Example 1.7. vorsque X est l vriété de l9exemple PFIH Y nous vons lulé l vleur exte de l plus grnde orne inférieure de iiD qui est X R(X) = 1046175339 1236719713 0.

Résumé

ve résultt prinipl de ette thèse est l9otention d9une ondition néessire et su0snte pour l9existene d9une métrique de uählerEiinstein sur une omptiE (tion iEéquivrinte lisse et pno d9un groupe omplexe rédutif onnexeF ges vriétés omprennent les vriétés toriques et les ompti(tions mgni(ques de groupes semisimples djointsF hns l première prtie de e trvil sont développés les outils néessires à l9étude de l9existene de métriques de uählerEiinstein sur es vriétésF xous lulons en prtiulier l ressienne omplexe d9une fontion K × KEinvrinte sur l omplexi(tion d9un groupe ompt KF xous ssoions églementD à toute métrique invrinte à ourure positive sur un (ré linérisé mple sur une ompti(tion de groupeD une fontion onvexe dont le omportement sympE totique est presritF gei est utilisé une première fois pour otenir une formule pour l9invrint lph d9un (ré en droite mple sur une ompti(tion de groupe pnoF gette formule est otenue pr le lul des seuils log noniques des métriques hermitiennes invrintes à ourure positiveD et induitD dns le s prtiulier des vriétés toriquesD un résultt otenu uprvntD (gurnt dns l9rtile pr illeurs inlus en ppendie de l thèseF xous prouvons ensuite le résultt prinipl en otennt des estimées C 0 le long de l méthode de ontinuitéD en se rmennt à une éqution de wongeE empère réelle sur un ôneF v ondition otenue est que le ryentre du polytope ssoié à l ompti(tion de groupeD pr rpport à l mesure de huistermtErekmnD doit être dns une zone prtiulière du polytopeF gette ondition peut être véri(ée sur les exemplesD donne de nouveux exemples de vriétés de uählerEiinstein pnoD et donne ussi un exemple qui n9dmet uun soliton de uählerEiiF xous lulons de plus l plus grnde orne inférieure de ii lorsqu9il n9y ps de métrique de uählerEiinsteinF Abstract he min result of this work is neessry nd su0ient ondition for the exE istene of uählerEiinstein metri on smooth nd pno iEequivrint omE pti(tion of omplex onneted redutive groupF ixmples of suh vrieties inlude wonderful ompti(tions of djoint semisimple groupsF he tools needed to study the existene of uählerEiinstein metris on these vrieties re developed in the (rst prt of the workD inluding omputtion of the omplex ressin of K × KEinvrint funtion on the omplexi(tion of ompt group KF enother step is to ssoite to ny nonEnegtively urved invrint hermitin metri on n mple linerized line undle on group ompti(tion onvex funtion with presried symptoti ehviorF his is used (rst time to derive formul for the lph invrint of n mple line undle on pno group ompti(tionF his formul is otined through the omputtion of the log nonil thresholds of ny nonEnegtively urved invrint hermitin metriD nd gives the sme resultD for tori mnifoldsD s the one we otined eforeD in n rtile tht is inluded in this thesis s n ppendixF hen we prove the min result y otining C 0 estimtes long the onE tinuity methodD using the tools developed to redue to rel wongeEempère eqution on oneF he ondition otined is tht the ryenter of the polyE tope ssoited to the group ompti(tionD with respet to the huistermtE rekmn mesureD lies in ertin zone in the polytopeF his ondition n e heked on exmplesD gives new exmples of pno uählerEiinstein mnifoldsD nd lso gives n exmple tht dmits no uählerEii solitonsF e lso omE pute the gretest ii lower ound when there re no uählerEiinstein metrisF

F F F F F F F F F F F F F F F F R IFP ésumé de l thèse F F F F F F F F F F F F F F F F F F F F F F F F F F II 2 Introduction 20 PFI pno uählerEiinstein metris F F F F F F F F F F F F F F F F F F F F PH PFP esults nd orgniztion of the work F F F F F F F F F F F F F F F F PU
3 Reductive groups and invariant functions 32

QFI edutive groups F F F F F F F F F F F F F F F F F F F F F F F F F F F QP QFP K × KEinvrint funtions on G F F F F F F F F F F F F F F F F F F F RI
4 Group compactications and metrics 56 RFI vinerized line undles on redutive groups F F F F F F F F F F F F SU RFP olrized tori vrieties F F F F F F F F F F F F F F F F F F F F F F F SV RFQ qroup ompti(tions F F F F F F F F F F F F F F F F F F F F F F F TI RFR onderful ompti(tions F F F F F F F F F F F F F F F F F F F F F TQ RFS purther exmples of pno group ompti(tions F F F F F F F F TV RFT rermitin metris on line undles F F F F F F F F F F F F F F F F F TW RFU olume forms nd the huistermtErekmn mesure F F F F F F UR 5 Alpha invariants of polarized group compactications 78 SFI vog nonil thresholds on ompt mnifolds F F F F F F F F F F UV SFP xewton ody of hermitin metri

F F F F F F F F F F F F F F F F F VH SFQ sntegrility riterions F F F F F F F F F F F F F F F F F F F F F F F F VQ SFR vog nonil thresholds on group ompti(tions F F F F F F F VS SFS elph invrint on group ompti(tions F F F F F F F F F F F F VU SFT ixmples F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F WH
6 Existence of Kähler-Einstein metrics on group compactications 91

