, which is in fact used as a key tool in [START_REF] Di Nezza | Uniqueness and short time regularity of the weak Kähler-Ricci flow[END_REF]. We expect to obtain a Hermitian analogue of the result of Di Nezza-Lu.

Question 5.2.1. Assume that 1/2c(T 0 ) < T max . Is the weak Chern-Ricci flow ω for t ∈ [0, T max [ starting form T 0 smooth in a Zariski open subset U of X ?

Recall that c(T 0 ) denotes the integrability index of T 0 (or φ 0 ) is defined by c(T 0 ) = c(φ 0 ) = inf x c(φ 0 , x) where c(φ 0 , x) := sup{c > 0 : e -2cφ 0 is L 1 on a neighborhood of x}.

Note that c(T 0 ) > 0 since X is compact.

The strategy is that one can apply the approximation process to produce a decreasing sequence of smooth solutions whose limit is defined to be the weak solution. The mild assumption 1/2c(T 0 ) < T max insures that the latter solution is well-defined, ie not identically -∞. To study partial regularity of the weak solution outside analytic subsets, we could establish a priori estimates together with Demailly's equisingular regularization. Of course, the main difficulty is in establishing a priori C 0 bounds. We must show the following C 0 -estimate which corresponds to the one in [DL17b, Lemma 3.3].

Lemma 5.2.2. Assume that φ ∈ PSH(X, ω) ∩ L ∞ (X) satisfies (ω + dd c φ) n ≤ e Aφ-f ω n , where A > 0, f is a measurable function such that e Aψ-f ∈ L q (X) with q > 1, for some ψ ∈ PSH(X, δω), δ ∈ (0, 1). Then we have the following estimate φ ≥ ψ -C where C is a positive constant only depending on n, A, δ, ω, q and a upper bound for ´X e q(Aψ-f ) ω n .

Idea of the proof :

In the Kähler case, Di Nezza and Lu have adapted Kolodziej's approach using their generalized capacities [START_REF] Di Nezza | Complex Monge-Ampère equations on quasi-projective varieties[END_REF][START_REF] Di Nezza | Generalized Monge-Ampère capacities[END_REF]. But this method breaks down in the Hermitian setting.

Recently, Guedj and Lu [START_REF] Guedj | Quasi-plurisubharmonic envelopes 3 : Solving Monge-Ampère equations on hermitian manifolds[END_REF] have found another method for uniform estimates of solutions of complex Monge-Ampère equation. This allows us to deal with singularities of quasi-psh functions. Set u := P (1-δ)ω (φψ). Denote D := {u + ψ = φ} the contact set. Arguing as in [GL21, Theorem 2.4], we have that sup X u is uniformly bounded from above and u is bounded from below (assuming that ψ is bounded). It also follows that

((1δ)ω + dd c u) n ≤ 1 D e A sup X u e Aψ-f ω n .

This implies that the oscillation of u is uniform bounded, the desired estimate thus follows.

In connection with the MMP, it is desirable to treat the case of mildly singular varieties. We expect to start the (twisted) Chern-Ricci flow from an initial positive current (which might have positive Lelong numbers).

Question 5.2.3. Let V be a complex variety with log terminal singularities. Let T 0 be a positive (1,1) current such that T 0 = ω V + dd c φ 0 with ω V a positive form on V and φ 0 a ω V -psh function. Assume that 1/2c(T 0 ) < T max . Does there exist a family (ω t ) t∈[0,T max [ of positive (1,1) currents such that 1. ω t = ω V -tRic(ω V ) + dd c φ t , where the function φ t is (ω V -tRic(ω V ))-psh ; 2. ω t → T 0 weakly as t → 0 + ; 3. for each ε > 0 there is an analytic subset E ε ⊂ X such that the function φ is smooth on [ε, T max [×(X \ E ε ) where (ω t ) [ε,T max [ restricts to a smooth path of positive forms satisfying ∂ω t ∂t = -Ric(ω t )?

  Proposition 3.1.11. Let D be a bounded open subset in R m , J ⊂ R be a bounded open interval, and 0

  0, T] × Ω. The right-hand side is finite thanks to Lemma 3.1.18. Letting λ → 0 we obtainφ ≤ ψ + 3εt in [0, T ′ ] × Ω, hence in [0, T ′ ] × X.We thus conclude the proof by letting ε → 0 and T ′ → T. Proposition 3.3.3. Fix a nonempty open subset of D ⋐ Ω. Let u, v be θ-psh functions, which are bounded in a neighborhood of D, such that lim sup D∋x→∂D (uv)(x) ≥ 0.

Remark 3. 4 . 3 .

 43 The uniqueness of the flow (3.4.2) follows directly from Theorem 3.3.7.

Definition 4 .

 4 1.1. A quasi-psh function u : X → [-∞, +∞) is called θ-plurisubharmonic (θ-psh for short) if it satisfies θ + dd c u ≥ 0 in theweak sense of currents. Here d = ∂ + ∂ and d c = i

Proposition 4 . 1 . 4 .

