Vincent Guedj

Henri Guenancia

Ahmed Zeriahi

currents. Ahmed Zeriahi Strict Positivity Of Kähler-Einstein

Strict positivity of Kähler-Einstein currents

published or not. The documents may come

Introduction

Introducting the main problem. -Einstein metrics are a central object in differential geometry. A Kähler-Einstein metric on a complex manifold is a Kähler metric whose Ricci curvature is proportional to the metric tensor. A foundational result of Yau [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I[END_REF] allows one to construct many examples of these fundamental objects.

In connection with the Minimal Model Program, singular Kähler-Einstein metrics on mildly singular Kähler varieties X have been constructed in [EGZ09, BG14, BBE + 19] and further studied by many authors (see [START_REF]Degenerate complex Monge-Ampère equations[END_REF][START_REF] Boucksom | Variational and non-archimedean aspects of the Yau-Tian-Donaldson conjecture[END_REF][START_REF] Donaldson | Some recent developments in Kähler geometry and exceptional holonomy[END_REF][START_REF] Li - | G-uniform stability and Kähler-Einstein metrics on Fano varieties[END_REF] and the references therein). These are Kähler forms ω KE = ω + dd c ϕ KE on the regular part X reg of X, where c 1 (X) = λ[ω] is proportional to a reference Kähler class [ω], such that Ric(ω KE ) = λω KE , and which admit bounded (or mildly unbounded) local potential near the singularities X sing . In particular ω KE uniquely extends as a positive closed current to X.

One constructs ω KE = ω + dd c ϕ KE by solving a complex Monge-Ampère equation

(KE) (ω + dd c ϕ KE ) n = e -λϕ KE µ X ,
where n = dim C X and µ X is an appropriate volume form. Due to the presence of singularities, the geometry of these Kähler-Einstein currents is quite mysterious despite recent important progress [START_REF] Donaldson | Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry[END_REF][START_REF] Hein | Calabi-Yau manifolds with isolated conical singularities[END_REF]. Understanding the asymptotic behavior of ϕ KE near X sing is a major open problem.

In this note we partially address the following basic question:

Main Question. -Is the Kähler-Einstein metric ω KE solving (KE) a Kähler current, i.e. does ω KE dominate a Kähler form ?

Note that unless X is smooth, ω KE is never dominated by a Kähler form, cf Remark 1.6.

Known cases.

Although the question sounds much easier than asking for asymptotics of ω KE , it has not yet been addressed in full generality and only a handful of particular cases seem to be understood, which we briefly survey below.

• Orbifold singularities. If X has only finite quotient singularities (i.e. X is an orbifold) then ω KE is a smooth orbifold Kähler metric (i.e. it is a Kähler metric in the local smooth uniformizing charts). In particular, ω KE is a Kähler current. Since 2-dimensional log terminal singularities are quotient singularities, it follows that the main Question admits a positive answer in dimension two for log terminal singularities. More generally, ω KE is orbifold-smooth near any quotient singularity of X at least when X is projective [START_REF] Li | Orbifold regularity of weak Kähler-Einstein metrics[END_REF]; in particular it is a Kähler current on the orbifold locus of X.

• Limits of smooth spaces. Other examples include when X admits a crepant resolution, or when X can be suitably obtained as limit of smooth Kähler-Einstein manifolds [RZ11b, DS14, HS17]. The common thread in the situations appearing in loc. cit. is that one can embed ι : X → P N with ω KE = ω FS | X + dd c ϕ and approximate (X, ω KE ) by a sequence of compact Kähler manifolds (X k , ω k ) such that there are -embeddings ι k :

X k → P N with ω k = ω FS | ι k (X k ) + dd c ϕ k and ϕ k L ∞ (X k ) C,
-uniform Ricci lower bounds Ric ω k -Cω k for some uniform constant C > 0. From these estimates, an easy application of Chern-Lu formula yields ω k C -1 ω FS | ι k (X k ) from which the strict positivity of ω KE follows.

In summary, the ideas above allow to treat the case of singular Kähler-Einstein varieties that are degenerations of smooth Kähler manifolds with uniform lower Ricci bound and uniform L ∞ bound for its potential.

Unfortunately, general singular Kähler-Einstein metrics ω KE on a singular space X arise by construction as limits of smooth Kähler metrics ω ε on a desingularization π : X → X such that Ric ω ε → -∞ along any divisor E ⊂ X with positive discrepancy. Moreover, a general singular variety cannot be obtained as a degeneration of smooth varieties, since there might be (even local and topological) obstructions to smoothability. This prevents one from directly applying the above ideas to a general singular Kähler-Einstein space. This paper grew out as an attempt to find the largest possible field of application of the general technique recalled above. More precisely, the goal of the present paper is threefold:

A. Formulate a general framework where the above global strategy applies mutatis mutandis, including non-projective Kähler spaces as well as singular pairs. B. Provide a local version of the approach that enables to treat isolated smoothable singularities. C. Develop a systematic use of the technique in order to enlarge the class of singularities (beyond the smoothable or crepant ones) for which one can answer positively the Main Question above, using a several step degeneration process. So far, most (general) results about singular Kähler-Einstein metrics have been derived by establishing uniform a priori estimates on smooth approximants (space and metric). The novelty of our approach is that given a singular space/metric (X, ω KE ), we are able to answer the Main Question positively for (X, ω KE ) as soon as one can suitably approximate our space/metric by a possibly singular space/metric on which we can qualitatively answer the Main Question, as long as we have a lower bound on the Ricci curvature and a uniform bound on a suitable potential. We refer to the last paragraph of this introduction or § 5 for an explicit application of this principle.

Statement of the results. -Let us now get a bit more explicit and expand what we mean by the above stated goals. Our first main result is as follows:

Theorem A. -Let X be a normal compact Kähler space with log terminal singularities such that K X ∼ Q O X , and let α be a Kähler class. If X is smoothable then the unique singular Ricci-flat metric ω KE ∈ α dominates a Kähler form.

This strict positivity result actually holds in more general contexts (canonically polarized varieties, Q-Fano varieties, klt pairs) as we explain in Theorem 2.8. The existence of a global smoothing is a rather restrictive assumption, although it holds e.g. at the boundary of the moduli space of positively curved Kähler-Einstein metrics. The latter has a natural compactification arising from the Gromov-Hausdorff topology, and the geometric meaning of the boundary points was elucidated in [START_REF] Donaldson | Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry[END_REF][START_REF] Spotti | Existence and deformations of Kähler-Einstein metrics on smoothable Q-Fano varieties[END_REF].

An isolated singularity is more likely to admit a local smoothing. Shifting perspective we use the local singular theory developed in [START_REF] Guedj | Continuity of singular Kähler-Einstein potentials[END_REF] to establish a positivity result of solutions to local Monge-Ampère equations at isolated smoothable singularities. As a consequence we obtain our second main result. Theorem B. -Let X be a normal compact Kähler space with log terminal singularities such that K X ∼ Q O X , and let α be a Kähler class. The unique singular Ricci-flat metric ω KE ∈ α dominates a Kähler form near any smoothable isolated singularity.

The result is more general and we refer the reader to Theorem 4.1 and Corollary 4.6 for more precise results, including the case of klt pairs with isolated singularities.

As recalled earlier, the basic idea is not new. Deforming X in a smooth Kähler approximant X t , we would like to use Chern-Lu formula [START_REF] Chern | On holomorphic mappings of hermitian manifolds of the same dimension[END_REF][START_REF] Lu | On holomorphic mappings of complex manifolds[END_REF] and establish a uniform lower bound for smooth approximants ω KE,t

C -1 ω t on nearby smooth fibers. This requires to establish uniform a priori bounds for families of degenerate complex Monge-Ampère potentials, a theme which has known important progress in the last decade (see [RZ11a, SSY16, DGG20]), but which still requires further understanding in order to extend Theorem B to the positively curved setting.

Our last result answers the Main Question positively in dimension 3, in non-positive curvature, unconditionally to any smoothability assumptions on the singularities.

Theorem C. -Let X be a normal compact Kähler space of dimension three with log terminal singularities such that K X ∼ Q O X , and let α be a Kähler class. The unique singular Ricci-flat metric ω KE ∈ α dominates a Kähler form.

Here again, there is a version of this statement in negative curvature, cf Theorem 5.3, which surprizingly requires significantly more work than the Calabi-Yau case, cf below. The general idea behind the proof of Theorem C is that one can reduce the situation to the smooth case via a two step degeneration. More precisely it goes as follows.

Step 1. One can reduce to the case where X has canonical singularities, up to passing to the index one cover.

Step 2. The first degeneration amounts to considering a terminalization π : X → X. Since X is canonical, π is crepant, i.e. K X = π * K X is trivial. Now, one can realize π * ω KE as limit of singular Ricci flat metrics ω ε ∈ π * α + εβ where β is a Kähler class on X.

Step 3. Now, X has terminal singularities of index one, hence these are (locally) smoothable by a classification result of Reid and we can apply Theorem B to ( X, ω ε ) and use Chern-Lu inequality on the singular space X to conclude.

In the case where K X is ample, the terminalization map is not crepant anymore. Instead, a boundary divisor ∆ arises on X so that K X + ∆ = π * K X and we are then required to generalize Theorem B to the case of klt pairs. This is not as innocuous as it may sound since even though X has only isolated singularities, it is not the case for the pair ( X, ∆) anymore! Taking care of this difficulty involves an additional (third) degeneration process (cf. Theorem 4.5), highlighting the guiding principle of this article.

Contents. -We recall basic facts from analysis on complex spaces in Section 1. We prove Theorem A in Section 2, using uniform a priori estimates from [SSY16, DGG20]. We study holomorphic families of Dirichlet problems for the complex Monge-Ampère equation in Section 3, using a priori estimates from [GL10, GGZ23], and prove Theorem B in Section 4. We use previous techniques, together with some classical facts from the MMP in dimension 3, to establish Theorem C in Section 5. We gather in the Appendix family versions of known estimates that are needed to establish our main results.

The Monge-Ampère operator on complex spaces

In this section we let X be a reduced complex analytic space of pure dimension n 1. We will denote by X reg the complex manifold of regular points of X. The set X sing := X \ X reg of singular points is an analytic subset of X of complex codimension 1.

1.1. Plurisubharmonic functions. -By definition for each point x 0 ∈ X there exists a neighborhood U of x 0 and a local embedding j : U → C N onto an analytic subset of C N for some N 1.

Using these local embeddings, it is possible to define the spaces of smooth forms of given degree on X. The notion of currents on X is then defined by duality by their action on compactly supported smooth forms on X. The operators ∂ and ∂, d, d c and dd c are then well defined by duality (see [START_REF] Demailly | Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines[END_REF] for a careful treatment).

In the same way one can define the analytic notions of holomorphic and plurisubharmonic functions. There are essentially two different notions :

Definition 1.1. -Let u : X -→ R ∪ {-∞} be a given function.
1. We say that u is plurisubharmonic on X if it is locally the restriction of a plurisubharmonic function on a local embedding of X onto an analytic subset of C N .

2. We say that u is weakly plurisubharmonic on X if u is locally bounded from above on X and its restriction to the complex manifold X reg is plurisubharmonic.

Fornaess and Narasimhan proved in [START_REF] Fornaess | The Levi problem on complex spaces with singularities[END_REF] that u is plurisubarmonic on X if and only if for any analytic disc h : D -→ X, the restriction u • h is subharmonic on D or identically -∞.

If u is weakly plurisubharmonic on X, u is plurisubharmonic on X reg , hence upper semi-continuous on X reg . Since no assumption is made on u at singular points, it is natural to extend u to X by the following formula :

(1.1) u * (x) := lim sup

X reg y→x u(y), x ∈ X.
The function u * is upper semi-continuous, locally integrable on X and satisfies dd c u * 0 in the sense of currents on X. By Demailly [START_REF] Demailly | Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines[END_REF], the two notions are equivalent when X is locally irreducible. More precisely we will need the following result :

Theorem 1.2. -[Dem85]
Assume that X is a locally irreducible analytic space and u : X -→ R ∪ {-∞} is a weakly plurisubharmonic function on X, then the function u * defined by (1.1) is plurisubharmonic on X.

Observe that since u is plurisubharmonic on X reg , we have u * = u on X reg . Hence u * is the upper semi-continuous extension of u| X reg to X.

Following [START_REF] Fornaess | The Levi problem on complex spaces with singularities[END_REF] we say that X is Stein if it admits a C 2 -smooth strongly plurisubharmonic exhaustion. We will use the following definition :

Definition 1.3. -A domain Ω
X is strongly pseudoconvex if it admits a negative C 2 -smooth strongly plurisubharmonic exhaustion, i.e. a function ρ strongly plurisubharmonic in a neighborhood Ω of Ω such that Ω := {x ∈ Ω ; ρ(x) < 0} and

Ω c := {x ∈ Ω ; ρ(x) < c} Ω
is relatively compact for any c < 0.