TFI gontinuity method F F F F F F F F F F F F F F F F F F F F F F F F F F WP TFP reliminries F F F F F F F F F F F F F F F F F F F F F F F F F F F F F WS TFQ istimtes on |m t | F F F F F F F F F F F F F F F F F F F F F F F F F F F IHH TFR ystrutionD nd upper ound on R(X) F F F F F F F F F F F F F F IHR I TFS esene of estimtes on |x t | F F F F F F F F F F F F F F F F F F F F F IHV TFT ixmples F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IIU
A Alpha invariants of toric line bundles 124 eFI vine undles on smooth tori mnifolds

F F F F F F F F F F F F F F IPS eFP vog nonil thresholds F F F F F F F F F F F F F F F F F F F F F F F IPV eFQ elphEinvrint F F F F F F F F F F F F F F F F F F F F F F F F F F F F IQP P Remerciements
wes remeriements vont d9ord à mon direteur de thèse hilippe iyssiE dieuxD pour m9voir proposé de trviller dns e domine de reherhe pssioE nnt et très tifF sl su trouver un sujet très riheD et orienter mes reherhes u ours des nnées de questions ssez simples vers des ojetifs plus mitieuxF g9est un grnd honneur pour moi que imon honldson et éstien foukE som ient epté d9être rpporteurs pour m thèseF te les remerie hleureuE sement pour l9interêt qu9ils ont porté à mes trvuxD insi que wihel frionD tenEierre hemillyD iveline vegendreD nn ollin et endrei elemn qui ont epté de fire prtie de mon jury de thèseF ves remrques de wihel frion sur mon trvil ont déjà permis d9méliorer e doument et je lui en suis reonnisE sntF ns pouvoir tous les nommerD je dois remerier les orgnisteurs et prtiiE pnts des di'érentes renontres de l9ex weguD des renontres nnuelles du qh qeqgD et des éoles ueeD qui ont pris une ple importnte lors de m thèse et ont été l9osion de déouvrir euoup de lieuxD de personnesD de mthémtiquesFFF rmi euxEiD ile et ghinh oupent une ple prtiulière pour les ons moments prtgésD insi que inent quedj qui m9 idé à me sentir à m pleF te remerie églement l9snstitut pourier pour son ueil pendnt es trois dernières nnéesD et pour ette qutrième nnée qui ommeneF in prtiulierD les diverses génértions de dotorntsD pour les séminires ompréhensilesD les disussions mthémtiquesD ou psD etF X wikëlD olndD onnD tunynD tenE wthieuD tulienD wroD fininD itienne @qui nous lâhement ndonnésAD quillumeD eddyD enhoD pederioD glémentD imonD vouisEglémentD edroD phëlD hizhongD et eux que j9oulieF

wes misD enore à vyonD ou illeurs X ophie et entoineD ghrlotte et wiE këlD voren et hvidD ominD wrionD ierreEwrieD vurineD imeriD golineD gorentinF g9étit toujours un plisir de vous voirD et d9oulier le resteF te remerie tout prtiulièrement mes prentsD m soeurD mon frêreD mes grndEmèresF i je suis rrivé jusqu9iiD 9est en grnde prtie grâe à euxF it (nlementD 9est à m femme férénie que je dois le plusD pour son soutienD et s présene à mes ôtés dns les moments di0ilesF xotre mrige été le meilleur moment de es nnées de thèseF Chapitre 1 Introduction et Résumé (Français) 1.1 Métriques de Kähler-Einstein Fano 1.1.1 Métriques de Kähler-Einstein ne métrique iemnnienne g est dite d'Einstein lorsque s ourure de ii Ric(g) véri(e g = λRic(g) pour une ertine onstnte réelle λF ne méE trique de Kähler-Einstein sur une vriété omplexe X est une métrique ieE mnnienne qui est à l fois de uähler et d9iinsteinF v donnée d9une métrique de uähler g est équivlente à l donnée de s forme de uähler ssoiée ωF he mnière similireD l ourure de ii d9une métrique de uähler peut être onsidérée omme une (1, 1)Eforme que nous noterons Ric(ω)F hérivons ette forme lolementF ne forme de uähler ω sur X peut s9érire sur une rte su0smment petite omme i∂∂ϕ pour une ertine fontion lisse et stritement plurisoushrmonique ϕ sur un ouvert de C n F xotons Hess C (ϕ) l ressienne omplexe de ϕD 9estEàEdire l mtrie dont les oe0ients sont les ∂ 2 ϕ ∂zi∂zj F ppelons qu9une fontion ϕ est strictement plurisousharmonique @pshA si s ressienne omplexe est dé(nie positive prtoutF v forme de ii de ω est lors dé(nie dns ette rte pr Ric(ω) = i∂∂ (-ln det Hess C (ϕ)) .