 414 Let D ⊂⊂ Ω be a nonempty open set. Fix

  |e ae b | ≤ |a -b|e a+b , for all a, b ≥ 0.

  in the weak sense of measures in D. It follows from the minimum principle (see Proposition 4.1.4) that min{x

  is a subsolution to (CMAF) on [ε, T) with the data (G, g), where B > 0 depends only on n, inf X T φ and inf [ε,T)×X φ, and M > 0 is under control. The comparison principle ensures that, for all

  [0,T]×X |φ|, and sup [ε 0 ,T]×X | φ| such that sup [ε 0 ,T]×K |∆φ t | ≤ Be -δψ - .

( 4 .

 4 4.5) using ωt ≤ Aω X and -dd c ψ + ≤ Cω X . Note here that ∆ := tr ω X dd c denotes the Laplacian with respect to ω X .From standard arguments as in [GL21, Eq. (4.5), p. 29], we have|∂ tr ω X ( ωt )| 2 ωt (tr ω X ( ωt )) 2 ≤ 1 tr ω X ( ωt ) ∑ i,j

jt

  on [ε, T] × K. Namely, for each m ∈ N there exists C m,ε,K > 0 such that ∥φ j ∥ C m ([ε,T]×K) ≤ C m,ε,K .

  If we choose σ a local generator of mK Y defined on an open subset U of Y, then (i mn 2 σ ∧ σ) 1/m (4.5.1) defines a smooth positive volume form on U ∩ Y reg . If f i is a local equation of E i around a point π -1 (U), then we can see thatπ * i mn 2 σ ∧ σ 1/m = ∏ i | f i | 2a i dV (4.5.2) locally on π -1 (U)for some local volume form dV. Since ∑ E i are simple normal crossing, this implies that Y has lt singularities if and only if each volume of the form (4.5.1) has locally finite mass near singular points of Y. The previous construction leads to the following definition of adapted measure which is introduced in [EGZ09, Sect. 6] : Definition 4.5.2. Let h be a smooth hermitian metric on the Q-line bundle O Y (K Y ). The corresponding adapted measure µ Y,h on Y reg is locally defined by choosing a nowhere vanishing section σ of mK Y over a small open set U and setting µ Y,h := (i mn 2 σ ∧ σ) 1/m |σ| 2/m h m .

  trouve être indépendante du générateur σ et est bien définie sur Y reg , puis sur Y en la prolongeant par 0 sur Y sing . On obtient alors l'équation (η + i∂ ∂ϕ) n = e ϕ µ Y,h où le produit étant à considérer au sens non-pluripolaire. Cela est équivalent à l'équation de Kähler-Einstein pour la pair (Y, ∆). Travaillant dans une désingularisation π : (X, D) → (Y, ∆), on est ainsi amené à étudier des équations de Monge-Ampère complexes du type(θ + i∂ ∂φ) n = e φ dV ∏ i |s i | 2a i (1.3.1) où θ est une (1,1)-forme fermée lisse représentant une classe big, dV est une forme volume sur X, D = ∑ i a i D i est un diviseur à support snc dont les a i ≤ 1 par la condition que (Y, ∆) est lc, les sections s i définissent D i . On est ramené au résultat suivant :

	1.3. PRÉSENTATION DES RÉSULTATS
	(σ∧ σ) 1/m h m |σ| 2/m	qui se

(courbure négative et nulle). Partons d'une paire (Y, ∆) telle que K Y + ∆ est Q-Cartier et big. On considère un générateur local σ de m(K Y + ∆) (sur un ouvert dans Y reg ), puis une métrique hermitienne h sur K Y + ∆ dont sa courbure est une forme lisse η. On introduira alor la mesure adaptée µ Y,h := Théorème B. Soient X une variété kählerienne compacte et θ une (1,1)-forme fermée réelle représentant une classe de cohomologie big. Soit φ ∈ E (X, θ) une unique solution de

  une variéte kählerienne compacte de type général, i.e., le fibré canonique K X est big. Alors le flot de Kähler-Ricci pluripotentiel normalisé, à partir d'une métrique kählerienne ω 0 , En outre, nous pouvons démarrer le flot à partir d'un courant positif fermé à singularités minimales. Pour l'autre application géométrique, nous étudions le flot de Kähler-Ricci pluripotentiel sur les variétés stables (les singularités sont semi-log canoniques et le fibré canonique est ample). Il a été démontré par Berman et Guenancia [BG14] que sur une variété stable Y il existe un courant de Kähler-Einstein ω KE dans la classe c 1 (K Y ) qui est lisse sur le lieu régulier Y reg . Nous appliquons nos résultats pour montrer que le flot de Kähler-Ricci pluripotentiel (normalisé) sur Y existe pour tout temps et converge vers une métrique de Kähler-Einstein singulière lorsque le temps tend vers l'infini. Plus précisément, nous obtenons : Théorème F. Soit Y un variété projective complexe à singularités (semi)-log canoniques telle que le fibré canonique K Y est ample. Alors le flot de Kähler-Ricci normalisé, à partir d'une métrique kählerienne ω 0 ,

	∂θ t ∂t = -Ric(θ ∂θ t

t )θ t existe pour tout temps. Il coïncide avec le flot lisse sur [0, T max [ et déforme ω 0 vers la métrique de Kähler-Einstein singulière sur Amp(K X ) lorsque t → +∞. Nous obtenons en fait un résultat plus général lorsque K X n'est que pseudoeffectif. ∂t = -Ric(θ t )θ t existe pour tout temps. Il déforme ω 0 vers la métrique de Kähler-Einstein singulière ω KE sur Y reg lorsque t → +∞.