Our complex spaces will be assumed to be reduced, locally irreducible of dimension n 1. We denote by PSH(X) the set of plurisubharmonic functions on X.

1.2. Dirichlet problem on singular complex spaces. -The complex Monge-Ampère measure (dd c u) n of a smooth psh function in a domain of C n is the Radon measure

(dd c u) n = c det ∂ 2 u ∂z i ∂z j dV eucl ,
where c > 0 is a normalizing constant. The definition has been extended to any bounded psh function by Bedford-Taylor, who laid down the foundations of pluripotential theory in [START_REF] Bedford | The Dirichlet problem for a complex Monge-Ampère equation[END_REF][START_REF]A new capacity for plurisubharmonic functions[END_REF].

The Dirichlet problem for the complex Monge-Ampère operator has been studied extensively by many authors (see [START_REF]Degenerate complex Monge-Ampère equations[END_REF] and the references therein).

The complex Monge-Ampère operator has been defined and studied on complex spaces by Bedford in [START_REF] Bedford | The operator (dd c ) n on complex spaces[END_REF] and Demailly in [START_REF] Demailly | Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines[END_REF]. It turns out that if u ∈ PSH(X) ∩ L ∞ loc (X), the Monge-Ampère measure (dd c u) n is well defined on X reg and can be extended to X as a Borel measure with zero mass on X sing . Thus all standard properties of the complex Monge-Ampère operator acting on PSH(X) ∩ L ∞ loc (X) extend to this setting (see [START_REF] Bedford | The operator (dd c ) n on complex spaces[END_REF][START_REF] Demailly | Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines[END_REF]).

The Dirichlet problem has been studied only recently in that context. We recall the following which is a combination of [START_REF] Guedj | Continuity of singular Kähler-Einstein potentials[END_REF] and [START_REF] Datar | Kähler-Einstein metrics near an isolated logcanonical singularity[END_REF][START_REF] Fu | Dirichlet problem for complex Monge-Ampère equation near an isolated KLT singularity[END_REF].

Theorem 1.4. -Let X be a Stein space of dimension n 1, reduced and locally irreducible, with an isolated log terminal singularity X sing = {p}. Let Ω ⊂ X be a bounded strongly pseudoconvex domain with smooth boundary. Fix a smooth volume form dV on X, φ ∈ C ∞ (∂Ω) and 0 < f ∈ C ∞ (Ω \ {p}) with f ∈ L p (Ω, dV) for some p > 1. Fix λ ∈ R + . Then there exists a unique plurisubharmonic function u in Ω wich satisfies the following:

• u is continuous on Ω with u |∂Ω = φ; • u is smooth in Ω \ {p} • u satisfies (dd c u) n = e λu f dV.
As shown further in [START_REF] Guedj | Continuity of singular Kähler-Einstein potentials[END_REF], one has a uniform a priori bound on u L ∞ (Ω) which depends on n, p > 1 and f L p (Ω) . This bound only weakly depends on the geometry of Ω → X, as we indicate in Theorem 3.6 so as to establish a uniform family version of this estimate.

1.3. Canonical measure of a Q-Gorenstein germ. -Let (X, x) be a germ of normal complex space of dimension n such that mK X is Cartier for some integer m 1. If σ is a trivialization of mK X over X reg , then the expression

i n 2 (σ ∧ σ) 1 m
defines a positive measure µ X,σ on X reg ; we still denote by µ X,σ its trivial extension to X. If τ is a trivialization of m K X , then there exists g ∈ O X (X) * such that µ X,τ = |g| 2 µ X,σ so that the qualitative behavior of the measure on the singular germ does not depend on the choice of σ. In the following, one will just write µ X for µ X,σ .

Let ω be a smooth hermitian form on X, restriction of a smooth hermitian form under an embedding (X, x) → C N . We denote by f the density of µ X with respect to ω n , that is, µ X = f ω n . In the following, the L p spaces are considered with respect to ω n .

Recall that (X, x) has log terminal singularities if and only if µ X has finite mass, i.e. f ∈ L 1 . It is standard to see that the latter condition is actually equivalent to having f ∈ L p for some p > 1, cf e.g. [START_REF]Singular Kähler-Einstein metrics[END_REF]Lemma 6.4].

The following result shows that the measure µ X has a singular density with respect to a smooth volume form unless (X, x) is smooth, compare [START_REF] Li | On Skoda's theorem for Nadel-Lebesgue multiplier ideal sheaves on singular complex spaces and regularity of weak Kähler-Einstein metrics[END_REF].

Lemma 1.5. -Let (X, x) be a germ of normal complex space of dimension n such that mK X is Q-Cartier and let ω be a smooth hermitian metric. Let f be the density of µ X with respect to ω n , i.e. µ X = f ω n . We have f ∈ L ∞ ⇐⇒ (X, x) is smooth.

Proof. -In order to lighten the notation, we will assume that m = 1. The proof of the general case is essentially identical. One direction of the lemma is obvious, so we need to show that boundedness of the density implies smoothness of the germ. We can assume that we have an embedding X → C N such that for all n-tuple I ⊂ {1, . . . , N}, the linear projections p I : X → C n are finite maps. Since ω and ω C N are equivalent on X, there is no harm assuming that ω is the restriction of the euclidean metric to X.

Given any I, there exists a holomorphic function f I on X reg such that dz I | X reg = f I σ. By normality of X, f I extends to a holomorphic function on X. Moreover, we have f = (∑ I | f I | 2 ) -1 up to some positive constant. If f is bounded on X, then there exists I such that f I (x) = 0. In particular, f I is non-vanishing on the germ (X, x), and dz I | X reg is a trivialization of K X reg . In other words, the map p I : X → C n is étale on X reg , hence everywhere by purity of the branch locus. This shows that (X, x) is smooth.

Remark 1.6. -Lemma 1.5 implies that given a Kähler-Einstein metric ω KE on a Stein or compact Kähler space (X, ω) with log terminal singularities, the inequality ω KE ω on X implies that X is smooth.

Remark 1.7. -Lemma 1.5 extends immediately to the setting of log pairs (X, D). Indeed, if D = ∅, let us write D = ∑ a i D i with a i = 0 and work at a general point y ∈ D i where both X and D are smooth so that D i = (g i = 0) for some holomorphic function g i defined near y. In a neighborhood of y, the density of µ (X,D) looks like |g i | -2a i hence it is not bounded.

Global smoothing

In this section we show that singular Kähler-Einstein metrics are Kähler currents when the variety X admits a global smoothing.

2.1. Kähler-Einstein currents. -Let X be a Kähler normal compact space. The study of complex Monge-Ampère equations in this context has been initiated in [START_REF]Singular Kähler-Einstein metrics[END_REF], providing a way of constructing singular Kähler-Einstein metrics and extending Yau's fundamental solution to the Calabi conjecture [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I[END_REF]. More precisely, it is proven there that given a Kähler metric ω on X, a non-negative number λ ∈ {0, 1} and a nonnegative function f ∈ L p (X) for some p > 1 (satisfying X f ω n = X ω n if λ = 0), then the equation

(2.1) (ω + dd c ϕ) n = f e λϕ • ω n has a unique solution ϕ ∈ PSH(X, ω) ∩ L ∞ (X) (with the additional normalization sup X ϕ = 0 if λ = 0).
Let us now explain the relation between the equation (2.1) above and the existence of singular Kähler-Einstein metrics.

We choose a pair (X, D) consisting of an n-dimensional compact Kähler variety X and a divisor D = ∑ a i D i with a i ∈ [0, 1] ∩ Q. We assume that there exists an integer m 1 such that m(K X + D) is a line bundle. More precisely, we mean by this that the reflexive hull of the coherent sheaf det(Ω 1 X ) ⊗ O X (D) ⊗m is locally free.

Given a hermitian metric h on K X + D and the singular metric e -φ D on X reg (unique up to a positive multiple), one can construct a measure µ (X,D),h on X as follows. If U is any open set where m(K X + D) admits a trivialization σ on U reg , then the expression

(σ ∧ σ) 1 m |σ| 2 m h ⊗m e -φ D
defines a measure on U reg which is independent of m as well as the choice of σ and can thus be patched to a measure on X reg . Its extension by 0 on X sing is by definition µ (X,D),h . We recall the following properties satisfied by the measure µ := µ (X,D),h , cf [START_REF]Singular Kähler-Einstein metrics[END_REF]Lemma 6.4].

-The Ricci curvature of µ on X reg is equal to -iΘ(h) + [D].

-The mass X dµ is finite if and only if (X, D) has klt singularities.

-If µ has finite mass, then the density

f of µ wrt ω n (i.e. µ = f • ω n ) satisfies f ∈ L p (X) for some p > 1.
From now on, we work in the following Setup 2.1. -Let (X, D) be a pair where X is a compact normal Kähler space and D is an effective Q-divisor. Assume that (X, D) has klt singularities, pick a Kähler metric ω and a hermitian metric h on K X + D, normalized so that X dµ (X,D),h = X ω n . We assume either • K X + D is ample and ω = iΘ(h); or • K X + D ≡ 0 and h satisfies iΘ(h) = 0; or else • K X + D is anti-ample and ω = -iΘ(h). where λ = 1, 0 or -1 according to whether we are in the first, second of third case. It satisfies

(2.3) Ric(ω KE ) = -λω KE + [D]
in the weak sense.

By the results [EGZ09] recalled above, (2.2) admits a unique solution ω KE whenever λ ∈ {0, 1}. Its potential ϕ KE is globally bounded on X and ω KE is a honest Kähler-Einstein metric on X reg \ Supp(D) and it has cone singularities along D generically [START_REF] Guenancia | Kähler-Einstein metrics with cone singularities on klt pairs[END_REF]. For λ = -1 we refer the reader to [BBE + 19, Bou18].

Kähler-Einstein theory in positive curvature is notoriously more complicated than in non-positive curvature. For that reason, we will make additional assumptions when working in the log Fano case and introduce the following Setup 2.3. -In the Setup 2.1 and in the case where -(K X + D) is ample, we assume additionally that

• There exists a Kähler-Einstein metric ω KE ,

• Aut • (X, D) = 0.
The assumption on the automorphism group is to ensure uniqueness of the KE metric. Actually, assuming existence of a KE metric, its uniqueness is equivalent to the discreteness of Aut(X, D).

In summary, in Setup 2.3 above, there exists in each three cases a unique Kähler-

Einstein metric ω KE ∈ [ω]. This is by [EGZ09] if K X + D 0 and by [BBE + 19, Theorem 5.1] if K X + D < 0.

Kähler currents. -Let us start by recalling the following terminology.

Definition 2.4. -Let (X, ω) be a compact Kähler space, and let T be a closed, positive (1, 1)-current. We say that T is a Kähler current if there exists ε > 0 such that the inequality T εω holds globally on X in the sense of currents.

In our main case of interest, T will have local potentials, i.e. there exist a finite open covering X = ∪ α∈I U α and functions u α ∈ PSH(U α ) such that T| U α = dd c u α . Up to refining the cover, one can assume that ω| U α = dd c ϕ α for some strictly psh functions ϕ α on U α . Then T is a Kähler current if and only if there exists ε > 0 such that for all α ∈ I, one has u αεϕ α ∈ PSH(U α ).

In the following, we will repeatedly use the following classical result: Lemma 2.5. -Let X be a normal complex space and let T be a closed (1, 1)-current on X admitting locally bounded potentials. Assume that there exists a closed analytic set Z X such that T| X\Z 0. Then, T 0 everywhere on X.

Proof. -The claim is an immediate application of Chern-Levine-Nirenberg inequality that ensures that T puts no mass on pluripolar sets. Alternatively, fix an open neighborhood U of a point x ∈ Z such that T| U = dd c ϕ. One can choose ϕ such that ϕ| U\Z is psh and locally bounded near Z, hence it extends to a psh function ψ on U. Since ϕ = ψ almost everywhere on U, we have T| U = dd c ψ 0. The purpose of this section is to study when ω KE is a Kähler current, by using a smoothability assumption. Our results are inspired by a result due to Ruan and Zhang [RZ11b, Lemma 5.2], and the main tool throughout the paper will be Chern-Lu inequality, which we recall below as it can be found in e.g. [Rub14, Proposition 7.2] Proposition 2.6 (Chern-Lu inequality). -Let X be a complex manifold endowed with two Kähler metrics ω, ω. Assume that there are constants

C 1 , C 2 , C 3 ∈ R such that Ric ω -C 1 ω -C 2 ω, and Bisec ω C 3 .
Then, we have the following inequality on X

∆ ω log tr ω ω -C 1 -(C 2 + 2C 3 ) tr ω ω.
Next, we introduce the following definition Definition 2.7. -Let (X, D) be a klt pair with X compact and let ω be a Kähler metric. We say that (X, D, [ω]) admits a Q-Gorenstein smoothing if there exists a triplet (X , D, [ω X ]) consisting of a normal complex space X , an effective Q-divisor D, and a smooth, (1, 1)-form ω X on X admitting a proper, surjective holomorphic map π : X → D satisfying:

1.