xous supposons dns l suite que l vriété X est ompteF sl est imporE tnt de noter que pour toute métrique de uähler ω sur une vriété de uähler ompte XD l forme de ii de ω est dns une lsse de ohomologie (xe qui dépend uniquement de X X l première classe de Chern c 1 (X) de XF gonsidérons mintennt l9éqution de uählerEiinstein X Ric(ω) = λω R pour une ertine onstnte λF upposons que ω soit une forme de uählerD solution de ette équtionF i λ est stritement négtiveD el signi(e qu9il y une forme de uähler -λω dns l lsse -c 1 (X) et don c 1 (X) < 0F i λ est égle à HD lors 0 = Ric(ω) ∈ c 1 (X) don l première lsse de ghern de X s9nnuleF he telle vriétés sont dites de gliEuF in(n si λ est stritement positiveD el signi(e qu9il y une forme de uähler dns c 1 (X)D don que ette lsse est positiveD et l vriété X est lors dite de FanoF gette disussion montre d9ord que l9éqution de uählerEiinstein ne peut ps voir de solution lorsque l première lsse de ghern de X n9est ps nulle ou de signe dé(niF he plusD el montre qu9étnt donnée une vriété X suseptile d9dmettre une métrique de uählerEiinstein ve λ non nulleD l9étude se réduit u s des métriques de uähler dns l lsse c 1 (X)D ou -c 1 (X) selon le signe de λF v onstnte λD lorsqu9elle n9est ps nulleD peut être (xée à I ou EID quitte à renormliser l métriqueF upposons mintennt que Ric(ω) et λω soient dns l même lsseF xous llons expliquer omment l9éqution de uählerEiinstein se rmène à une ih sur une fontion grâe u lemme du ∂∂F oit ω ref une métrique de référene (xée dns l lsse de ωF r le lemme du ∂∂D nous pouvons érireD d9une prtD ve s où c 1 (X) < 0 été résolu pr euin dns euUT X il existe touE jours une métrique de uählerEiinsteinD et le s où c 1 (X) = 0 été résolu pr uF lus générlementD u prouvé le théorème @de gliEuA suivntD onjeturé pr gliF Théorème. [Yau78] Soit [ω 0 ] la classe d'une forme de Kähler sur X et θ une forme représentant c 1 (X), alors il existe une métrique de Kähler ω ∈ [ω 0 ] telle que Ric(ω) = θ.

ω = ω ref + i∂∂ϕ
1.1.2 Métriques de Kähler-Einstein Fano hns le s pnoD l9éqution de uählerEiinstein n9dmet ps toujours de solutionsF Obstructions ne première ostrutionD otenue pr wtsushimD est que l omposnte onnexe de l9identité dns le groupe d9utomorphismes d9une vriété de pno qui dmet une métrique de uählerEiinstein doit être un groupe rédutifF wtE sushim prouve ussi que le groupe des isométries holomorphes d9une métrique de uählerEiinstein est ussi grnd que possileF Théorème. [Mat57] Soit X une variété de Fano admettant une métrique de Kähler-Einstein. Alors Aut 0 (X) est un groupe réductif complexe, et le groupe des isométries holomorphes d'une métrique de Kähler-Einstein est un sous-groupe compact maximal de Aut 0 (X).

putki introduit ensuite un invrint intégrl putVQD ppelé à présent l9invariant de FutakiD qui est un rtère d9lgères de vie entre l9lgère de vie des hmps de veteurs holomorphes η(X) et RF putki montré que e rtère s9nnule lorsque X dmet une métrique de uählerEiinsteinF emrE quons que η(X) est l9lgère de vie du groupe Aut 0 (X)D don que le théorème de wtsushim implique que ette lgère de vie est rédutive lorsqu9il existe une métrique de uählerEiinsteinF La méthode de continuité ne prtie du trvil ompli pour résoudre l9éqution de uählerEiinstein dns le s c 1 (X) ≤ 0 peut enore être utilisée dns le s pnoF in e'etD l9outil prinipl pour résoudre es s est l méthode de ontinuitéD qui demnde d9oE tenir des estimées priori des solutionsF v méthode de ontinuité onsiste à T onsidérerD à l ple de l9éqution de uählerEiinstein seuleD une fmille d9équE tions indexées pr t ∈ [0, 1]D telle que l9éqution pour t = 1 soit l9éqution de uählerEiinsteinF ve ut est ensuite de prouver que l9ensemle des t tels qu9il existe une solution à l9éqution orrespondnte est nonEvideD ouvert et ferméF r onnexité el implique qu9il existe une solution pour t = 1F ves résultts otenus pr euin et u peuvent être utilisés pour montrer queD dns l méthode de ontinuité que nous llons dérireD et ensemle est ouvertF hes ostrutions pprissent ependnt pour l fermeture de et enE semleD mis il est enore possile d9utiliser leur trvil pour se rmener à l reherhe seulement d9estimées C 0 sur les solutionsF v fmille d9équtions pprissnt dns l méthode de ontinuité lssique pour le s pno est l suivnte X Ric(ω t ) = tω t + (1 -t)ω ref .