  où dV est la forme volume lisse sur X et • (ω t ) t∈]0,T[ est une famille lisse des formes semi-positives vérifiant -Aω t ≤ ωt ≤ Aω t and ωt ≤ Aω t , ∀ t ∈ [0, T],

	• pour tout t ∈ [0, T[, ω t ≥ θ pour une (1,1)-forme semi-positive et big θ, i.e. il existe une fonc-tion quasi-plurisousharmonique ρ à singularités analytiques vérifiant que θ + dd c ρ domine
	une forme hermitienne ;
	• 0 ≤ f ∈ L p , pour p > 1 et f est presque partout strictement positive ;
	• F : [0, T] × X × R → R est continue ;
	• la fonction r → F(•, •, r) est quasi-croissante en r, i.e. il existe une constante λ F >0 telle que
	pour (t, x) ∈ [0, T] × X, la fonction
	r → F(t, x, r) + λ F r est croissante dans R;	(1.3.5)
	• F est localement uniformément Lipschitz, i.e. pout toute J ⋐ [0, T[×R il existe une constante
	κ J ≥ 0 telle que pour tout (t, r), (t ′ , r ′ ) ∈ J, x ∈ X,
	|F(t, x, r) -F(t ′ , x, r ′ )| ≤ κ J (|t -t ′ | + |r -r ′ |);	(1.3.6)
	• (t, r) → F(t, •, r) est localement uniformément semi-convexe, i.e. pout toute J ⋐ [0, T[×R il
	existe une constante C J > 0 telle que pour tout x ∈ X,
	(t, r) → F(t, x, r) + C J (t 2 + r 2 ) est convexe dans J.	(1.3.7)
		(1.3.4)
	pour A > 0 fixée ;

  Lemma 1.12]. Let φ : Ω T → R be a continuous function which is locally uniformly semi-concave in (0, T). Then (t, x) → ∂ -

	Proposition 3.1.10.

t φ(t, x) is upper semi-continuous while (t, x) → ∂ + t φ(t, x) is lower semicontinuous in Ω T . In particular, ∂ + t φ and ∂ - t φ coincide and are continuous in Ω T \E, where E is a Borel set with ℓ ⊗ µ measure zero.

  x) and Ctψ(t, x) are increasing in t ∈ J. These functions thus have derivatives in t almost everywhere on J (see e.g. [KK96, Theorem 1.2.8]). Hence the first equality follows from [GT83, Lemma 7.6].The second inequality is a simple consequence of the elliptic maximum principle in the local context (see e.g. [GZ17a, Theorem 3.27]).

  •, r) is locally uniformly semi-convex, i.e., for any J ⋐ [0, T) × R there exists a constant C J > 0 such that for every x ∈ X, (t, r) → F(t, x, r) + C J (t 2 + r 2 ) is convex in J.

	(4.1.8)
	Note that if F is C 2 -smooth then the local assumptions (4.1.7) and (4.1.8) are automatically satis-
	fied, while (4.1.6) is a global assumption.

  4, 2.5]. Note here that C depends explicitly on T, C 0 (defined in Proposition 4.2.1), and upper bounds for ∥ f ∥ L p ,

	sup [0,T)×X×[-C 0 ×C 0 ]	∂F ∂t	, and	sup [0,T)×X×[-C 0 ×C 0 ]	∂F ∂r	.

  We are going to use the stability result established in [GL21, Theorem 2.3]. For any t ∈ [ε, T ′ ], consider the densities

	f i t = e	φi t +F i (t,•,φ i t ) f i , i = 1, 2.
	It follows from Proposition 4.2.3 that for any t ∈ [ε, T ′ ], φi t are uniformly bounded by C 1 /ε, while Proposition 4.2.1 ensures that the φ i t are uniformly bounded. We thus deduce that the L p -norms of the densities f i t are uniformly bounded from above by
	A 1

and sup [ε,T]×X | φi t |. Proof.

  Applying ∂ ∂t -∆ t to H, ∂ ∂t -∆ t H = log tr ω X ( ωt ) + t tr ω X ( ωt+ε + dd c φt+ε ) tr ω X ( ωt ) t ∆ t tr ω X ( ωt ) tr ω X ( ωt ) + t |∂ tr ω X ( ωt )| 2 ωt (tr ω X ( ωt )) 2 γ ′ (u) ut + γ ′ (u)∆ t u t + γ ′′ (u)|∂u t | 2 ωt -φt+ε (φ t+ε + C 0 ) 2 + ∆ t φ t+ε (φ t+ε + C 0 ) 2 -

	(4.4.2)
	2|∂φ t+ε | 2 ωt (φ t+ε + C 0 ) 3 .