K X /D + D is a Q-line bundle. 2. Every irreducible component of D surjects onto D. 3. (X , D)| π -1 (0) (X, D). 4. For t = 0, (X t , D t ) = (X , D)| π -1 (t) is log smooth. 5. For t ∈ D, ω t := ω X | X t is Kähler and [ω 0 ] = [ω].
Let us make few remarks:

• If K X + D is ample or anti-ample, then so is K X t + D t for t small. Then, the last condition in the definition above is automatic if ω ∈ ±c 1 (K X + D), by using an embedding of

X into P N × D via sections of ±m(K X t + D t ) for m large. • If K X + D is numerically trivial, then so is K X t + D t for t small, cf e.g. [DG18, Lemma 2.12]. • The pair (X t , D t ) is automatically klt for any t ∈ D. Theorem 2.8. -Let (X, D, [ω]) as in Setup 2.3 and assume that (X, D, [ω]) admits a Q- Gorenstein smoothing. Then, the Kähler-Einstein metric ω KE is a Kähler current. That is, there exists C > 0 such that ω KE C -1 ω.
Remark 2.9. -Along the same lines, one can obtain the result above assuming instead that (X, D) admits a crepant resolution, i.e. a proper bimeromorphic map p : X → X such that K X + D = p * (K X + D) where D is the proper transform of D.

Proof. -We consider the smoothing π : (X , D) → D. Up to shrinking D slightly and adding π * dd c |t| 2 to ω X , one can assume that the later form is strictly positive.

Since ω X is the restriction of a smooth positive (1, 1)-form under local embedding X → loc C N , one can assume that the bisectional curvature of ω X | X reg is bounded above by a given constant C 1 , maybe up to shrinking D just a little more. This is because the bisectional curvature decreases when passing to holomorphic submanifolds. By the same argument, we see the bisectional curvature of ω t = ω X | X t is bounded above by C 1 , for any t ∈ D.

For t ∈ D, we consider ω t = ω t + dd c ϕ t the unique Kähler-Einstein metric of the pair (X t , D t ). For t = 0, existence and uniqueness of ω t is due to [START_REF] Kołodziej | The complex Monge-Ampère operator[END_REF] when λ 0 and to [SSY16, LWX19], cf also [START_REF] Pan | Kähler-Einstein metrics on families of Fano varieties[END_REF] when λ < 0. In the latter case, we need t to be small enough. The Kähler-Einstein metric solves

(2.4) (ω t + dd c ϕ t ) n = e λϕ t f t ω n t
for some non-negative function f t on X t satisfying X t f p t ω n t C(p, ω) for some p > 1 and some constant C(p, ω) independent of t ∈ D. This is proved in [DGG20, Lemma 4.4] assuming additionally X 0 has canonical singularities, but the klt case can be proved with minimal changes, cf also (3.6) in the next section.

If λ = 0, we normalize ϕ t by sup X t ϕ t = 0. We claim that up to shrinking D, there exists a constant

C 2 > 0 such that (2.5) ϕ t L ∞ (X t ) C 2 for any t ∈ D. If λ > 0, this is a consequence of [DGG20, Theorem E]. If λ = 0,
this is a consequence of the proof of [DGG20, Theorem 6.1]. Indeed, in loc. cit. the fibers are assumed to be canonical but this is a mostly cosmetic assumption since going from canonical to klt singularities leaves the proof resting on [DGG20, Lemma 4.4] unchanged, as explained above. Finally, if λ < 0, the estimate (2.5) is proved in [SSY16, Proposition 2.23], [LWX19, Theorem 1.2 (iii)] when D is pluri-anticanonical and in [PT23, Theorem 5.10] in general.

Set E := Supp(D). We choose a smooth hermitian metric h on O X (E), a section s of that line bundle cutting out E, and we introduce the function ψ := log |s| 2 h . It satisfies

dd c ψ = -iΘ h (E) on X \ E, hence there exists a constant C 3 > 0 such that (2.6) ψ C 3 and dd c ψ -C 3 ω on X \ E.
We introduce a number ε > 0 (meant to go to zero) and we assume wlog that εC 3 1. We also define v t := log tr ω t ω t , which is smooth on X t \ E and globally bounded (for t = 0). We want to quantify that bound.

On X t \ E, we have Ric ω t ω t and Bisec(X t , ω t ) C 1 . By Chern-Lu inequality, i.e. Proposition 2.6, we get ∆ ω t v t -1 -2C 1 e v t Using the identity ω t = ω t + dd c ϕ t and the inequality (2.6), we get

∆ ω t (v t -Aϕ t + εψ) e v t -C 4 .
where A = 2(C 1 + 1) and C 4 = An + 1. The maximum of the term inside the Laplacian is attained on X t \ E, and an easy application of the maximum principle shows

v t (x) C 5 -εψ(x)
for any x ∈ X t \ E and ε > 0, and where C 5 = log C 4 + 2AC 2 + 1. Passing to the limit when ε → 0, we find (2.7)

ω t C -1 5 ω t on X t \ E, for any t ∈ D * .
Next, we choose a continuous family of smooth maps F t : X reg → X t inducing a diffeomorphism onto their image and such that F 0 = Id X reg . We claim that F * t ω t converges locally smoothly to ω KE on X reg \ E when t → 0. Thanks to (2.7), this would imply that

ω KE C -1
5 ω on X reg \ E hence everywhere on X by Lemma 2.5. Set X • := X reg \ E. In order to show the convergence, we claim successively:

-The family of functions F * t ϕ t is precompact in the C 2 loc (X • )-topology. -Each cluster value ω ∞ = ω + dd c ϕ ∞ of F * t ω t solves the Monge-Ampère equation (ω + dd c ϕ ∞ ) n = e λϕ ∞ f 0 ω n on X • .
-Each cluster value ϕ ∞ is globally bounded on X • , hence its unique ω-psh extension to X solves the equation above on X.

By uniqueness of the Kähler-Einstein metric, it would then follow that ω ∞ = ω KE , concluding the proof of the convergence. Let us briefly justify each of the items above.

The first point follows from the local Laplacian estimate on X • obtained from (2.7). Indeed, once the laplacian estimates are obtained, one can invoke Evans-Krylov and Schauder estimates, since F * t ω t (resp. F * t J t , F * t f t ) converges locally smoothly on X • to ω (resp J, f 0 ) when t → 0.

The second item is an immediate consequence of the first one. As for the last one, it is a consequence of (2.5).

Isolated singularities

3.1. Families of Monge-Ampère equations in a local setting. -Throughout this section, we will work in the following geometric context.

Setup 3.1. -Let X C N be a bounded normal Stein space of dimension n + 1 endowed with a surjective holomorphic map π : X → D such that • X is Q-Gorenstein and K X /D ∼ Q O X .
• For every t ∈ D, the schematic fiber X t = π -1 (t) is irreducible and reduced.

• X 0 has klt singularities. In the following, we fix a basepoint 0 ∈ X 0 .

The first item means that there exists an integer m 1 such that the reflexive hull of K ⊗m X is trivial. Once and for all, we pick a trivialization Ω ∈ H 0 (X , mK X ), that is, Ω| X reg is non-vanishing. We set Ω t := Ω (dπ) ⊗m X t ∈ H 0 (X t , mK X t ); this induces a trivialization of mK X t and we define the measure

µ X t := µ X t ,Ω t = i n 2 (Ω t ∧ Ω t ) 1 m on X t , cf § 1.3.
Recall that X t has klt singularities if and only if µ X t has finite mass on each compact subset of X t . Since X 0 is klt, inversion of adjunction [KM98, Theorem 5.50] (cf also § 3.2.1) shows that X is klt in the neighborhood of X 0 hence for any X X there exists δ > 0 such that X ∩ X t is klt for |t| < δ. We will therefore shrink X so that each X t is relatively compact in a klt space for |t| small enough. Definition 3.2. -We will use the following terminology.

• A holomorphic map π : X → D as in Setup 3.1 is a smoothing of X 0 if π is smooth over D * . • A normal Stein space X is smoothable if there exists a family π : X → D as in Setup 3.1 such that X X 0 = π -1 (0) and X → D is a smoothing of X 0 .

In the geometric context provided by Setup 3.1, one can consider a natural family of Monge-Ampère equations which we now describe, and whose analysis will take up most of this section. We pick a smooth, strictly psh non-positive function ρ on X such that ∂X = {ρ = 0} and set ω = dd c ρ. The restriction ω| X t of ω to the fiber X t will be denoted by ω t . Next, we extend F (resp. h) to a smooth function on X ) (resp. ∂X ) which we still denote by F (resp. h). We also choose a (small) neighborhood V of ∂X and set V t = V ∩ X t . We are interested in the Dirichlet problem, i.e. finding a plurisubharmonic function u t ∈ PSH(X t ) ∩ C 0 (X t ) solution of

(MA t ) (dd c u t ) n = e λu t +F t µ X t on X t u t | ∂X t = h t .
with λ ∈ {0, 1}. If X t is smooth, then the existence and uniqueness of u t is classical and provides a solution which is smooth in X t (see [START_REF] Caffarelli | The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations[END_REF][START_REF] Guan | Complex Monge-Ampère equations and totally real submanifolds[END_REF]). In this more general setting, the existence and uniqueness of u t is provided by [GGZ23, Theorem A] for λ = 0 since X t is klt up to the boundary so that the density of µ X t with respect to ω n t belongs to L p (X t , ω n t ) for some p > 1. The case λ = 1 can be treated along very similar lines.

The main technical contribution of this section is summarized in the following result.

Proposition 3.3. -Let π : X → D be a family as in Setup 3.1 and let u t be the solution of the Monge-Ampère equation (MA t ). We have the following. 1. There exist constants δ, C > 0 such that for any |t| < δ, we have u t L ∞ (X t ) C. 2. If π is smooth outside the basepoint 0 ∈ X 0 , then for any t = 0 the equation (MA t ) admits a smooth subsolution u t such that dd c u t εω t on X t , and

u t C k ( Vt ) C(k)
for any k ∈ N, where ε, C = C(k) > 0 are constant that do not depend on t.

The proof of Proposition 3.3 is quite lengthy and will be provided in § 3.2 below.

Building upon Proposition 3.3, one can prove the following local version of Theorem 2.8, which will also be useful for understanding global problems further along in the paper.

Theorem 3.4. -Let X C N be a normal, connected n-dimensional Stein space with an isolated klt singularity at the origin. Let X X be a strongly pseudoconvex domain containing the origin and such that K

X ∼ Q O X . Consider the solution u ∈ PSH(X) ∩ L ∞ (X) of (MA) (dd c u) n = e λu+F µ X on X u| ∂X = h where F ∈ C ∞ (X), h ∈ C ∞ (∂X) and λ ∈ {0, 1}. If X is smoothable, then dd c u is a Kähler current. More precisely, there exists C > 0 such that dd c u C -1 ω C N | X .
The proof of Theorem 3.4 is provided in § 3.3. Its flavor is very similar to the proof of its global counterpart, but the a priori estimates in the local setting require significantly more work. The latter are quite classical on a fixed manifold and one only needs to check that the arguments can be made to work in families, which we have chosen to do carefully in the Appendix, cf. § 6.

Proof of Proposition 3.3. -

The proof involves a careful analysis of the behavior of the measure µ X t which is achieved using the semi-stable reduction theorem following the lines of [DGG20, § 4]. The two items in the theorem are then proved separately.

3.2.1.

Semi-stable model. -By [START_REF] Kempf | SAINT-DONAT -Toroidal embeddings. I[END_REF], one can find a semi-stable model of π. More precisely, up to shrinking D, there exists a finite cover ϕ : t → t k of the disk for some integer k 1 and a proper, surjective birational morphism g := Y → X := (X × ϕ D) ν from a smooth manifold Y, where ν stands for the normalization, as below

(3.1) Y X X D D p f g h π π
ϕ such that around any point y ∈ p -1 (0), there exists an integer n + 1 and a system of coordinates (z 0 , . . . , z n ) centered at y and such that p(z 0 , . . . ,

z n ) = z 0 • • • z . Moreover, π is smooth (resp. g is étale) away from h -1 (Sing(π))
, and that set has π -relative codimension at least two.

Thanks to the generic smoothness theorem, one can shrink D so that p is smooth away from 0. In particular, the induced morphism f | Y t : Y t → X ϕ(t) is a resolution of singularities for any t = 0, where Y t = p -1 (t). We want to understand the behavior of the measures (or volume forms) µ X t when t → 0. This can be achieved quite explicitly on Y via pull back by f as we now explain.