À t = 0D ette éqution est résolule pr le théorème de gliEuF our montrer l9existene de métriques de uählerEiinsteinD il est su0snt d9otenir des estimées |φ t | ≤ C pour les solutionsD ve une onstnte C indépendnte de tF L'invariant α de Tian in déterminé dns iVU une ondition su0snte pour l9otention d9esE timées C 0 F gelleEi implique un invrint de l vriétéD ppelé l9invariant αD qui enode les singulrités possiles des métriques hermitiennes singulières à ourE ure positive sur -K X F i et invrint stisfit l9inéglité α(X) > n/(n + 1) lors il existe une métrique de uählerEiinstein sur XF hns le s où un groupe ompt git sur l vriétéD et invrint peut être r0né en onsidérnt uniE quement les métriques hermitiennes invrintes pr l9tion de e groupeF v ondition su0snte d9existene reste l même pour et invrint reltif à un groupeF lus générlementD il est possile de dé(nir et invrint pour n9imE porte quel (ré en droite mpleF K-stabilité v9vnée mjeure réente dns le prolème des métriques uählerEiinstein pno est l résolution de l onjeture de uEinEhonldson pr ghenD hoE nldson et un ghISD ghISD ghIS @voir ussi iAF ge résultt relie l9existene d9une métrique de uählerEiinstein sur une vriété de pno à une ondition lgéroEgéométrique de stilitéF ge résultt fournit une ondition néessire et su0snte d9existeneF our otenir des exemples de métriques de uählerEiinstein ependntD ette ondition n9est ps file à véri(er en prtiqueD et n9est ps enore très ien ompriseF U 1.1.3 Exemples de variétés Kähler-Einstein Fano in dimension unD le seul exemple de vriété de uähler ompte de pno est l droite projetive P 1 D qui dmet pour métrique de uählerEiinstein l métrique de puiniEtudyF our les surfes de pnoD ppelées surfes de hel ezzoD in iWH montré que l9existene d9une métrique de uählerEiinstein dns e s est équivlente à l9nnultion de l9invrint de putkiD ou enore u fit que l9lgère de vie des hmps de veteurs holomorphes soit rédutiveF ge n9est plus vri en dimension plus grndeD omme in l9 montré sur un exemple dns iWUF xous dérivons dns l suite quelques fmilles d9exemples de vriétés de uählerEiinstein de pnoF ve premier exemple de vriété de uählerEiinstein pno en n9importe quelle dimension est ien sûr l9espe projetif P n muni de l métrique de puiniE tudyF lus générlementD tout vriété de pno ompte homogène dmet une métrique de uählerEiinsteinF v ondition su0snte donnée pr in en termes d9invrint α est un outil très puissnt qui permis de trouver de nomreuses fmilles de métriques de uählerEiinstein pno @voir pr exemple fegHPD iD üÿIQAF in introduit initilement et invrint pour prouver qu9une hypersurfe de dimension n et degré supérieur à n -1 dmet une métrique de uählerEiinsteinF hns le s des vriétés toriquesD 9estEàEdire des vriétés de dimension n sur lesquelles le tore (C * ) n git ve une orite ouverte denseD le premier résulE tt mjeur d9existene été otenu pr ftyrev et elivnov fWW grâe à l9invrint αF sls ont montré que les vriétés toriques pno symétriquesD 9estE àEdire elle pour lesquelles il existe un groupe ompt d9utomorphismes ne lissnt uun hmps de veteurs non nul invrintD dmettent une métrique de uählerEiinsteinF ge résultt ne su0t ps à résoudre le prolème uählerEiinstein pour les vriétés toriquesF our les petites dimensions @n ≤ 6AD toute vriété torique pno est soit symétriqueD soit sns métrique de uählerEiinsteinF wis à prtir de l dimension UD il y des exemples de vriétés torique uählerEiinstein nonE symétriquesF ges exemples ont été trouvés pr xill et 'enholz xIID en utilisnt le théorème de ng et hu qui ont omplétement résolu l question de l9existene de métriques de uählerEiinstein sur les vriétés toriquesF Théorème. [WZ04] Soit X une variété torique lisse Fano. Alors X admet une métrique de Kähler-Einstein si et seulement si l'invariant de Futaki de X s'annule. v9invrint de putki d9une vriété torique pno est donné pr le ryentre du polytope ssoié à ette vriétéD omme l9 montré wuhi wVUF in prtiulierD l9invrint de putki s9nnule si et seulement si le ryentre du polytope est l9origineF ve ryentre d9un polytope ve des sommets entiers peut être lulé exE pliitementD don e ritère peut être véri(é en prtique étnt donné le polytope d9une vriété toriqueF xill nd 'enholz ont utilisé une lssi(tion des poE lytopes orrespondnt jusqu9à l dimension V pour otenir un exemple dont le V ryentre ssoié est l9origine mis tel que l vriété ssoiée ne soit ps syméE triqueF h9près le lul pr ong onHS des invrints α des vriétés toriques pnoD non seulement les exemples ne sont ps symétriquesD mis ussi le ritère de in en termes d9invrint α n9urit ps pu être utilisé sur es vriétésF v méthode de démonstrtion de ng et hu est d9otenir diretement des estimées C 0 le long de l méthode de ontinuitéF uisque nous suivrons l même strtégie qu9euxD nous expliquons le point de déprt de leur méthodeF n ingrédient prinipl est d9utiliser le théorème de wtsushim