  C 3 tr ωt (ω X ) -C 4 tr ω X ( ωt ) tr ωt (ω X ), X ( ωt + dd c φt ) tr ω X ( ωt ) -∆ t tr ω X ( ωt ) tr ω X ( ωt ) ≤ C 5 + ∆ψ --∆F tr ω X ( ωt ) -1 tr ω X ( ωt ) ∑ i,jgi ī gj j gi jj gj ī j+ C 3 tr ωt (ω X ) tr ω X ( ωt ) + C 4 tr ωt (ω X ),

	(4.4.4)
	which yields
	tr ω

  tr ωt (ω X ) (tr ω X ( ωt )) 2 + (-γ ′′ (u)) X ) t(tr ω X ( ωt )) 2 + |∂φ t+ε | 2 ωt t(φ t+ε + C 0 ) 3 + 2C tr ωt (ω X ) (tr ω X ( ωt )) 2where we have used that |T ij j| ≤ C. Together with (4.4.6) this yields at (t 0 , x 0 ),|∂ tr ω X ( ωt )| 2 ωt (tr ω X ( ωt )) 2 ≤ 1 tr ω X ( ωt ) ∑ i,j gi ī gj j gi jj gj ī j + |∂φ t+ε | 2 ωt t(φ t+ε + C 0 ) 3 X ) (tr ω X ( ωt )) 2 + (-γ ′′ (u))On the other hand, we observe thatγ ′ (u)∆ t u = γ ′ (u)(ntr ωt (ω t+ε + dd c (ρ U + δψ -)) ≤ γ ′ (u)(nδ tr ωt (2ω X + dd c ψ -)).(4.4.8) Plugging (4.4.5), (4.4.7) and (4.4.8) into (4.4.2), we obtain at(t 0 , x 0 ) ∆ t H ≤ log tr ω X ( ωt ) + t C 5 + ∆ψ --∆F tr ω X ( ωt ) δγ ′ (u) tr ωt (ω X + dd c ψ -) + (C 7 Tδγ ′ (u)) tr ωt (ω X ) + C 6 γ ′ (u) 2 -γ ′′ (u) + 1 + 2T tr ωt (ω X ) (tr ω X ( ωt )) 2 + nγ ′ (u)γ ′ (u) φt+ε -φt+ε (φ t+ε + C 0 ) 2 + ∆ t φ t+ε (φ t+ε + C 0 ) 2 -|∂φ t+ε | 2 ωt (φ t+ε + C 0 ) 3 ,(4.4.9)

	0 ≤	∂t ∂	-
				|∂u| 2 ωt
				t
			+ C tr ωt (ω + C 6 γ ′ (u) 2 -tγ ′′ (u) + 1 t + 2	ωt tr ωt (ω |∂u t | 2 t	.	(4.4.7)

  X T |φ| such that -∂F ∂r ∆φ ≤ Ctr ω X ( ωt )since ω t is semi-positive, and we have assumed that tr ω X ( ωt ) ≥ 1. Next, there exists a constant C > 0 depending on sup X T |φ| such that-2Re ∑ g i j ∂F ∂r i φ j ≤ |∂φ| 2 ωt + Ctr ω X ( ωt ),which can be proved by using Arithmetic-Geometric Mean inequality. Altogether, we thus obtain-∆F(t, •, φ) tr ω ( ωt ) ≤ Ctr ωt (ω X ) + C + |∂φ| 2 ωt tr ω X ( ωt )using again tr ω X ( ωt )tr ωt (ω X ) ≥ n 2 . Together with (4.4.10), we obtainC 5 + ∆ψ --∆F tr ω X ( ωt ) ≤ tr ωt (ω X + dd c ψ -) + C 8 tr ωt (ω X ) + + ∆ψ --∆F tr ω X ( ωt ) δγ ′ (u) tr ωt (ω X + dd c ψ -) ≤ (Tδγ ′ (u)) tr ωt (ω X + dd c ψ -) + C 8 Ttr ωt (ω X ) + T |∂φ| 2 ωt tr ω X ( ωt ) .We now choose the function γ as followsγ(s) := Bs -1 s with B > 0 so large that B ≥ 3 + (C 7 + 1)Tδ -1 . Since u ≥ 1, we have B ≤ γ ′ (u) ≤ 1 + B, C 6 γ ′ (u) 2 -γ ′′ (u) + 1 + 2T ≤ C 9 u 3 .Plugging this together with (4.4.11) into (4.4.9) we obtain0 ≤ log tr ω X ( ωt ) -3 tr ωt (ω X ) + C 9 u 3 tr ωt (ω X ) (tr ω X ( ωt )) 2 + T tr ω X ( ωt ) -1 (φ t+ε + C 0 ) 3 |∂φ t+ε | 2 ωt + C10 (4.4.12) with C 10 only depending on n, sup X T |φ t | and sup X T | φt+ε |, where have used ∆ t φ t+ε ≤ n. If we have at (t 0 , x 0 ), tr ω X ( ωt ) ≤ C 9 u 3 t + T(φ t+ε + C 0 ) 3 then at the same pointH ≤ T log( C 9 u 3 + 8TC 3 0 )γ(u) + 1 ≤ C 11 ,with C 11 > 0 under control, and we are done. Otherwise at (t 0 , x 0 ) we have tr ω X ( ωt )≥ C 9 u 3 + T(φ t+ε + C 0 ) 3 ,and then (4.4.12) becomes 0 ≤ log tr ω X ( ωt ) -2 tr ωt (ω X ) + C 10 . (4.4.13) Now it follows from [BG13, Lemma 4.1.1] that log tr ω X ( ωt ) ≤ (n -1) log tr ωt (ω X ) + C 12ψ - (4.4.14) with C 12 depending on sup |φ|, sup | φ| and sup X ψ + . Together with (4.4.13) this yields at (t 0 , x 0 ) 0 ≤tr ωt (ω X )ψ -+ C 13 (4.4.15) since (n -1) log x -2x ≤ -x + O(1) for x > 0. Plugging this into (4.4.14), it follows that log tr ω X ( ωt ) ≤ (n -1) log(C 13ψ -) + C 12ψ - at (t 0 , x 0 ). Therefore, H(t 0 , x 0 ) ≤ t 0 (n -1) log(C 13ψ -) + C 12 + (Bδ -1)ψ --B(φ