As we mentionned above, π and π are smooth in codimension one (even in codimension two); this implies that X → X × ϕ D is isomorphic in codimension one, hence X 0 → (X × ϕ D) 0 is finite and generically 1 -1. Next, (X × ϕ D) 0 → X 0 is finite and 1 -1 on points. Therefore, X 0 → X 0 is finite and generically 1 -1, hence it is isomorphic since X 0 is normal. In particular, X 0 is irreducible and has klt singularities.

Next, we write (3.2)

K Y + Y 0 = g * (K X + X 0 ) + ∑ i∈I a i E i
where the E i 's are g-exceptional divisors and Y 0 is the strict transform of X 0 ; it is irreducible. One should observe that some of the divisors E i 's may be irreducible components of p -1 (0) so that the inclusion Y 0 p -1 (0) is strict in general. The divisors E i not included in the fiber p -1 (0) surject onto D since we have assumed that p -1 (t) is irreducible for t = 0. In other words, we have

(3.3) ∀i, g(E i ) ∩ X 0 = ∅.
The divisor E := ∑ i∈I E i is the exceptional locus of g and E + Y 0 has simple normal crossing support. Now restrict (3.2) to each irreducible component of Y 0 and use adjunction to obtain (3.4)

K Y 0 = g * K X 0 + ∑ i∈I a i E i | Y 0 Since E + Y 0 is SNC, g| Y 0 : Y 0 → X 0
is a log resolution of X 0 and we find that

(3.5) ∀i, a i > -1.
Indeed, since X 0 is klt, the inequality above holds for for any i such that E i ∩ Y 0 = ∅. And that set of indices i coincides with the full set I thanks to (3.3) and the connectedness of the fibers of g.

3.2.2.

Analysis of µ t . -Let ω be a fixed Kähler metric on X , and let us define the function γ on X reg = X \ Sing(π) by

i n 2 (Ω ∧ Ω) 1 m = e -γ ω n ∧ dπ ∧ dπ.
The main result in this section is the following:

Lemma 3.5. -Up to shrinking D, there exists p > 1 and a constant C > 0 such that for any t ∈ D, one has

(3.6) X t e -pγ ω n t C.
Proof of Lemma 3.5. -Equation (3.2) can be reformulated by saying that the form f * ( 1 π m Ω) is a holomorphic section of mK Y on Y \ p -1 (0) with a (possibly negative) vanishing order ma i along E i and a pole or order m along Y 0 . Given y ∈ Y 0 , pick a coordinate chart (U, z) centered at y such that p(z 0 , . . . , z n ) = z 0 • • • z for some n + 1. We can relabel the coordinates so that (z 1 , . . . , z n ) are a system of coordinates on Y 0 and U ∩ Y 0 = (z 0 = 0). Note that (z 1 , . . . , z n ) remains a system of coordinates on Y t for |t| small. One can choose an injection {1, . . . , } → I such that U ∩ E i = (z i = 0) for 1 i and E i ∩ U = ∅ for the other indices i ∈ I. On U \ p -1 (0), one can write

(3.7) f * 1 π m Ω = a(z 0 , . . . , z n ) dz 0 z 0 ∧ dz 1 ∧ • • • ∧ dz n ⊗m for some holomorphic function a on U \ E such that b(z) := z ma - 1 1 • • • z ma -
a(z) extends holomorphically across E, where one defines x -:= -min{0, x} for any x ∈ R. Since ma i > -m from (3.5), we get 0 ma - i < m. We have

f * Ω (dπ) ⊗m = f * Ω (dp) ⊗m = f * ( 1 π m Ω) ( dp p ) ⊗m
as well as

dz 0 z 0 ∧ dz 1 ∧ • • • ∧ dz n = dp p ∧ dz 1 ∧ • • • ∧ dz n on U.
Combining those two identities with (3.7) and recalling that we defined Ω t = Ω (dπ) ⊗m | X t , we get on U ∩ Y t :

f * Ω t = a(z)(dz 1 ∧ • • • ∧ dz n ) ⊗m
and therefore there exists C > 0 such that

(3.8) f * µ X t C idz 1 ∧ d z1 ∧ • • • ∧ idz n ∧ d zn ∏ i=1 |z i | 2a - i .
Arguing as in the proof of [RZ11a, Thm. B.1(i)] we can shrink X further so that there exist bounded holomorphic functions (σ 1 , • • • , σ r ) on X satisfying V(σ 1 , . . . , σ r ) ⊂ X sing and (3.9)

γ = log ∑ j |σ j | 2 + O(1)
It follows from (3.9) that there exists a constant A > 0 such that f * γ A log |s E | 2 where s E ∈ H 0 (X , O X (E)) cuts out the exceptional divisor E and | • | is a smooth hermitian metric on O X (E). Next, since f : Y t → X t is generically finite (with degree bounded independently of t), we get for any p = 1 + δ:

X t e -pγ ω n t Y t e -δ f * γ f * (i n 2 Ω t ∧ Ω t ) 1 m Y t |s E | -2δA dµ X t .
Now, one can cover Y t by finitely many open sets U t = U ∩ Y t as above. On U, the system of coordinates (z 0 , . . . , z n ) induces a system of coordinates (z 1 , . . . , z n ) on U t such that we have

|s E | -2δA µ X t C ∏ i=1 |z i | -2(δA+a - i ) idz 1 ∧ d z1 ∧ • • • ∧ idz n ∧ d zn
for some uniform constant C thanks to (3.8). Since U C n+1 , the U t live in a fixed compact subset of C n and the lemma follows by taking δ < 1-max a - i A .

3.2.3.

Proof of the uniform estimate. -The first item of Proposition 3.3 is a consequence of (3.6) and the following more general statement.

Theorem 3.6. -Let π : X → D be a family as in Setup 3.1. Fix p > 1, λ ∈ R + , h ∈ C ∞ (∂X ) and f t ∈ L p (X t ). There exists a unique plurisubharmonic function u

t ∈ PSH(X t ) ∩ C 0 (X t ) solution of (dd c u t ) n = f t e λu t ω n t on X t u t | ∂X t = h t . Moreover u t L ∞ (X t ) C f t 1/n L p (X t )
, where C only depends on p, n and h t L ∞ (∂X t ) .

Proof. -The existence and uniqueness of u t is proved in [GGZ23, Theorem A] when λ = 0; the case λ > 0 can be treated similarly.

The key point here is to show that the solutions u t are uniformly bounded on X t when t ∈ D varies, as soon as the densities f t are uniformly bounded in L p . This follows from the analysis developed in [START_REF] Guedj | Continuity of singular Kähler-Einstein potentials[END_REF], which is an extension of Kolodziej's technique [START_REF] Kołodziej | The complex Monge-Ampère operator[END_REF]. Indeed [GGZ23, Proposition 1.8] (applied with v = 0) shows that u t is globally bounded on X t , while the uniform bound (3.6) allows one to show that this bound is also independent of t. We provide a sketch of the proof as a courtesy to the reader.

Step 1. We claim that there exists m 1 , C 1 1 such that (3.10)

X t exp(-2 -m 1 v t )ω n t C 1 ,
for all t ∈ D and for all v t ∈ F t , where

F t := w ∈ PSH(X t ) ∩ L ∞ (X t ), w |∂X t = 0 and X t (dd c w) n 1 .
Indeed the family F t is relatively compact, and any function w in F t belongs to the domain of definition of the complex Monge-Ampère operator, with zero boundary values and Monge-Ampère mass less than 1. It follows from Demailly's comparison theorem that the Lelong number ν(w, x) is less than 1 at a smooth point, and less than mult(X t , x) -1/n if x is singular.

If p t : X t → X t is the blow up of a (single) smooth subvariety, then ν(w, p t (y)) ν(w • p t , y) 2ν(w, p t (y)) if y belongs to the exceptional set E t of p t , while ν(w • p t , y) = ν(w, p t (y)) otherwise. This can be checked by embedding X t locally in C N and using the explicit expression of the blow up of a smooth subvariety in the euclidean space. We infer that there exists

m 1 ∈ N such that ν(w • f , y) 2 m 1 ν(w, f (y)) 2 m 1 for all y ∈ Y t .
We thus have a uniform control of the Lelong numbers of the compact family f * F t . Using the subextension trick [GGZ23, Lemma 1.7], we further reduce to controlling

X t exp(-2 -m 1 v t )ω n t = Y t exp(-2 -m 1 v t • f )ω n t
on a relatively compact subset X t X t . We finally invoke Skoda's uniform integrability theorem which holds for holomorphic families, see [DGG20, Theorem 2.9].

Step 2. We claim that for all compact subsets K ⊂ X t ,

(3.11) Vol ω t (K) C 1 exp - 1 2 m 1 Cap(K, X t ) 1/n ,
where Cap(K, X t ) := sup{ K (dd c w) n , w ∈ PSH(X t ) with 0 w 1} denotes the Monge-Ampère capacity.

Indeed set λ = Cap(K, X t ) -1/n and v t = λh * K,X t where h * K,X t denotes the relative extremal function of the compact set K (see [GZ17, Definition 4.30]). It follows from [GZ17, Theorem 4.34] that X t (dd c v t ) n = 1 and v t + λ = 0 a.e. on K, hence

Vol ω t (K) = K exp -2 -m [λ + v t ] ω n t exp(-2 -m λ) X t exp -2 -m v t ω n t C 1 exp - 1 2 m 1 Cap(K, X t ) 1/n ,
where the last inequality follows from (3.10).

Step 3. Let Φ denotes the maximal psh extension of h to X : this is the largest psh function in X which lies below h at the boundary. It is uniformly bounded in X , satisfies u t Φ t , and coincides with h at the boundary. We claim that for all s, δ > 0 (3.12)

δ n Cap{u t -Φ t < -s -δ -1} c n,p f t L p (X t ) 2 m 1 q Cap{u t -Φ t < -s -1} 2 ,
where 1/p + 1/q = 1. It follows indeed from [GKZ08, Lemma 1.3] that

δ n Cap{u t -Φ t < -s -δ -1} {u t -Φ t <-s-1}
(dd c u t ) n .

Since (dd c u t ) n = f t ω n t , we can apply Hölder inequality, together with (3.11) and the elementary inequality exp(-x -1/n ) c n,p x 2q , valid for all x > 0, to conclude. Conclusion. It follows from (3.12) that the function g(s) := Cap{u t -Φ t < -s -1} 1/n satisfies δg(s + δ) Bg(s) 2 for all s, δ > 0, with B = c 1/n n,p f t 1/n L p (X t ) 2 -m 1 q/n . We can invoke DeGiorgi's lemma (see [GKZ08, Lemma 1.5] with τ = 1) to conclude that g(s) = 0 for s 4Bg(0). Thus u t Φ t -4Bg(0) -1, which yields a uniform lower bound on u t if we can uniformly bound g(0) fom above.

To estimate g(0) = Cap{u t -Φ t < -1} 1/n we let w t denote the extremal function of the set {u t -Φ t < -1}. Recall that -1 w t 0 hence

w t dd c (Φ t -u t ) n n(-w t )(Φ t -u t ) n-1 dd c u t n(Φ t -u t ) n-1 dd c u t .
Using Stokes theorem n times we thus obtain, following [START_REF] Błocki | Estimates for the complex monge-ampère operator[END_REF],

Cap{u t -Φ t < -1} X t (Φ t -u t ) n (dd c w t ) n n! X t (dd c u t ) n n! f t L p (X t ) Vol ω t (X t ) 1/q ,
which shows that g(0) is uniformly bounded from above by c n f t 1/n L p (X t ) .

3.2.4.

Existence of a suitable subsolution. -The exceptional divisor E of g satisfies that there exist positive rational numbers (b i ) i∈I such that -∑ i∈I b i E i is g-ample, hence fample as well. On each O Y (E i ), one can pick a section s i cutting out E i as well as a smooth hermitian metric h i such that ρ := f * (Aρ) + ∑ i∈i b i log |s i | 2 h i is strictly psh on Y for A 1. From now on, we assume that π is smooth away from our distinguished point 0 ∈ X 0 ⊂ X . This ensures for all t = 0, ρ is bounded on Y t X ϕ(t) , and that ρ is smooth on ∂Y ∂X.