pour montrer qu9une métrique de uählerEiinstein sur une vriété torique doit être invrinte sous l9tion du tore ompt (S 1 ) n ⊂ (C * ) n @à onjugison prèsAF sl est de plus possile de supposer que si l métrique de référene est invrinte pr le tore omptD lors toutes les solutions le long de l méthode de ontinuité seront ussi invrint pr le tore omptF in se restreignnt ensuite à l9orite ouverte (C * ) n D ils trduisent l9éqution de uählerEiinstein en une éqution de wongeEempère réelleD impliqunt une fontion réelle onvexe dont le omportement symptotique à l9in(ni est presritF our voir elD l première remrque à fire est que le quotient de (C * ) n pr (S 1 ) n est isomorphe à R n F heuxièmementD l9éqution de uählerEiinstein peut être exprimée lolement omme une éqution de wongeEempère omplexe en le potentiel lol de l métriqueD qui est une fontion pshF roisièmementD dns un hoix de oordonnées ppropriées @logrithmiquesAD le wongeEempère omplexe d9une fontion (S 1 ) n Einvrinte est égl u wongeEempère réel dns le quotient R n D et une fontion psh devient une fontion onvexe sur le quotientF pinlementD l fontion onvexe otenue sur R n doit stisfire ertines onditions sur son omportement symptotique si elle provient d9une métrique dé(nie sur l vriété toute entièreF hns e dreD ng et hu prviennent à otenir des estimées pour les solutions de telles équtions de wongeEempère sur R n F ne vriété presque-homogène est une vriété munie de l9tion d9un groupe de vie omplexe ve une orite denseF r exempleD les vriétés toriques sont presqueEhomogènes sous l9tion du toreF ves premiers exemples de vriétés de uählerEiinstein pno non homogènes furent otenus pr kne et uoiso uVTD uVV omme (rés en P 1 uEdessus d9utres vriétés uählerEiinstein pnoF ves exemples qu9ils exhièrent insi étient presqueEhomogènesF ne générlistion des exemples donnés pr uoiso et kne été étudiée pr odest et piro X ils ont trité l question de l9existene de métriques de uählerEiinstein sur des (rés toriques homogènes IHF sl s9git de (rés homogènes sur des vriétés de drpeux de groupes semisimples omplexesD dont l (re est une vriété toriqueF sls ont déterminé qund es vriétés sont pnoD et qundD dns e sD l9invrint de putki s9nnuleF emrquons que es vriétés sont presqueEhomogènes pour un groupe rédutifD et que le quotient de l9orite ouverte pr un sousEgroupe ompt mximl est enore isomorphe à R r D où ette fois r est l dimension de l (re toriqueF gomme dns le s toriqueD il est possile d9utiliser l9invrine sous l9tion du groupe ompt pour simpli(er l9éqution de uählerEiinsteinD et el donne enore une éqution de wongeE empère réelle sur R r D ve le même type de omportement symptotique presrit que dns le s toriqueF in ppliqunt le trvil de ng et hu ve quelque W modi(tions mineuresD odest et piro onluent que leur vriétés dmettent une métrique de uählerEiinstein si et seulement si leur invrint de putki s9nnuleF our onlure ette liste nonEexhustive d9exemplesD mentionnons le trvil de qun sur l9existene de métriques de uählerEiinstein @et plus générlement de métriques noniquesA sur les vriétés presqueEhomogènes de ohomogénéité unD e qui signi(e qu9un sousEgroupe ompt mximl git ve une orite de odimension unF ur es vriétésD qu9il est di0ile de lssi(er omplétementD l9éqution de uählerEiinstein doit se trduire en un ertin sens en une éqution di'érentielle ordinire grâe à l9tion du groupe omptF ves derniers rtiles de qun sur le sujet sont quIID quIID quIID quIQF 1.1.4 Compactications de groupes ves vriétés de pno que nous étudierons dns ette thèse sont des ompE ti(tions de groupesF oit G un groupe lgérique omplexe rédutif onnexe @pr exemple GL n (C)D SL n (C)D (C * ) n D etFAF ne variété torique compacte Z de dimension r est une ompti(tion du tore T = (C * ) r F ve fit que le tore T gisse sur l vriété Z entière et que T ⊂ Z soit une orite pour l multiplition à guhe signi(e que Z est en fit une ompti(tion T Eéquivrinte de T F in d9utre termesD il s9git de l donnée d9un plongement j de T dns une vriété ompt Z telle que Z dmette une tion de T et que j soit une pplition équivrinteD 9estEàEdire j(t•z) = t•j(z)F he mnière équivlenteD il est ussi possile d9utiliser l9tion de T sur luiEmême pr multiplition à droite pr l9inverseF gomme T est élienD les deux tions sont équivlentesF i l9on onsidère un groupe rédutif non élien GD es deux tions ne sont plus équivlentesF xous pourrions toujours onsidérer des ompti(E tions équivrinte pr l multiplition à guhe de GD ou pr l multiplition à droiteD mis nous llons ii supposer que les ompti(tions sont équivE rintes simultnément pour les deux tionsF xous onsidérons don des omE pti(tions G × GEéquivrintes d9un groupe rédutif omplexe onnexe GF our simpli(erD nous ppellerons dns e texte es vriétés des compactications de GD ou des compactications de groupes lorsque le groupe n9est ps (xéF ne des risons priniples pour onsidérer es