		|∂φ| 2 ωt tr ω ( ωt )	.
	From this, we have
	t	C 5 (4.4.11)

  for some uniform constant A > 0. Fix ε > 0. From the previous section, we have a uniform bounds for φ , for some constant C ε,T . Fix a compact set K ⋐ U. We apply Theorem 4.4.1 to have a uniform Laplacian estimates for φ

		j j ≤ e	C-ψ -
	j t ) satisfies
		θ ≤ ω	j t ≤ Aω X ,		-Aω	j t ≤	ωj t ≤ Aω t j
	j t and	φj t on [ε, T] × X :
		sup [ε,T]×X	|φ	j t | +	∂φ ∂t	j t	≤ C ε,T

j is uniformly bounded in L p and the family of Hermitian form (ω

j t on [ε, T] × K : sup [ε,T]×K |∆φ j t | ≤ C ε,T,K .

Théorème 1.1.3[START_REF] Chen | Kähler-Einstein metrics on Fano manifolds. I,II[END_REF]). Une variété de Fano admet une métrique de Kähler-Einstein si et seulement si elle est K-stable.Il existe de nombreuses méthodes pour prouver l'existence de métriques canoniques : approche parabolique, approche de viscosité, approche variationnelle, etc. On va introduire brièvement l'approche parabolique dans le paragraphe suivant, qui est aussi appelée la méthode de l'équation d'évolution d'Hamilton.

Perelman a exposé une lecture privée à MIT, en Avril, 2003. On pourra consulter les textes[START_REF] Sesum | Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman)[END_REF][START_REF] Cao | The Kähler-Ricci flow on Fano manifolds[END_REF][START_REF] Guedj | Convergence of the Kähler-Ricci flow on a Kähler-Einstein Fano manifold[END_REF].

Remerciements

Tout d'abord, je souhaiterais adresser mes très sincères remerciements à mes directeurs de thèse, Hoang-Chinh Lu et Vincent Guedj d'avoir accepté de diriger mes recherches.

Hoang-Chinh, pour sa disponibilité, sa générosité et sa patience depuis mes études en France.

Using the arguments as in [GLZ20, Proposition 3.16], we will show that u ε is a pluripotential subsolution to (CMAF) which is C 1 in t. Indeed, since V s is a pluripotential subsolution to (CMAF), Lemma 3.1.16 yields, for any t ∈ (0, T),

We know that the function A → (det A) 1/n is concave on the convex cone of non negative hermitian matrices. It follows from Jensen's inequality that, for any t ∈ (0, T), (det(ω t +dd c u ε )) 1/n = det ˆR(ω t + dd c V s )η ε (s -1)ds

The second line follows from the Main Theorem in [START_REF] Guedj | Weak subsolutions to complex Monge-Ampère equations[END_REF], the third and fourth ones follow from the convexity of the exponential and F, and the last one follows from the monotonicity of F. Using Lemma 3.1.16 again, we infer that u ε is a subsolution to (CMAF).

On the other hand, it follows from the proof of Theorem 3.2.7 that V s (0, x) ≤ φ 0 on X. Hence u ε (0, •) ≤ φ 0 on X by taking B > 0 large enough. We can thus apply Proposition 3.3.4 to obtain u ε ≤ Φ on [0, T ′ ] × Ω, hence on [0, T ′ ] × X. Letting ε → 0 and T ′ → T we get U ≤ Φ on [0, T) × X. Hence the equality holds.

Applications

We apply the tools we have developed in related geometrical settings. We first define and study the longtime behavior of the normalized Kähler-Ricci flow on manifolds of general type. We next prove the existence of a longtime solution of the flow on manifolds with nonnegative Kodaira dimension. We then analyze the normalized Kähler-Ricci flow on varieties with semi-log canonical singularities and ample canonical bundle (stable varieties).

Manifolds of general type

We study in this section the (normalized) Kähler-Ricci flow on a manifold of general type. We try to run such flow in a weak sense beyond the maximal existence time by using the results obtained in the previous section. We then study the long-time behavior of this flow.

Let (X, ω 0 ) be a compact Kähler manifold of general type i.e. the canonical divisor K X is big, and ω 0 is a Kähler form. The normalized Kähler-Ricci flow is the evolution equation :

Let T be the maximal existence time of smooth flows which is defined by

Note that T = +∞ if and only if the canonical divisor K X is nef. In this case, the normalized Kähler-Ricci flow exists in the classical (smooth) sense and converges to a singular Kähler-Einstein metric (cf. [START_REF] Tsuji | Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type[END_REF][START_REF] Tian | On the Kähler-Ricci flow on projective manifolds of general type[END_REF]).