Next, one defines for δ > 0 small enough (to be determined later) the function

v = v δ := ρ + ∑ i∈I |s i | 2δ h i .
Up to scaling the metrics h i , we find that v is strictly psh. More precisely, we can cover Y with finitely many coordinate charts (U α ) α such that

U α ∩ E = (z 1 • • • z α = 0)
for some number α n and in these charts we have

(3.13) dd c v| U α c ω Y | U α + α ∑ k=1 idz k ∧ d zk |z k | 2(1-δ)
for some c > 0 and some fixed Kähler metric ω Y on Y. In particular, it follows from (3.13) above and (3.8) that

(dd c v| Y t ) n c f * µ X t on Y t
for some uniform c > 0. Since v is uniformly bounded above on Y and F is bounded on X , we can scale up v so that

(dd c v| Y t ) n e λv+ f * F t f * µ X t on Y t .
Given t ∈ D * , there exists s such that ϕ(s) = t and we denote by v t the function on X t defined by v| Y s under the identification Y s X t via f . Clearly, v t satisfies (dd c v t ) n e λv t +F t µ X t on X t .

At this point, v t is not a subsolution of (MA t ) because the boundary condition is not satisfied. So we pick a cut-off function χ compactly supported on X and satisfying χ ≡ 1 near 0. Next, we still denote by h an arbitrary smooth extension of h from ∂X to X . Finally, we set u t := χv t + Bρ + (1χ)h. One can easily see that for B large enough, u t is a subsolution of (MA t ). Moreover, it is obvious on the shape of u t that the estimates claimed in the second item of Proposition 3.3 are satisfied.

Proof of Theorem 3.4. -

In this final subsection, we borrow the notation of Setup 3.1 and assume that π is smooth outside of 0 ∈ X 0 . We consider the solution u t of (MA t ).

First, we are going to derive higher order estimates near ∂X t of u t of (MA t ). We will then conclude the proof of Theorem 3.4 by using Chern-Lu inequality as in the proof of Theorem 2.8. Lemma 3.7. -There exists C 1 > 0 such that for all t = 0, u t C 1 (∂X t ) C 1 .

Proof. -Let H be an arbitrary smooth extension of h to X . For A large enough, the function v := Aρ -H is a smooth psh function near X such that v t = -h t on ∂X t . Thus u t + v t is psh in X t with zero boundary values. It follow from the maximum principle that u t + v t 0 in X t . Using the subsolution constructed in the previous subsection we obtain a two-sided bound u t u t -v t , with u t = u t = -v t = h t on ∂X t . The desired uniform C 1 -bound on ∂X t follows.

Once u t C 1 (∂X t ) is under control, one can obtain a global control of u t C 1 (V t ) in a neighborhood V t of ∂X t that avoids the singular point. The proof of the following proposition in given in the Appendix, cf Proposition 6.1. Proposition 3.8. -There exists C 1 > 0 such that for all t = 0, u t C 1 (V t ) C 1 .

In the Appendix, we explain how to derive Laplacian estimates near the boundary from lower order ones, cf. Theorem 6.3. It is then straighforward to obtain the following Proposition 3.9. -There exists C 2 > 0 such that for all t = 0, we have

∆u t L ∞ (∂X t ) C 2 .
Proof. -It follows from Proposition 3.3 and Lemma 3.7 that the assumptions of Theorem 6.3 are met and that we have uniform upper bounds on

u t C 1 (V t ) , v t C 2 (V t ) , f t C 1 (V t ) , h C 4 (∂X t ) , ε -1 , δ -1 . The proposition follows.
End of the proof of Theorem 3.4. -Pick a Kähler metric ω on X and set ω t := ω| X t . Since (dd c u t ) n is uniformly comparable to ω n t in a small neighborhood of ∂X t , the uniform bound ∆u t L ∞ (∂X t ) C 2 actually yields a uniform constant c 2 > 0 such that (3.14) c -1 2 ω t dd c u t c 2 ω t on ∂X t . Indeed, let σ := inf t inf p∈∂X t lim inf z→p,z∈∂X t θ n t (z) ω n t (z) ; we have σ > 0 since π is smooth along ∂X and u t L ∞ (X t ) is uniformly bounded below by Theorem 3.6. Given p ∈ ∂X t , we have lim sup z→p,z∈X t tr ω t θ t (z) C by our boundary Laplacian estimate hence lim sup z→p,z∈X t tr θ t ω t (z) σ -1 C n-1 and (3.14) follows.

Arguing as in the proof of Theorem 2.8, we consider v t = log tr θ t ( ω t ), where θ t = dd c u t and deduce from the Chern-Lu formula that (3.15)

∆ θ t (v t -Au t ) e v t -C 3 ,
for uniform constants A, C 3 > 0. We infer that v t C 4 is uniformly bounded from above. Indeed, either the maximum of the function v t -Au t is reached in X t and the bound follows from (3.15) and the uniform bound on u t or the maximum of v t -Au t is reached on ∂X t and we conclude from (3.14).

Therefore dd c u t C -1 4 ω t in X t for all t = 0. Similarly to what we have done at the end of the proof of Theorem 2.8, we conclude by letting t → 0 that dd c u 0 C -1 4 ω 0 , hence ω KE = dd c u 0 is a Kähler current. The proof of Theorem 3.4 is complete.

Kähler-Einstein currents near isolated smoothable singularities

We now use the previous analysis to establish the strict positivity of singular Kähler-Einstein metrics of non-positive curvature near smoothable isolated singularities.

4.1. The case of klt spaces X. -Theorem 4.1. -Let X be a compact Kähler normal space with klt singularities such that either K X is ample or K X ∼ Q O X . Then a Kähler-Einstein metric ω KE in the sense of Definition 2.2 is a Kähler current near an isolated smoothable singularity of X.

Remark 4.2. -In the case where X is a Q-Fano Kähler-Einstein variety (i.e. -K X is ample), we expect a similar result to hold as well but this requires a better understanding of local families of Kähler-Einstein metrics of positive curvature.

Proof. -We work near an isolated singular point a. We let B denote a small strictly pseudoconvex neighborhood of a in X, isomorphic to the trace of a ball in some local embedding in C N , and let ρ denote a local smooth potential for ω = dd c ρ in B.

Recall from [START_REF]Singular Kähler-Einstein metrics[END_REF][START_REF] Ȃun | Regularity properties of the degenerate Monge-Ampère equations on compact Kähler manifolds[END_REF] that the Kähler-Einstein potential ϕ KE is smooth in B \ {a}. Define λ = 1 if K X is ample, and λ = 0 if K X ≡ 0, and set F = λρ. The local theory recalled in Section 1.2 shows that ψ = ρ + ϕ KE is the unique solution of the Dirichlet problem (dd c w) n = e λw+F µ (X,a),h in B, with w |∂B = ψ |∂B .

We assume that (X, a) is a smoothable singularity in the sense of Definition 3.2 and we let π : X → D denote a smoothing so that B = π -1 (0) and X t = π -1 (t) is smooth for all t = 0. We let h denote a smooth extension of (ρ + ϕ KE ) |∂B to ∂X , and still denote by F a smooth extension of F to X . It follows from Proposition 3.3 that there exists a unique function u t ∈ PSH(X t ) ∩ C 0 (X t ) such that (4.1) (dd c u t ) n = e λu t +F t µ t on X t u t | ∂X t = h t .

together with a uniform bound u t L ∞ (X t ) C 0 . We can thus apply Theorem 3.4 and conclude that dd c u 0 = dd c ψ = ω KE is a Kähler current near a.

The case of klt pairs (X, ∆).

-We would now like to investigate the strict positivity of Kähler-Einstein currents ω KE associated to compact klt pairs (X, ∆) near a smoothable isolated singularity x ∈ X. There are two possibilities.

Case 1. If x /

∈ Supp(∆), then one can find a neighborhood U of x such that ∂U ∩ Supp(∆) = ∅ so that ω KE is smooth on ∂U, and the same arguments used in the proof of Theorem 4.1 will carry over mutatis mutandis to show that ω KE is a Kähler current near x. Case 2. If x ∈ Supp(∆), then as a singularity of the pair (X, ∆), it is not isolated anymore. This reflects on the metric side as well since on the boundary ∂U of a small neighborhood of x, ω KE is not smooth anymore. More precisely ω KE has conic singularities, to be understood in a generalized sense since ∆ may not have snc support near ∂U. Even if ∆ were smooth (or snc) away from x, the local analysis developed so far could not be applied directly and one would have to derive boundary laplacian estimate in this singular conic setting which probably requires a lot of work. Instead we can regularize the conic singularities globally to avoid boundary problems when applying Chern-Lu inequality. This will require us to assume that each component ∆ i of ∆ is Q-Cartier and that any singularity of X is isolated and smoothable.

We will state the main result of this section with a slightly weaker assumption than local smoothability, which will be useful later when we work with threefolds.

Definition 4.3. -We say that a germ of normal complex space (X, x) is Q-smoothable if there exists a finite Galois quasi-étale cover p : Y → X with Y normal and connected such that for all y ∈ p -1 (x), (Y, y) is smoothable in the sense of Definition 3.2.

In the definition above, one can always shrink Y and assume that p -1 (x) is a singleton.

It will be convenient to introduce the following setup.

Setup 4.4. -Let (X, ω X ) be a n-dimensional compact Kähler space endowed with an effective Q-divisor ∆ = ∑ a i ∆ i such that (X, ∆) has klt singularities. We assume that each component ∆ i of ∆ is a Q-Cartier divisor and that X has only Q-smoothable, isolated singularities.

We are now ready to state the main result.

Theorem 4.5. -Let (X, ∆) be as in Setup 4.4 and consider the unique (normalized

) solution ϕ ∈ PSH(X, ω X ) ∩ L ∞ (X) of the Monge-Ampère equation (ω X + dd c ϕ) n = e λϕ+F dµ (X,∆),h
where F ∈ C ∞ (X), h is a smooth hermitian metric on the Q-line bundle K X + ∆ and λ ∈ {0, 1}.

Then, the current ω ϕ = ω X + dd c ϕ is a Kähler current.

As an immediate consequence of the theorem above, we get:

Corollary 4.6. -Let (X, ∆) be as in Setup 4.4 such that K X + ∆ is ample (resp. K X + ∆ ∼ Q O X ). Then the unique Kähler-Einstein metric ω KE (resp. ω KE ∈ {ω X }) solving Ric ω KE = -ω KE + [∆] (resp. Ric ω KE = [∆])
is a Kähler current.

Remark 4.7. -We would like to add two comments on the result above • It is conceivable that one could remove the assumption on the components ∆ i of ∆ being Q-Cartier by considering a Q-factorialization Y → X of X, but it is not completely clear how the smoothability assumption would lift to Y. • There is a noticeable difference between the assumptions of Theorem 4.1 and Corollary 4.6, as in the latter one we need to assume that all singularities are isolated. It has to do with the fact that we deal with isolated singular points that may not be isolated as singularity of the pair (X, ∆) as explained in the beginning of § 4.2, and this requires a subtle combination of local and global methods. If one is only interested in isolated singular points in X \ Supp(∆), then we would not need any global assumptions on the singularities of X elsewhere, cf ibid.

Proof of Theorem 4.5.

-By assumption, one can find an integer m > 0 and sections

s i ∈ H 0 (X, O X (m∆ i )) such that div(s i ) = m∆ i . We pick hermitian metrics h i on the Q-line bundle O X (∆ i ) and define |s i | 2 := |s i | 2 h ⊗m i . Since K X is Q-Cartier (as a difference K X = (K X + ∆) -∆) of Q-Cartier divisors), one can find a metric h X on K X such that h = h X ⊗ i h ⊗a i i .
Setting b i := a i m , one can rewrite the Monge-Ampère equation solved by ϕ as

(ω X + dd c ϕ) n = e λϕ+F dµ X,h X ∏ i |s i | 2b i .
We consider the unique (normalized) solution ϕ ε of the regularized equation

(ω X + dd c ϕ ε ) n = e λϕ ε +F dµ X,h X ∏ i (|s i | 2 + ε 2 ) b i .
and we set ω ε := ω X + dd c ϕ ε . It follows from [START_REF]Singular Kähler-Einstein metrics[END_REF][START_REF] Ȃun | Regularity properties of the degenerate Monge-Ampère equations on compact Kähler manifolds[END_REF] that ω ε is a Kähler form on X reg for any ε > 0, and moreover there is a uniform constant

C 0 > 0 such that (4.2) ϕ ε L ∞ (X) C 0 and ϕ ε -→ ε→0 ϕ in L 1 (X).
thanks to [GZ12, Theorem C] applied to a desingularization of X. Therefore, we can reduce the proof of the theorem to showing that there exists a constant C > 0 independent of ε such that

(4.3) ω ε C -1 ω X on X.
Claim 4.8. -The uniform inequality (4.3) holds if for any ε > 0, one has the qualitative inequality (4.4) ω ε C -1 ε ω X on X, where C ε > 0 is a positive constant that may depend on ε.

Proof of Claim 4.8. -Here again, the key tool is Chern-Lu inequality, Proposition 2.6. The bisectional curvature of ω X is bounded above since ω X can be extended to a Kähler metric in local embeddings in C N , and we need to bound Ric ω ε from below.