deux tions est que de telles vriétés sont sphériquesF ne vriété X est dite sphérique lorsqu9elle est munie d9une tion d9un groupe rédutif G telle qu9un sousEgroupe de forel de G gisse sur X ve une orite ouverteF gel implique que G dmet une orite dense et ouverte dns XD ette orite étnt un espe homogène pour GF in prtiulierD une vriété sphérique est presqueEhomogèneF ge sont les vriétés presque homogènes pour lesquelles l9étude est l plus vnéeD et pour lesquelles euoup de prolèmes résolus pour les vriétés toriques ont une hne d9voir une solution similireF in fitD l pluprt des exemples de vriétés uählerEiinstein pno mentionE nées préédemment sont des vriétés sphériquesF sl est file de voir que les vriétés omptes pno homogènes sont sphériquesD et nous vons vu que les IH vriétés toriques l9étient églementF ves exemples étudiés pr odest et piro font prtie de l fmille des vriétés horosphériques @prmi les vriétés sphéE riquesAD et ontiennent à l fois les vriétés homogènes et les vriétés toriquesF gomme nous l9vons vuD les vriétés toriques sont ussi des ompti(tions de groupesD pour les groupes rédutifs éliensF À l9opposéD les groupes réduE tifs les 4moins éliens4 sont les groupes semisimplesF ves exemples les plus onnus de ompti(tions de groupes semisimples sont les ompti(tions mgni(ques de groupes semisimples djointsD onstruites pr he gonini et roE esi hgVQF in résuméD e sont des vriétés lisses et pnoD onstruites omme l9dhérene de l9imge de G dns une représenttion irrédutile G → GL N (C) ssoiée à un poids régulier et dominntF honldson suggéré dns son survey honHV d9étudier l9existene de méE triques de uählerEiinsteinD et plus générlementD l9existene de métriques exE trémles ou à ourure slire onstnteD pour les vriétés sphériquesF in e'etD elles entrent dns l première tégorie de vriétés présentée pr honldson dns honHVD etion R r elles sont sns multipliité @multipliity freeAF eE mrquons que l lsse plus petite des ompti(tions de groupes que nous onsidérons ii entre églement dns l deuxième tégorie de vriétés présentées dns honHVD etion RF 1.2 Résumé de la thèse 1.2.1 Chapitre 3 ve ut du hpitre Q est de donner les outils néessires pour trviller sur un groupe rédutif omplexe GD et don sur l9orite ouverte et dense d9une ompE ti(tion de GF v première setion fournit l dé(nition d9un groupe rédutifD et rppelle plusieurs outils usuels pour les étudierF ve système de rines Φ ssoié à un groupe rédutif ser très importnt pour nous puisque nos résultts seront toujours exprimés en termes de e système de rinesD pour e qui onerne l9orite denseF v deuxième setion se onentre sur l9tion d9un sousEgroupe ompt mximl K sur G pr multiplition à droite et à guheF oit G un groupe rédutif omplexe onnexeD et K un sousEgroupe ompt mximl de GF ghoisissons T un tore mximl de GD tel que T ∩ K soit un tore mximl de KF xotons g l9lgère de vie de GD k elle de K et t elle de T F ve fit que G soit rédutif omplexe est équivlent u fit que G soit isomorphe à l complexication de KF eu niveu des lgères de vieD on peut érire g = k ⊕ ikF xotons a l sousElgère de vie iLie(T ∩ K)F v9outil de se lorsque l9on onsidère l9tion de K à guhe et à droite est l décomposition KAK X tout élément g de G s9érit sous l forme k 1 exp(a)k 2 où k 1 , k 2 ∈ K et a ∈ aF yn peut même être plus préisF oit Φ le système de rines de (G, T )D et hoisissons un système de rines positives Φ + F elors el détermine une hmre de eyl positive fermée a + dns aF hns l déomE position KAKD il est en fit possile d9imposer que a ∈ a + D et et élément a est uniquement déterminé pr gF eutrement ditD ette déomposition fournit un II domine fondmentl pour l9tion de K × K sur G X l9imge pr exp de a + F ppelons que a est un espe vetoriel réel de dimension r le rng de GD 9estEàEdire l dimension @omplexeA de T D sur lequel git le groupe de eyl W = N G (T )/T de (G, T )F ve ône a + est ussi un domine fondmentl pour l9tion du groupe de eyl sur aF xotre ojetif est d9étudier les fontions K × K invrintes sur GF oit ψ une telle fontionF Étnt donnée l déomposition KAKD il est évident que l donnée de ψ est équivlente à l donnée de l fontion f dé(nie sur a pr f (a) = ψ(exp(a))F ezd et voe evWP ont montré que l fontion ψ est pluE risoushrmonique si et seulement si l fontion f est convexeF einsi l9étude des fontions plurisoushrmoniques K × KEinvrintes sur G se rmène à l9étude des fontions onvexes W Einvrintes sur aF n utre outilD qui ser utilisé dns les hpitres ultérieursD est l formule d'intégration KAKD dptée à l déomposition KAKF réisémentD si dg est une mesure de rr sur GD dx est une mesure de veesgue orretement normE lisée sur a @pr le réseu des sousEgroupes à un prmètreAD et J est l fontion dé(nie sur a pr J(x) = α∈Φ + sinh 2 (α(x))D lors il existe une onstnte C dépendnt uniquement du hoix de l mesure de rr dgD telle que pour toute fontion ψ K × KEinvrinte et dgEintégrleD