Set u := (1λ)φ(t 0 , •) + λ(ρ + χ) and v = Φ(t 0 , •). Since φ is a pluripotential solution to (3.4.12), using Lemma 3.2.6 we infer that

On the other hand, Φ is a pluripotential solution to (3.4.12), hence

in the weak sense of measures in D. The last two inequalities yield

We then repeat the arguments as in the proof of Proposition 3.3.2 to obtain a contradiction. Therefore, we must have t 0 = 0, hence

We thus finish the proof by letting ε → 0 and T → +∞.

Step 4.2. We next remove the continuity assumption on Φ in Step 4.1. Fixing s > 0 small enough, we set

where h is defined as in the proof of Theorem 3.4.1. We observe that

where the last inequality follows from Lemma 3.2.6. Since h(s) ≤ 0 for s > 0 we have (ω t+s + dd c u s ) n ≥ e ∂ τ u s +u s dµ.

On the other hand, (ω t+s + dd c v s ) n = e ∂ τ v s +v s dµ, where v s (t, x) := Φ(t + s, x) for (t, x) ∈ (0, +∞) × X. By Lemma 3.4.14 we have u s (0, x) ≤ v s (0, x) for all x ∈ X. Since v s is continuous on [0, +∞) × U, it follows from Step 4.1 that

Letting s → 0 we thus obtain φ ≤ Φ on [0, +∞) × X.

Lemma 3.4.14. For all (t, x) ∈ (0, T) × X,

.23)

where h is the unique solution to the ODE : h

Definition 4.1.14. A family F ⊂ P (X T , ω) is uniformly semi-concave in (0, T) if, for any compact interval J ⋐ (0, T), there exists a constant κ = κ(F , J) > 0 such that any φ ∈ F is uniformly κconcave in J, i.e., for each x ∈ X, t → φ(t, x)κt 2 is concave in t ∈ J.

Fix µ a Borel measure on X, and let ℓ denote the Lebesgue measure on R.

Theorem 4.1.15. Let ( f j ) be a sequence of positive functions which converges to f in L 1 (X T , ℓ ⊗ µ). Let (φ j ) ⊂ P (X T , ω) be a sequence of parabolic potentials which

• converges ℓ ⊗ µ-almost everywhere in X T to a function φ ∈ P (X T , ω),

• is uniformly semi-concave in (0, T).

Then lim j→+∞ φj (t, x) = φ(t, x) for ℓ ⊗ µ-almost every (t, x) ∈ X T , and

in the weak senses of measures on X T , for all h ∈ C 0 (R, R).

We refer the reader to [GLZ20, Theorem 1.14] for a proof.

Assumptions and Notation

In the whole article, we let X denote a compact complex manifold of complex dimension n ≥ 1. We always denote by ω X a Hermitian metric on X.

We fix T ∈ (0, +∞]. We are mainly concerned with finite time intervals, i.e., T < +∞, and we implicitly assume that our data are possibly defined in slightly large time interval, i.e., on (0, T + ε) for some ε > 0. We let X T denote the real (2n + 1)-dimensional manifold X T = (0, T) × X with parabolic boundary

We fix a smooth semi-positive (1, 1) form θ whose equivalence class is big, i.e., contains a positive (1,1) current which dominates a hermitian form. We fix ρ a θ-psh function with analytic singularities such that θ + dd c ρ ≥ δω X for some δ > 0. (4.1.2)

We let Ω denote the positive locus of the big form θ,

Assumptions on the forms

We assume throughout the chapter that (ω t ) t∈[0,T) is a smooth family of semi-positive (1,1) forms on X satisfying θ ≤ ω t for all t ∈ [0, T). For finite times we can also assume, up to multiplying by a positive constant, that ω t ≤ ω X for all t ∈ [0, T). In Section 4.2, we need to assume that t → ω t moreover satisfies for t ∈ [0, T), for some constant A > 0. The lower bound (4.1.4) is equivalent to the fact that t → e At ω t is increasing. In particular,

The latter will be used on several occasions in the sequel.

Definitions

holds in the sense of measures in (0, T) × X.

In the sequel one can interpret these notions by considering a family of inequalities on slices :

Then 1) φ is a pluripotential subsolution of (CMAF) if and only if for all t ∈ (0, T),

in the sense of measures in X.

2) φ is a pluripotential supersolution of (CMAF) if and only if for all t ∈ (0, T),

in the sense of measures in X.

Proof. [GLZ20, Lemma 3.11]

We use properties of solutions to complex Monge-Ampère equations to show that parabolic supersolutions automatically have continuity properties : Proposition 4.3.6. Let ψ ∈ P (X T , ω) ∩ L ∞ (X T ) be a pluripotential supersolution to (CMAF). Then ψ is continuous on (0, T) × Ω.

Recall here that Ω is defined by Ω := {ρ > -∞}, where ρ is an θ-psh function with analytic singularities such that θ + dd c ρ dominates a Hermitian form.

Proof. Fix J ⋐ (0, T). By definition, for every t ∈ (0, T) we have

Since ψ is locally uniformly Lipschitz in t and F is bounded from above, there exists a constant M = M(J) > 0 for almost every t ∈ J, ψt + F(t, •, ψ t ) ≤ M. We thus obtain

for almost every t ∈ J. This inequality actually holds for all t ∈ J by the (weak) continuity of the left-hand side. It follows from [GL21, Theorem 3.7] that ψ t is continuous on Ω for each t ∈ J. Let us emphasize that in the proof of [GL21, Theorem 3.7, p. 25] we only require ω t to be merely big since the supersolution ψ t is bounded on X, for each t > 0. Since ψ is uniformly Lipschitz in J it follows that ψ is continuous on J × Ω as follows from standard arguments ; see e.g. [GLZ20, Proposition 3.12]. Remark 4.5.3. In [ST17], Song and Tian defined an adapted measure of the form µ Y,h for a smooth metric h on K Y as a smooth volume form. We would like to avoid this terminology, which has the drawback that ω n might not be smooth in this sense even when ω is smooth positive (1, 1)-form on X. This is in fact already the case for quotient singularities.