A classical computation shows that if s is a holomorphic section cutting out a divisor D and if h D is a smooth hermitian metric on L := O X (D), then

Θ h D (L) + dd c log(|s| 2 h D + ε 2 ) = ε 2 (|s| 2 h D + ε 2 ) 2 • |D s ∧ D s| 2 h D - ε 2 |s| 2 h D + ε 2 • Θ h D (L). In particular, if one choses a constant C D > 0 such that Θ h D (L) C D ω X , then (4.5) Θ h D (L) + dd c log(|s| 2 h D + ε 2 ) -C D ω X . Since Ric ω ε = -λω ε + λω X -dd c F + ∑ i b i Θ h ⊗m i (O X (m∆ i )) + dd c log(|s i | 2 + ε 2 )
and F is smooth (so that its Hessian is bounded with respect to ω X ), inequality (4.5) ensures that one can find

C 1 > 0 such that Ric ω ε = -ω ε -C 1 ω X .
Finally, let ψ ∈ PSH(X, ω X ) be such that (ψ = -∞) = X sing and ψ| X reg ∈ C ∞ (X reg ). For any δ > 0, the smooth quantity log tr ω ε ω X + δψ on X reg tends to -∞ near X sing thanks to our assumption (4.4). In particular, the quantity above achieves its maximum on X reg . Moreover, our curvature estimates coupled with Proposition 2.6 yield a constant C 2 > 0 such that

∆ ω ε [log tr ω ε ω X + δψ] -1 -C 2 tr ω ε ω X on X reg . Since -∆ ω ε ϕ ε -n + tr ω ε ω X , we infer that ∆ ω ε [log tr ω ε ω X + δψ -(C 2 + 1)ϕ ε ] tr ω ε ω X -C 3 where C 3 = -1 + n(C 2 + 1). A classical application of the maximum principle shows that tr ω ε ω X C 4 e δ(sup X ψ-ψ)
with C 4 = C 3 e 2C 0 (C 2 +1) . Passing to the limit when δ → 0, we get

ω ε C -1
4 ω X on X reg , hence everywhere on X by Lemma 2.5. The claim follows.

In order to prove the theorem, we are left to establishing the qualitative estimate (4.4). In order to achieve that, we use a local deformation argument. From now on, ε > 0 is fixed and all subsequent constants are allowed to depend on ε. Since ω ε is smooth on X reg , it is enough to work in small neighborhoods of the finitely many singular points in X. From now on, we pick a small Stein neighborhood U of x admitting a quasi-étale cover p : V → U such that p -1 (x) = {y} is a singleton and (V , y) admits a smoothing V → D whose fibers V t satisfy V 0 V .

One can assume that U is small enough so that ω X | U = dd c ρ for some smooth strictly psh function ρ on U . Next, since p is quasi-étale, we have K V = p * K U and the smooth hermitian metric h U := h X | U on K U pulls back to a smooth hermitian metric

h V on K V satisfying p * (dµ U ,h X ) = dµ V ,h V .
Next, we fix U U strongly pseudoconvex and set V := p -1 (U). The function

v ε := p * (ρ + ϕ ε ) satisfies (dd c v ε ) n = e λv ε +p * (F-λρ) dµ V,h V ∏ i (|p * s i | 2 +ε 2 ) b i on V v ε | ∂V = p * ((ρ + ϕ ε )| ∂U )
hence by Theorem 3.4 the current dd c v ε is a Kähler current. In particular, it dominates a multiple of p * (ω X )| U . Pushing forward, we get that in restriction to U, we have ω ε C -1 ε ω X and the theorem is proved.

Kähler-Einstein currents on threefolds

In this final section, we provide two results in dimension three (zero and negative curvature) ensuring that Kähler-Einstein metrics on a compact klt space are Kähler currents without any extra assumption on the singularities. Although the proofs of the two results follow the same lines, we have chosen to write them separately to highlight the non-trivial simplifications occuring in the zero curvature case (or dually to insist on the consequential additional difficulties popping up in the negative curvature case).

The crucial input specific to dimension three is Reid's classification of terminal singularities:

Theorem 5.1. -([Rei80]
) Let (X, x) be an (isolated) terminal singularity of dimension three such that K X ∼ Z O X . Then (X, x) is a compound du Val singularity. In particular, any terminal singularity of dimension three is Q-smoothable.

Recall that a compound du Val singularity is a hypersurface singularity isomorphic to

( f + tg = 0) ⊂ C 3 × C where f , g ∈ C[x, y, z] are such that ( f = 0) ⊂ C 3 is
( f + tg + s = 0) ⊂ C 3 × C × C,
where the total space is a smooth hypersurface of C 5 .

The second statement follows from the first after considering the quasi-étale index one cover Y → X making K Y Cartier [KM98, Definition 5.19].

Calabi-Yau threefolds with klt singularities. -

Theorem 5.2. -Let (X, ω X ) be a normal compact Kähler space of dimension three with klt singularities such that K

X ∼ Q O X . Then the Kähler-Einstein metric ω KE ∈ [ω X ] is a Kähler current.
Proof. -Let p : Y → X be the quasi-étale index one cover of X. Clearly, p * ω KE is the Kähler-Einstein metric in the Kähler class p * [ω X ], so that if we prove the statement for Y, it will follow for X by push-forward (since any Kähler form on Y dominates a multiple of p * ω X .

From now on, one can assume that K X ∼ Z O X and, in particular, X has canonical singularities. Let us consider a terminalization π : X → X of X, cf [KM98, Theorem 6.23]. The complex space X is Kähler and it has terminal singularities, hence isolated singularities. Moreover, one has K X = π * K X , i.e. π is crepant (but in general, the exceptional locus of π might have codimension one components). This implies that K X ∼ Z O X so that X has smoothable singularities by Theorem 5.1. Moreover, since π is crepant, it is automatically isomorphic over X reg .

We choose ω a Kähler metric on X, and one considers the singular Ricci flat metric

ω ε ∈ [π * ω X + ε ω]. One can write ω ε = π * ω X + ε ω + dd c ϕ ε with sup X ϕ ε = 0, where ϕ ε is a solution of (π * ω X + ε ω + dd c ϕ ε ) n = c ε µ X = c ε f ω n , where c ε = [π * ω X +ε ω] n µ X ( X) and f ∈ L 1+δ ( X, ω n ) with δ > 0, since X has terminal singularities. It follows from the techniques in [EGZ08, DP10] that (5.1) ϕ ε L ∞ ( X) C 0 , and ω ε -→ ε→0 π * ω KE weakly.
where C 0 > 0 is independent of ε. The discerning reader will have noticed that the situation is slightly different from the one in loc. cit. since X is singular. To patch this little gap, one could e.g. appeal to [DGG20, Theorem A] applied to a desingularization of X → X with reference form the pull-back of π * ω X + ε ω to X and this would yield the uniform estimate. As for the stability statement, it is a classical consequence of the uniform estimate: one first gets higher order estimates locally on X reg using Tsuji's trick and then one uses uniqueness of the Kähler-Einstein metric to conclude that the relatively compact family (ϕ ε ) ε>0 has a single cluster value in L 1 ( X) which is nothing but ϕ 0 .

The main point to establish is that there is a constant C > 0 independent of ε such that (5.2)

ω ε C -1 π * ω X
Indeed, passing to the limit when ε → 0 would then imply that π * ω KE C -1 π * ω X and the theorem would follow by pushing forward by π. Now we know that inequality (5.2) holds for any ε > 0 with a constant C = C ε depending a priori from ε, as this is the content of Theorem 4.1. As before, we will use Chern-Lu inequality to make this qualitative control quantitative.

Choose a function ψ ∈ PSH(X, ω X ) be such that (ψ = -∞) = X sing and ψ| X reg ∈ C ∞ (X reg ). For any δ > 0, the smooth quantity log tr ω ε π * ω X + δψ on π -1 (X reg ) ⊂ X \ Exc(π) tends to -∞ near π -1 (X sing ) since ω ε is a Kähler form. In particular, the quantity above achieves its maximum on X reg .

Next, the bisectional curvature of π * ω X is well-defined and bounded above uniformly on π -1 (X reg ), while Ric ω ε = 0. By Chern-Lu inequality (Proposition 2.6) we find a constant C 2 > 0 such that

∆ ω ε [log tr ω ε π * ω X + δψ] -C 2 tr ω ε π * ω X on π -1 (X reg ). Since -dd c ϕ ε = -ω ε + π * ω X + ε ω, we have -∆ ω ε ϕ ε -n + tr ω ε π * ω X , hence ∆ ω ε [log tr ω ε π * ω X + δψ -(C 2 + 1)ϕ ε ] tr ω ε π * ω X -C 3
where C 3 = -1 + n(C 2 + 1). Using (5.1) we obtain in the end

tr ω ε π * ω X C 4 e δ(sup X ψ-ψ)
with C 4 = C 3 e C 0 (C 2 +1) , holding on π -1 (X reg ) for any δ > 0. Equivalently, we have

ω ε
C 4 e -δ(sup X ψ-ψ) π * ω X . After passing to the limit when δ → 0, one can appeal to Lemma 2.5 to obtain (5.2), hence the theorem.

Canonically polarized threefolds with klt singularities. -

Theorem 5.3. -Let X be a normal projective variety of dimension three with klt singularities such that K X is ample. Then the Kähler-Einstein metric ω KE is a Kähler current.

Proof. -The reduction to the terminal case is a bit more involved here. More precisely, we cannot easily assume that X has canonical singularities without resorting to a local argument which would collapse since the singularities of X are not isolated. Instead, we use a Q-factorial terminalization π : X → X whose existence (in any dimension) is guaranteed by [BCHM10, Corollary 1.4.3]. The map π is such that there exists an effective divisor ∆ on X such that K X + ∆ = π * K X and ( X, ∆) is a terminal pair, hence X has isolated Q-smoothable singularities by Theorem 5.1. Moreover, X is Q-factorial.

Next, let ω be a Kähler metric on X and let us consider the twisted Kähler-Einstein metric

ω ε ∈ c 1 (π * K X ) + ε[ ω], i.e. the solution of Ric ω ε = -ω ε + ε ω + [ ∆].
If ω X ∈ c 1 (K X ) is a Kähler metric, say the curvature of an hermitian metric h on K X , one can write where C > 0 is independent of ε. The only difference with the Calabi-Yau case is that we first need to show that there exists C > 0 independent of ε such that

ω ε = π * ω X + ε ω + dd c ϕ ε where ϕ ε is a solution of (π * ω X + ε ω + dd c ϕ ε ) n = e ϕ ε µ ( X, ∆),π * h = e ϕ ε f ω n , with f ∈ L 1+δ ( X, ω n ) with δ > 0 since X
(5.4) sup X ϕ ε C
This is certainly classical, but we recall the argument for the reader's convenience. First, one can assume without loss of generality that µ := µ ( X, ∆),π * h is a probability measure. By Jensen's inequality, we infer that

X ϕ ε dµ log X (π * ω X + ε ω) n C 1
for some C 1 > 0 independent of ε. Let p : X → X be a resolution of singularities and let ω X be a Kähler metric on X. Up to scaling the latter metric, one can assume that p * (π * ω X + ε ω) ω X and p * µ ω n X where the latter follows since X has terminal (hence canonical) singularities. By the usual compactness properties of ω X -psh functions, we find a constant C 2 such that

X (sup X ψ -ψ)ω n X C 2 for any ψ ∈ PSH( X, ω X ). Now, sup X ϕ ε = X (sup X ϕ ε -ϕ ε )dµ + X ϕ ε dµ X (sup X p * ϕ ε -p * ϕ ε )ω n X + C 1 C 1 + C 2 since p * ϕ ε ∈ PSH( X, ω X ). Hence (5.4) follows.
Coming back to proof of the main result, the remaining point to establish is that there is a constant C > 0 independent of ε such that (5.5)

ω ε C -1 π * ω X .
Thanks to Theorem 4.5, inequality (5.5) holds for any ε > 0 with a constant C = C ε depending a priori from ε. In order to make this bound quantitative, the exact same strategy as in the Calabi-Yau case applies.

Appendix: Gradient and Laplacian estimates in families

In this Appendix we provide the proofs of the family version of the gradient estimate (Proposition 6.1) and the Laplacian estimate (Theorem 6.3) as a courtesy to the reader. 6.1. Gradient estimate in families. -Let us consider the solution u t of (MA t ), and recall that f : Y → X be a log resolution of singularities after base change. We recall that Y is smooth and that E := Exc( f ) ⊂ Y is a divisor satisfying E ⊂ Y 0 since we assume that π is smooth outside of 0 ∈ X 0 . Moreover, the map f t : Y t → X t is a finite étale cover of constant degree for any t ∈ D * . Up to picking a connected component of Y t , there is no loss of generality in assuming that the latter map is actually isomorphic.