G ψdg = C a + f (x)J(x)dx.
xotre ontriution dns e hpitre est le lul de l Hessienne complexe d9une fontion K ×KEinvrinte ψ sur G dns un hoix de oordonnées dptées en termes de l ressienne réelle Hess R (f ) de l fontion orrespondnte f sur aF Théorème 1.1. Soit ψ une fonction K × K-invariante sur G, et f la fonction associée sur a. Alors dans un choix de coordonnées approprié et pour a ∈ a + , la Hessienne complexe de ψ est diagonale par blocs, égale à :

Hess C (ψ)(exp(a)) =         1 4 Hess R (f )(a) 0 0 0 M α1 (a) 0 0 0 . . . . . . . . . . . . . . . 0 0 0 M αp (a)        
où les (α i ) i∈{1,...,p} parcourent les racines positives de Φ et M α est déni par :

M α (a) = 1 2 α(∇f (a)) coth(α(a)) i -i coth(α(a))
et ∇f est le gradient de f par rapport à un produit scalaire xé qui étend la forme de Killing sur la partie semisimple de a.

IP

gomme nous l9vons vu dns ette introdutionD e lul ser très importnt pour exprimer l9éqution de uählerEiinstein sur l9orite ouverte de l omptiE (tionF lus préisément nous utiliserons le déterminnt de l ressienneD ppelé le Monge-Ampère complexe et noté MA C (ψ)F Corollaire 1.1. Soit ψ une fonction K × K-invariante sur G, et f la fonction associée sur a. Alors dans un choix de coordonnées approprié, et pour a ∈ a + , si r est le rang de G et p le nombre de racines positives, nous avons :

MA C (ψ)(exp(a)) = 1 4 r+p MA R (f )(a) α∈Φ + α(∇f (a)) 2 sinh 2 (α(a))
réisons dns quelles oordonnées e lul est vlleF our el rppelons que l9lgère de vie g dmet une déomposition en espes de rines

g = t ⊕ α∈Φ g α , où hque g α = {x ∈ g; ad(h)(x) = α(h)
x ∀h ∈ t} est un sousEespe vetoE riel omplexe de dimension IF in hoisissnt une se de tD et un générteur pour hque g α D on otient une se de g dptée à ette déompositionF our prendre en ompte le groupe ompt KD nous utilisons une vrinte de ette déomposition en espes de rines sur k X

k = ia ⊕ α∈Φ + k α , où k α = {x ∈ k; ad(h) 2 (x) = α(h) 2 x ∀h ∈ t}F xous vons lors k α ⊕ ik α = g α ⊕ g -α D et
pouvons modi(er l se préédente pour otenir une se de g qui est ussi une se réelle de kF sl est ussi possile d9imposer ertines reltions entre les éléments de l se que nous ne préisons ps iiF in notnt l 1 , . . . , l n les éléments de l se hoisieD nous otenons des oorE données loles utour de exp(a) ∈ G pour a ∈ aD données pr X

(z 1 , . . . , z n ) → exp(z 1 l 1 + • • • + z n l n ) exp(a).
xous lulons lors les oe0ients de l ressienne dns es oordonnées lolesD 9estEàEdire que pour toute pire d9éléments (l 1 , l 2 ) de l seD nous lulons X

∂ 2 ∂z 1 ∂z 2 z1,z2=0 ψ(exp(z 1 l 1 + z 2 l 2 ) exp(a)).
ve prinipe du lul est d9otenir une déomposition KAK su0smment exE pliite sur l9rgument exp(z 1 l 1 + z 2 l 2 ) exp(a)D pour utiliser l K × KEinvrineF xous utilisons pour el l formule de fkerEgmpellErussdorfD qui donne le logrithme d9un produit d9exponentielles sous forme d9une série en les rguments de es exponentiellesF hns le lul que nous fisonsD nous n9vons esoin que des termes d9ordre deuxD et le hoix de l se dptée permet d9e'etuer le lul du terme entrl dns l déomposition KAKD à l9ordre deuxF IQ 1.2.2 Chapitre 4 ve hpitre R est une introdution ux ompti(tions de groupe où nous nous onentrons sur le polytope ssoié à une polristion d9une telle vriétéF ve polytope ontient toute l9informtion sur l frontière de l ompti(tionD et sur le (ré mple hoisiF g9est une générlistion du polytope ssoié à une vriété torique polriséeD et nous utilisons en fit ette orrespondne usuelle pour présenter le s des ompti(tions de groupesF in e'etD étnt donnée une ompti(tion X de GD onsidérons T un tore mximl dns G et Z son dhérene dns XF elors Z est une vriété toriqueD ontennt toute l9informtion de l ompti(tion @si G est (xéAF he mêmeD si L est un (ré mple G × GElinérisé sur XD l restrition L| Z de L à l sousEvriété torique est enore un (ré mpleD linérisé sous l9tion du normlisteur de T dns GF ve polytope P ssoié à l ompti(tion polrisée (X, L) est lors dé(ni omme le polytope ssoié à l vriété torique polrisée (Z, L| Z )F xous donnons quelques exemples de ompti(tions lisses et pno de groupesF sl y de nomreux tels exemples X tout groupe semisimple djoint dmet une ompti(tion nonique ve une unique orite ferméeD s omE pti(tion mgni(queD et elle est lisse et pnoF ve polytope ssoié à une telle ompti(tion polrisée pr le (ré ntinonique est l9enveloppe onvexe des imges pr le groupe de eyl W de 2ρ + r i=1 α i où 2ρ est l somme des rines positives de Φ et les α i sont les rines simples de ΦF xous onsidérons ensuite les métriques hermitiennes à ourure positive sur les (rés en droites mples linérisés sur les ompti(tions de groupesF À une telle métrique h nous ssoions une fontion plurisoushrmonique ψ sur GD son potentiel pr rpport à une trivilistion G × {e}Eéquivrinte s de L sur l9orite isomorphe à G X ψ(g) = -ln(|s(g)| 2 h )F vorsque h est K × KEinvrinteD l fontion ψ l9est églement et el détermine prD le hpitre QD une fontion onvexe sur a + D ppelée le potentiel convexe de hF xous prouvons que le fit que h soit une métrique à ourure positive sur une polristion de X impose des onditions sur le omportement symptotique de ette fontion onvexeD et nous dérivons es onditions en terme du polytope ssoiéF uisque l orrespondne entre une métrique et son potentiel onvexe est ijetiveD el fournit une desription de l9espe des métriques hermitiennes singulières K × KEinvrintes à ourure positive sur le (ré onsidéréF Théorème 1.2. Soit (X, L) une compactication lisse polarisée de G, de polytope associé P . Les métriques hermitiennes K × K-invariante sur L, à courbure positive au sens des courants, sont en bijection avec les fonctions convexes Winvariante ϕ : a -→ R telles qu'il existe une constante C 1 ∈ R avec ϕ(x) ≤ f P (x) + C 1 sur a où f P est la fonction support du polytope 2P . De plus, h est localement bornée si et seulement si il existe aussi une constante C 2 avec