Given a Hermitian form ω Y on Y, there exists a unique hermitian metric h = h(ω Y ) of K Y such that ω n Y = µ Y,h . We have the following definition. We now define the notion of the weak Chern-Ricci flow on compact complex varieties with log terminal singularities. Definition 4.5.5. Let Y be a Q-Gorenstein compact complex variety with log terminal singularities and let θ 0 be Hermitian metric such that

Fix S 0 = θ 0 + dd c ϕ 0 a positive (1,1)-current with ϕ 0 a bounded θ 0 -psh function. A family (θ t ) t∈[0,T max ) of semi-positive (1,1)-currents on Y is said to be a solution of the weak Chern-Ricci flow starting with S 0 if the following conditions hold :

2. θ t → S 0 as t → 0, 3. (θ t ) t∈(0,T max ) restricts to a smooth path of Hermitian forms on Y reg satisfying ∂θ t ∂t = -Ric(θ t ).

Existence and uniqueness of the weak Chern-Ricci flow

Our aim is to establish the existence and uniqueness for the weak Chern-Ricci flow on complex variety with log terminal singularities. Furthermore, we obtain smoothing properties for the weak Chern-Ricci flow even if the initial data is not smooth. Theorem 4.5.6. Let Y be a Q-Gorenstein compact complex variety with log terminal singularities and let θ 0 be Hermitian metric such that

If S 0 = θ 0 + dd c ϕ 0 for some bounded θ 0 -psh function ϕ 0 , then there exists a unique solution (θ t ) t∈[0,T max ) of the weak Chern-Ricci flow starting with S 0 .

Proof. It is classical that solving the (weak) Chern-Ricci flow is equivalent to solving a complex Monge-Ampère equation flow. Let χ be a closed smooth (1,1) form that represents c BC 1 (K Y ). Given T ∈ (0, T max ), we set θt := θ 0 + tχ which defines an affine path of semipositive (1,1)-forms with local potentials. Since χ is a smooth representative of c BC 1 (K Y ), one can find a smooth metric h on the Q-line bundle O Y (K Y ) with curvature form χ. We obtain µ Y,h the adapted measure corresponding to h. The Chern-Ricci flow is equivalent to the following Monge-Ampère flow

Chapitre 5

Future projects

We introduce in this chapter some problems that we plan to tackle. These are motivated by the Analytic Minimal Model Program proposed by Song and Tian.

Monge-Ampère flows with prescribed initial data

The study of the (long-term) behaviour of the Kähler-Ricci flow on mildly singular varieties in relation to the MMP was undertaken by Song and Tian [START_REF] Song | Canonical measures and Kähler-Ricci flow[END_REF][START_REF] Song | The Kähler-Ricci flow through singularities[END_REF]. It requires a theory of weak solutions for certain degenerate parabolic complex Monge-Ampère equations.

On a compact Kähler manifold X, the (weak) Kähler-Ricci flow can be written as a parabolic complex Monge-Ampère equation of the following type :

where T > 0, F is a continuous increasing function in the last variable, (ω t ) t≥0 is a smooth family of closed real (1, 1)-forms, and 0 ≤ f is a density on X. When f ∈ L p (X, dV) for some p > 1 and ω t are semi-positive and big forms, these have been studied in [START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF].

Our goal in [START_REF] Dang | Pluripotential Monge-Ampère flows in big cohomology classes[END_REF] is to extend the previous analysis to the case of big cohomology classes, i.e. when the references form ω t for t > 0, are merely big and not necessarily semi-positive. The latter situation appears naturally in the MMP as volume non-collapsing of the Kähler-Ricci flow finite time singularities. The existence of the weak Kähler-Ricci flow is often proved by using approximation arguments and a priori estimates (see [START_REF] Song | The Kähler-Ricci flow through singularities[END_REF][START_REF] Guedj | Pluripotential Kähler-Ricci flows[END_REF]). Unfortunately, big cohomology classes can not be approximated by Kähler ones so this approach breaks down in our case. We instead use the Perron method, inspired by [START_REF] Guedj | The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows[END_REF], considering the upper envelope of all pluripotential subsolutions to the Cauchy problem. We prove in [START_REF] Dang | Pluripotential Monge-Ampère flows in big cohomology classes[END_REF] that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semiconcave in time, and provides a unique pluripotential solution with such regularity and minimal singularities. We have required the initial data φ 0 to have minimal singularities.

We expect to generalize the latter result and prove the following Question 5.1.1. For any initial φ 0 (not necessarily having minimal singularities), does there exist a unique weak solution φ to (CMAF) which is continuous in some Zariski open subset of X.