Recall that the function ρ = f * (Aρ) + ∑ i∈I b i log |s i | 2 h i is strictly psh on Y, where E i = (s i = 0) are the irreducible components of the exceptional locus E of f . Without loss of generality, one can assume that the positive numbers b i satisfy max i b i 1 and that log |s i | 2 h i 0 for any i. Next, one can write

dd c ρ = ω + ∑ b i [E i ] so that ω is a Kähler form on Y. Setting v := f * (u -Aρ) -∑ i b i log |s i | 2 h i ,
we see that the following equation holds on Y t for any t = 0:

(ω t + dd c v t ) n = (dd c f * u t ) n = e λ f * u t + f * F t f * µ t = e λv t +G t ∏ i |s i | 2(a i -b i ) ω n t v t | ∂Y t = ( f * h -∑ b i log |s i | 2 h i )| ∂Y t with G smooth
on Y, and a i < 1 (not necessarily positive) as explained in § 3.2.2. In the following, one sets |s| 2 := ∏ i∈I |s i | 2 h i .

Proposition 6.1. -In Setup 3.1, we assume that π is smooth outside of the basepoint 0 ∈ X 0 and we let u t be the solution of (MA t ). There exist constants C > 0 and N > 0 such that for all t = 0, (6.1)

|s| N ∇ ω t v t L ∞ (Y t ) C.
for some constant C > 0 independent of t.

Proof. -This is a family version of [DFS23, Proposition 2.2], which itself is an extension of previous similar estimates of Blocki [START_REF]A gradient estimate in the Calabi-Yau theorem[END_REF] and Phong-Sturm [START_REF] Phong | On pointwise gradient estimates for the complex Monge-Ampère equation[END_REF].

In order to increase the readability of the proof, we will soon skip the subscripts t appearing in all the quantities involved in our problem, but before we do that, let us emphasize explicitly what are the uniform bounds that we will later rely on. In order to do so, one rewrites the equation solved by v t as (ω t + dd c v t ) n = e λv t + f t ω n t and we have uniform constants C i (independent of t ∈ D * ) such that one has the a priori estimates (6.2)

v t + ∑ i b i log |s i | 2 h i L ∞ (Y t ) C 1 , ∇ ω t v t L ∞ (∂Y t ) C 2 , ∇ ω t f t C 3 |s|
as well as the following inequalities

(6.3) log |s| 2 C 4 , |∇ ω |s| 2 | 2 C 5 , dd c log |s| 2 -C 6 ω, |Bisec ω t | C 7
holding on Y t for all t = 0. The a priori estimates (6.2) have already been established, cf § 3.2.3 and Lemma 3.7. From now on, we drop the subscript t.

Define β := |∇v| 2 ω and α := log βγ(v) + k log |s| 2 , where k ∈ N >0 is number that we will fix later (we will actually see that any number k 2 is suitable) and γ is of the following form

γ(t) = A(t - 1 t + B ),
where A, B > 0 are constants to be chosen later. We should however observe already that the first estimate in (6.2) shows that for B := C 1 + 1, one has (6.4)

-A(C 1 + 1) γ(v) A(-log |s| 2 + C 1 )
as well as (6.5) A γ (v) 2A, and

-γ (v) 2A (2B -log |s| 2 ) 3 .
The function α attains its maximum at a point x 0 ∈ Y t . We can assume that x 0 ∈ Y t , otherwise we are done by (6.2). We choose normal complex coordinates (z 1 , • • • , z n ) centred at x 0 so that ω x 0 = ∑ dz j ∧ d zj and θ :

= ω + dd c v = √ -1 ∑ i,j θ i jdz i ∧ d zj is diagonal at x 0 i.e. θ i j(x 0 ) = δ ij θ i ī.
By [Bło09, eq. (1.14)] (cf. also [PS12, eq. (2.7)]) we have

∆ θ (log β -γ(v)) = 1 β |∇∇v| 2 ω,θ + |∇∇v| 2 ω,θ + 2 (∂ p (λv + f ) ∂ p v) + 1 β θ k l ∂ p vR p k l m∂ mv - 1 β 2 |∇β| 2 θ -γ (v)|∇v| 2 θ -γ (v)∆ θ v.
We have the following easy estimates

• θ k ¯ ∂ p vR p k ¯ l m∂ mv (inf Bisec ω ) θ k ¯ ω k ¯ ω p m∂ p v∂ mv -C 7 β tr θ ω, • ∂ p λv ∂ p v = λβ, • |∂ p f ∂ p v| |∇ ω f | β C 3 β|s| -1 , • -∆ θ v = -n + tr θ ω.
On the other hand, since ∆ θ log |s| 2 -C 6 tr θ ω, it follows that

∆ θ α ∆ θ (log β -γ(v)) -kC 6 tr θ ω.
All in all, we get

∆ θ α |∇∇v| 2 ω,θ β - ∇β β 2 θ -γ (v)|∇v| 2 θ (6.6) - 2C 3 β|s| + (γ -kC 6 -C 7 )tr θ ω -nγ
We now deal with the first summand of the RHS of (6.6). At the point x 0 ∈ Y t where α achieves its maximum, we have the critical equation

∇β β -γ (v)∇v + k ∇|s| 2 |s| 2 = 0.
On the other hand, by [Bło09, eq. (1.15)] (cf. also [PS12, eq. (2.12)]) we have,

1 β |∇∇v| 2 ω,θ ∇β β - 1 β ∇v • h + 1 β ∇v 2 θ ,
where h is the smooth section of End(T Y t ) defined by θ via the metric ω i.e. locally h i j = ω i ¯ θ¯ j . Set

a := γ (v)∇v -k ∇|s| 2 |s| 2 = at x 0 ∇β β , and b := 1 β (∇v -∇v • h).
Then at the point x 0 we have 1

β |∇∇v| 2 ω,θ |a + b| 2 θ |a| 2 θ + 2 a, b θ (6.7) = ∇β β 2 θ + 2 a, b θ •
Now we analyze the term a, b θ . At the point x 0 we have

• ∇v, ∇v • h θ = θ i j∇ i v∇¯ v ω ¯ p θ p j = ω p p∇ p v∇ pv = β • ∇|s| 2 , ∇v • h θ = θ i j∇ i |s| 2 ∇¯ v ω ¯ p θ p j = θ i ī∇ i |s| 2 ∇¯iv θ i ī = ∇|s| 2 , ∇v ω
Therefore we conclude that at x 0 , we have

a, b θ = γ (v) β |∇v| 2 θ -β + (6.8) + k β|s| 2 ∇|s| 2 , ∇v θ -∇|s| 2 , ∇v ω Observe that ∇|s| 2 2 θ = θ i ī ∇ i |s| 2 2 |∇ ω |s| 2 | 2
tr θ ω C 5 tr θ ω, so that (6.9) 2 ∇|s| 2 , ∇v θ C 5 tr θ ω + |∇v| 2 θ . By Cauchy-Schwarz inequality again, we have 2

β|s| 2 ∇|s| 2 , ∇v ω ∇|s| 2 2 ω + 1 (β|s| 2 ) 2 |∇v| 2 ω (6.10) C 5 + 1 β|s| 4 .
Combining (6.7)-(6.8) with (6.9)-(6.10), we get at x 0 (6.11) 1

β |∇∇v| 2 ω,θ - ∇β β 2 θ -γ (v) - k β|s| 2 C 5 tr θ ω + |∇v| 2 θ -k(C 5 + 1 β|s| 4 ).
On can now put together all our estimates. More precisely, if we combine (6.6) with (6.11), the maximum principle yields the following inequality at

x 0 0 γ (v) -(kC 6 + C 7 ) - kC 5 β|s| 2 tr θ ω + -γ (v) - k β|s| 2 |∇v| 2 θ -(n + 1)γ (v) + k C 5 + 1 β|s| 4 + 2C 3 β|s| .
Claim 6.2. -Given some number 1 p k, one can assume that at the point x 0 we have

β(x 0 )|s(x 0 )| 2p 1. Proof of Claim 6.2. -Assume that β(x 0 )|s(x 0 )| 2p
1, and consider the function G := log β + N log |s| 2 = α + γ(v) + (Nk) log |s| 2 for some N to fix later. We want to show that for some suitable N, G is uniformly bounded above. We have

G α(x 0 ) + γ(v) + (N -k) log |s| 2 -γ(v)(x 0 ) + (k -p) log |s| 2 (x 0 ) + γ(v) + (N -k) log |s| 2 A(2C 1 + 1) + (k -p) log |s| 2 (x 0 ) + (N -k -A) log |s| 2
thanks to (6.4) and the claim follows by taking N = k + A.

At this point, it is important to assume that k 2 so that one can ensure that β|s| 4 (x 0 ) 1 thanks to the claim. From now on, we fix k := 2 (but any number k 2 would do too). In particular, we have 1 β|s| 2 (x 0 ) 1 and β|s| 2 |s| -2 at x 0 . One chooses

A := max k(C 5 + C 6 ) + C 7 + 1, k • sup x∈[0,1]
x(2Blog x) 3 . Given (6.5), one gets at x 0

-γ (v) - k β|s| 2 2A (2B -log |s| 2 ) 3 -k|s| 2 > A (2B -log |s| 2 ) 3 .
Therefore we have at x 0 :

tr θ ω + A (2B -log |s| 2 ) 3 |∇v| 2 θ C 8 , with C 8 := 2C 3 + k(1 + C 5 ) + 2(n + 1)A.
In particular, we have tr θ ω(x 0 ) C 8 , and |∇v| 2 θ (x 0 ) C 8 (2Blog |s| 2 ) 3 . Since θ n |s| -2 ω n , we infer from the above estimate that tr ω θ(x 0 ) C 9 |s| -2 for some constant C 9 > 0. This implies that at x 0 , one has

β = |∇v| 2 ω C 9 |s| -2 |∇v| 2 θ C 8 C 9 |s| -2 (2B -log |s| 2 ) 3 Set C 10 := sup x∈[0,1] √ x(2B -log x) 3 .
Then at the point x 0 where α achieves his maximum, we have

α(x 0 ) = log β(x 0 ) + k log |s(x 0 )| 2 -γ(v(x 0 )) (k - 3 2 ) log |s(x 0 )| 2 + log(C 8 C 9 C 10 ) + A(C 1 + 1) C 11 .
In the end, we get

log β = α -k log |s| 2 + γ(v) C 11 -(k + A) log |s| 2 + AC 1
which proves the required inequality with N = k + A.

Laplacian estimate in families. -

The following result -in the context of holomorphic families-is a combination of the main results of [CKNS85, GL10].

Theorem 6.3. -In Setup 3.1, we assume that π is smooth outside the basepoint 0 ∈ X 0 . We fix V ⊂ X a small neighborhood of ∂X not containing the singular point 0.

Fix λ ∈ R, ω a Kähler form on X , h ∈ C ∞ (∂X) and f ∈ C ∞ (V) such that inf V f = σ > 0.
(V t ) , v t C 2 (V t ) , f t C 1 (V t ) , h C 4 (∂X t ) , ε -1 , σ -1 .
The proof is an adaptation of [GL10, Section 4] which deals with the case when f is globally smooth on X . We sketch the arguments for the convenience of the reader.

Proof. -We pick a smooth strictly plurisubharmonic non-positive function ρ on X such that X = {ρ < 0} and ∂X = {ρ = 0}.

We proceed in several steps. For any fixed point p ∈ ∂X , we will choose local com-

plex coordinates (z 1 , • • • , z n , • • • , z N ) centered at p (i.e. z(p) = 0) defined in a neighbor- hood V of p in C N such that z N = π -π(p) and V ∩ X = {z ∈ V ; z n+1 = • • • = z N-1 = 0}. Moreover we can arrange so that if z 1 = x 1 + iy 1 , • • • , z n = x n + iy n , ∂/∂x n
is the inner normal to ∂X t at the point p and ω t (p) = √ -1 ∑ j dz j ∧ d zj and (u t ) j k is diagonal at p. We set θ t := dd c u t .

Step 1 : Estimates of the tangential second order partial derivatives.

We consider the real coordinates near the point p defined as follows s 2j-1 = x j , s 2j := y j , 1 j n -1, and s 2n-1 := y n , s 2n := x n .

We will omit the subscript t and use the usual notations for the partial derivatives φ α = φ s α = ∂φ ∂s α and φ αβ := ∂ 2 φ ∂s α ∂s β , φ j k := ∂ 2 u ∂z j ∂ zk etc. We claim that there exists a constant M 0 > 0 depending on u

t C 1 (∂X t ) , v t C 2 (∂X t ) , h C 2 (∂X )
(u -v) αβ (p) = φ α (p)ρ β (p) + φ β (p)ρ α (p) + φ αβ (p).
Moreover for any α, β < 2n, we have

(u -v) αβ (p) = φ(p) ρ αβ (p) = -(u -v) x n (p)ρ αβ (p).
Therefore the estimates (6.12) follow the boundary C 1 estimates for u.

Step 2 : Estimates of tangential-normal second order partial derivatives.

These estimates rely on the Maximum Principle for the linearized operator (see [START_REF] Guan | Complex Monge-Ampère equations and totally real submanifolds[END_REF]Lemma 4.3]). With the same notations as before, for δ > 0 we define X t (δ) := X t ∩ B(p, δ), where B(p, δ) ⊂ C N is the euclidean ball centered at the point p ∈ ∂X t with radius δ. We will choose δ > 0 so small that X t (δ) ⊂ V t for any t = 0.

We will use a barrier function. For τ > 0 so that τ • ρ C 2 (V t ) < ε/4, we define (6.15) γ t := (u tv t )τρ -Nρ 2 , where N > 0 is large enough so that nN 1/n ε > 4(1 + n + ε) sup

V t ×[-M,M] ψ 1/n t ,
where ψ t (z, u) := e λu f t (z) and δ is so small that δ < τ/N. Here M := u t L ∞ (V t ) . Under these conditions the barrier function γ = γ t satisfies the following properties (see [GL10, Lemma 4.1]) (6.16) ∆ θ γ -ε 4 • (1 + tr θ ω) in X t (δ), γ 0 in ∂X t (δ).

Using these estimates we can easily derive the following form of the Maximum Principle (see [GL10, Lemma 4.3]).

Lemma 6.4. -Let w ∈ C 2 (X t (δ)). Suppose that w satisfies ∆ θ w -C 1 (1 + tr θ ω) on X t (δ), Then the normal derivative of w with respect to the inner unit normal vector ν at p satisfies the inequality D ν w(p) = w x n (p) C 2 , for a constant C 2 > 0 depending on ε -1 , C 0 , C 1 , u t C 1 (∂X t ) , v t C 1 (∂X t ) , sup X t (δ) w and the constants N, t, δ.

Proof of Lemma 6.4. -This lemma is an easy consequence of (6.16) coupled with the Maximum Principle for the second order linear elliptic operator ∆ θ in X t (δ) applied to the test function φ := Aγ + Br 2w, where A B 1 are both large enough so that ∆ θ φ 0 in X t (δ), φ 0 in ∂X t (δ) and φ(p) = 0. Indeed, it follows from (6.16) that

     ∆ θ φ (C 1 + Btr ω ω C N -Aε 4 )(1 + tr θ ω) in X t (δ), φ (B -C 0 )r 2 in B(p, δ) ∩ ∂X t , φ Bδ 2 -sup X t (δ) w in ∂B(p, δ) ∩ X t .
so that B := max{C 0 , δ -2 sup X t (δ) w} and A := 4ε -1 (C 1 + Btr ω ω C N ) are suitable. Then by the Maximum Principle we have φ 0 in X t (δ) and then D ν φ(p) 0. Hence D ν w(p) AD ν γ(p) = AD ν (uv)(p) -τD ν ρ(p) C 2 .

We next apply Lemma 6.4 to deduce the following estimates (6.17)

|u s α x n (p)| M 1 , ∀ α < 2n,
where M 1 > 0 is a constant under control.

In order to do that, one considers for any α < 2n the vector field T = T α defined near p by the formula T := D α -ηD x n where η := ρ α ρ x n and α < 2n.

Since T ρ = 0, it is a tangential operator (i.e. it cancels any function that vanishes on the boundary). Following [GL10, pp. 1205[GL10, pp. -1206]]) one applies Lemma 6.4 to the function w := (u y nh y n ) 2 ± T (uh),

to deduce an upper bound on its normal derivative at p i.e.

w x n (p) C,
where C > 0 depends on ε -1 , f t C 1 ( Vt ) , w C 0 (V t ) , v t C 1 ( Vt ) , η C 1 (X t (δ)) . Observe here that w C 0 (V∩ Xt ) can be estimated in terms of u t C 1 (V t ) and h C 1 (V t ) . From the tangential estimates (6.12), we deduce a uniform upper bound on ±D ν T (uh) since (D y n u -D y n h)(p) = 0. The estimates (6.17) follow with a constant M 1 depending on C as well as h C 2 (∂V t ) . Summarizing the above estimates we have proved so far that (6.18) max α,β<2n

|u αβ (p)|, |u αx n (p)| M 2 ,
where

M 2 = max{M 0 , M 1 } depends on u C 1 (V t ) , v C 1 (V t ) , v C 2 (∂X t ) , h C 2 (∂X t ) , ε -1 .
Step 3 : Estimates of the normal-normal second partial derivative

We want to estimate |u , and b is bounded by a constant depending only on the constant M 2 from (6.18). So we only have to estimate a from below, given our control on f t C 0 (V t ) , which follows from the result below. Proof of Lemma 6.5. -Let T C ∂X t ⊂ T C X t | ∂X t be the complex tangent bundle to ∂X t and set T 1,0 ∂X t = T 1,0 X t ∩ T C ∂X t = {ξ ∈ T 1,0 X t ; ∂ρ(ξ) = 0}. In local coordinates near p, we have

T 1,0 ∂X t = {ξ = ∑ ξ j ∂ ∂z j ∈ T 1,0 X t ; ∑ j ξ j ∂ρ ∂z j = 0}.
Let us define m 0 := min ξ 2 θ ; ξ ∈ T 1,0 ∂X t , ξ 2 ω = 1 , where the minimum is taken over X t (δ) ∩ ∂X t . Up to shrinking δ, we can assume that the minimum is attained at an interior point q ∈ ∂X t ∩ B(p, δ) and choose a new set of coordinates (z i ) centered at q as before (i.e. such that ∂ ∂x n is the inner normal vector at q such that ρ x n | ∂X t ≡ -1 near q) such that (6. [START_REF] Berman | Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties[END_REF])

m 0 = θ q ( ∂ ∂z 1 , ∂ ∂ z1 ) = u 1 1(q).
By (6.14), we have u 1 1(q) = v 1 1(q) -(uv) x n (q)ρ 1 1(q) Claim 6.6. -There exists a constant C > 0 such that u x n x n (q) C and the constant C depends only on ε -1 , C 0 , C 1 , u t C 1 (∂X t ) , v t C 1 (∂X t ) , h C 4 ( Xt (δ)) , the constants N, τ, δ and a uniform bound on ρ C 4 (V t ) .

Proof of Claim 6.6. -To prove this we will apply Lemma 6.4 to a well chosen test function. For fixed δ > 0 small enough, define the following vector field since ρ z n (q) = 0. Then ζ := κ -1 ξ is a unit vector field in T 1,0 X t defined on X t ∩ B(p, δ) such that ζ(q) ∈ T 1,0 q ∂X t and the minimum m 0 = u 1 1(q) is achieved at ζ(q).

ξ := -ρ z n ∂ ∂z 1 + ρ z 1 ∂ ∂z n ∈ T 1,
Define the following smooth function

Φ := ζ, ζ dd c h -(u -h) x n ζ 2 dd c ρ -u 1 1(q)
= ∑ j,k h j kζ j ζk -(u -h) x n ∑ j,k ρ j kζ j ζku 1 1(q), on X t ∩ B(p, δ). Similarly to earlier, the identity (uh)| ∂X t ≡ 0 ensures that we can write uh = ψρ. By (6.13), we know that (6.20)

ψ| ∂X t = -(u -h) x n .
Moreover, we have the following identity of forms on ∂X t :

∂ ∂(uh) = ψ∂ ∂ρ + 2 (∂ρ ∧ ∂ψ) hence for any ν ∈ T 1,0 ∂X t = T 1,0 X t ∩ ker(∂ρ), ν, ν dd c h = ν 2 θψ ν 2 dd c ρ = ν 2 θ + (uh) x n ν 2 dd c ρ thanks to (6.20). Hence since ζ(q) ∈ T 1,0 q ∂X t , by definition of m 0 := u 1 1(q), we have Φ = ζ 2 θu 1 1(q) 0, in B(p, δ) ∩ ∂X t and Φ(q) = 0.

A simple computation shows that the function w := (u y nh y n ) 2 -Φ satisfies the conditions of Lemma 6.4 which implies that w x n C and then Φ x n (q) -C , where C > 0 is a constant depending on ε -1 , u t C 1 (∂X t ) , v t C 1 (∂X t ) , h C 4 ( Xt (δ)) and the constants N, t, δ.

This implies that at q we have u x n x n ρ 1 1 C , where C depends on C as well as h C 3 ( Xt (δ)) and ρ C 3 (( Xt (δ)) . The claim now follows from the uniform positivity of dd c ρ.

In view of (6.18) and (6.6), it follows that we have an a priori upper bound on all the eigenvalues of the matrix (u j k(q)).

Since det(u j k(q)) = e λu(q) f t (q) e min ∂X t λu σ, the eigenvalues of θ q must have a uniform lower bound. In particular, u 1 1(q) is bounded away from 0 uniformly. Given (6.19), this proves Lemma 6.5.

The theorem is now proved.

  has terminal singularities. As in the Calabi-Yau case, one has(5.3) ϕ ε L ∞ ( X) C, and ω ε -→ ε→0 π * ω KE weakly.

and w C 0 r 2

 2 in B(p, δ) ∩ ∂X t , w(p) = 0, where r 2 := d(•, p) 2 = | • -p| 2 .

  x n x n (p)|. Since 0 u n n( p) = 1 4 (u x n x n (p) + u y n y n (p)), it is enough to estimate u n n( p) from above since u y n y n (p) is uniformly bounded by (6.18). This is the tricky part of the proof (see [GL10, Proposition 4.4]). Expanding det(u j k( p)) we obtain the formula det (u j k( p)) = au n n( p) + b, where a := det u j k( p) j,k n-1

Lemma 6. 5 .

 5 -There exists a positive c 0 > 0 such that for any ξ ∈ T 1,0 p ∂X t C n-1 , we haveξ 2 θ c 0 ξ 2 ω . Equivalently, ∑ α,β n-1 ξ α ξβ u α β( p) c 0 |ξ| 2 , ξ ∈ C n-1 .

Definition 2.2. -In

  the Setting 2.1 above, a Kähler-Einstein metric ω KE := ω + dd c ϕ KE is a solution of the Monge-Ampère equation (2.2)(ω + dd c ϕ KE ) n = e λϕ KE µ (X,D),h

  a du Val surface singularity, cf e.g. [KM98, § 4.2 & Definition 5.32]. Such a singularity can be smoothed out e.g. by

  For all t ∈ D, we let u t denote a smooth plurisubharmonic function in V t such that (dd c u t ) n = e λu t f t ω n t in V t and u |∂X t = h t . Assume that there exists a smooth plurisubharmonic function v t in V t such that dd c v t εω t in V t and u t v t in V t , with v t = u t = h t on ∂X t .Then ∆u t L ∞ (∂X t ) C,where C depends on upper bounds for u t C 1

  and ρ C 2 (∂X t ) such that (6.12) |u αβ (p)| M 0 , ∀ α, β < 2n. Indeed, since u = v on ∂X t , we can write uv = φρ near ∂X t , where φ = φ t is a smooth function in a neighborhood of ∂X t . Moreover, one can assume that (6.13) ρ x n | ∂X t ≡ -1 in a neighborhood of p e.g. by passing to a Moser normal form for the real-analytic boundary ∂X t = {x n = z 2 + ∑ >0,|α|,|β|>2 a ,α,β y n z α z

β }). Since ρ = 0 on ∂X t , we get (6.14) φ(p) = -(uv) x n (p). Similarly, we have for any α, β 2n

  0 X t

	and observe that
	κ := |ξ| ω := -ρ z n	∂ ∂z 1	+ ρ z 1	∂ ∂z n ω	> 0, in X

t ∩ B(p, δ)

Acknowledgements. -The authors are partially supported by the research project Hermetic (ANR-11-LABX-0040), and the ANR projects Paraplui and Karmapolis.

May 21, 2023

VINCENT GUEDJ • E-mail : vincent.guedj@math.univ-toulouse.fr, Institut de Math. de Toulouse; UMR 5219, Université de Toulouse; CNRS, UPS, 118 route de Narbonne, F-31062 Toulouse, France HENRI GUENANCIA • E-mail : henri.guenancia@math.cnrs.fr, Institut de Math. de Toulouse; UMR 5219, Université de Toulouse; CNRS, UPS, 118 route de Narbonne, F-31062 Toulouse, France AHMED ZERIAHI • E-mail : ahmed.zeriahi@math.univ-toulouse.fr, Institut de Math. de Toulouse; UMR 5219, Université de Toulouse; CNRS, UPS, 118 route de Narbonne, F-31062 Toulouse, France