It is also interesting to study higher regularity of the pluripotential solutions. When {ω t } are Kähler classes (or more generally big and semi-positive), the problem has been studied by many experts ; see e.g. [START_REF] Song | The Kähler-Ricci flow through singularities[END_REF][START_REF] Guedj | Regularizing properties of the twisted Kähler-Ricci flow[END_REF][START_REF] Di Nezza | Uniqueness and short time regularity of the weak Kähler-Ricci flow[END_REF][START_REF] Tô | Regularizing properties of complex Monge-Ampère flows[END_REF]. The general case however remains open.

Idea of existence of weak solution.

We follow the Perron method used in Chapter 3. The point is that we need to construct a pluripotential subsolution whose singularities are under control. We use the notation from Chapter 3. We assume that ω t , F, f satisfy all the assumptions as in CHAPITRE 5. FUTURE PROJECTS Chapter 3. We are going to deal with the initial function φ 0 with more general singularities. Recall that there exists a θ-psh function χ on X with analytic singularities such that, for some δ 0 > 0, θ + dd c χ ≥ 2δ 0 Θ. Moreover, χ is smooth in the ample locus Amp(θ), and χ(x) → -∞ as x → ∂Amp(θ). We see that χ has a model type singularity in the sense of [START_REF] Darvas | Monotonicity of nonpluripolar products and complex Monge-Ampère equations with prescribed singularity[END_REF][START_REF] Darvas | Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity[END_REF]. We will assume that φ 0 is less singular than ϕ := P θ [χ] in the sense that ϕ ≤ φ 0 + O(1). In the aforementioned papers, it was shown that there exists a unique ρ ∈ E (X, θ, ϕ) solving

Moreover, there exists a constant C > 0 only depending on p, dV, ´X(θ

We proceed as in Lemma 3.1.18 to obtain that λg(0) ρ ϕ +χ 2 is a ω 0 -psh function which is less singular than φ 0 , for λ close to 1 and in Lemma 3.2.2 to construct a subsolution to (CMAF) under the form λg(t) ρ ϕ +χ 2 -C(t + 1) for C > 0 under control. We then follow step by step Section 2.2.6 to show that the envelope of subsolutions is also a subsolution which is locally uniformly Lipschitz and locally uniformly semi-concave in time. It is also a pluripotential solution to (CMAF) by using the balayage process.

It remains to study the continuity of such a solution. This leads one to the following question :

Question 5.1.2. Fix µ = f dV a non-pluripolar positive measure on X, with f ≤ e -Aϕ g, where g ∈ L p (dV) for some p > 1 and assume ´X f dV = ´X θ n ϕ > 0. Let u ∈ E (X, θ, ϕ) be such that (θ + dd c u) n = µ and sup X u = 0. Is u continuous on Amp(θ) ?

We remark that [ϕ] is a model potential, ie. ϕ = P θ [ϕ] and locally bounded on Amp(θ) (since it is less singular than χ).

Partial regularity of the Chern-Ricci flow

Let (X, ω) be a compact complex manifold. Fixing η a smooth (1,1) form on X, the η-twisted Chern-Ricci flow is the following :

= -Ric(ω t ) + η, ω| t=0 = T 0 , (5.2.1)

where T 0 = ω + dd c φ 0 is a fixed positive current, for some ω 0 -plurisubharmonic function φ 0 (a global potential of T 0 ). We put here d = ∂ + ∂ and d c = i 2 ( ∂ -∂) so that dd c = i∂ ∂. If T 0 is a Hermitian metric, i.e. φ 0 is smooth, then it was proved in [START_REF] Tosatti | On the evolution of a Hermitian metric by its Chern-Ricci form[END_REF] that such a flow admits a unique solution on a maximal interval of time [0, T max [, where T max := sup{t > 0 : ∃ ρ ∈ C ∞ (X) with ω -tRic(ω) + dd c ρ > 0}.

For the Kähler case, attempts to run the Kähler-Ricci flow from a rough initial data have motivated several recent works [START_REF] Song | The Kähler-Ricci flow through singularities[END_REF][START_REF] Székelyhidi | Regularity of weak solutions of a complex Monge-Ampère equation[END_REF][START_REF] Guedj | Regularizing properties of the twisted Kähler-Ricci flow[END_REF][START_REF] Tô | Regularizing properties of complex Monge-Ampère flows[END_REF][START_REF] Di Nezza | Uniqueness and short time regularity of the weak Kähler-Ricci flow[END_REF]. The best results so far were obtained in [START_REF] Di Nezza | Uniqueness and short time regularity of the weak Kähler-Ricci flow[END_REF], where the authors succeeded to run the Kähler-Ricci flow from an initial current with "mild" condition.

Coming back to the (η-twisted) Chern-Ricci flow, Tô [START_REF] Tô | Regularizing properties of complex Monge-Ampère flows II : Hermitian manifolds[END_REF] succeeded to run the flow (5.2.1) starting from an initial current T 0 with zero Lelong numbers, extending his own work [START_REF] Tô | Regularizing properties of complex Monge-Ampère flows[END_REF] (also of [START_REF] Guedj | Regularizing properties of the twisted Kähler-Ricci flow[END_REF]) to the Hermitian case. Moreover, the evolved currents are smooth Hermitian metrics away from initial one.

We would like to show that for arbitrary initial positive currents (with "mild" condition) which might have positive Lelong numbers, one can still define and continue the weak Chern-Ricci