f P (x) + C 2 ≤ ϕ(x) ≤ f P (x) + C 1 .

IR

La fonction ϕ associée à h est son potentiel convexe.

hns le s d9une métrique lisse à ourure stritement positiveD le potenE tiel onvexe ϕ est une fontion lisse et stritement onvexeF ve hngement de vrile p = ∇ϕ(x) est don ien dé(ni et le théorème préédent ssure que l9imge pr ∇ϕ de a est l9intérieur du polytope 2P en identi(nt a et a * F in prennt en ompte l9tion de W D on même plus préisément que l9imge pr ∇ϕ de a + est l9intérieur du polytope 2P + F in utilisnt e hngement de vrile onjointement ve l formule pour le wongeEempère omplexeD on peut relier le degré du (ré mple L à une intégrle sur le polytope P X le volume de e polytope pour l mesure de huistermtE rekmnF Proposition 1.2. Soit (X, L) une compactication lisse polarisée de G, correspondant au polytope P . Alors

deg(L) = C 2P + α∈Φ + (α(p)) 2 dp
pour une constante C qui dépend seulement du groupe G.

ge résultt est ohérent ve l formule expliite otenue pr uzrnovskii uzVU et frion friVWD mis nous ne déterminons ps dns notre lul l onstnte C expliitementD puisque nous n9en vons ps esoin dns l suiteF 1.2.3 Chapitre 5 hns le hpitre SD nous lulons l9invrint α d9un (ré mple L sur une ompti(tion de groupe X pnoF lus préisémentD nous lulons l9invrint α pr rpport à l9tion d9un sousEgroupe ompt mximl K × K de G × GF ge résultt est otenu en lulnt le seuil log nonique de n9importe quelle métrique hermitienne K × KEinvrinte h sur L à ourure positive u sens des ourntsD en termes d9un orps onvexe ssoié à hF xotons P le polytope ssoié à LD Q le polytope ssoié à -K X D et H l9enveloppe onvexe des imges pr le groupe de eyl W de l somme des rines positives 2ρF oit h une métrique hermitienne K×KEinvrinte h sur L à ourure positive u sens des ourntsF ve orps onvexe N (h) ssoié à hD que l9on ppelle le corps de Newton de hD est le domine de l trnsformée de vegendreEpenhel du potentiel onvexe ϕ de h X

N (h) = {m ∈ a * ; ∃C, ∀x ∈ a, ϕ(x) -m(x) ≥ C}.
v9informtion dont nous vons esoin sur le omportement symptotique de ϕ se trduit en fontion de e orps onvexeF in prtiulierD les onditions otenues u hpitre préédent se trduisent pr le fit que N (h) est ontenu dns 2P où P est le polytope ssoié à LD ve églité pour les métriques lolement ornéesF v9vntge de trviller ve es orps de xewton est qu9ils sont ien IS dptés pour utiliser l déomposition en éventil donnée pr l sousEvriété torique ZF v méthode pour luler le seuil log nonique est de se rmener à un ritère d9intégrilité sur l9orite ouverte GD puis sur R r pr l formule d9intégrtion KAKF À e pointD nous devons déterminer qund l9exponentielle d9une fontion onve est intégrle pr rpport à l mesure de potentiel J ontre l mesure de veesgueF in utilisnt le ritère d9intégrilité otenu pr quenni dns une preuve nlytique du lul pr rowld des seuils log noniques d9idéux monomiuxD nous otenons un tel ritère en fontion du orps de xewtonF ve ritère permet d9otenir l9expression du seuil log noniqueF Théorème 1.3. Soit h une métrique hermitienne K × K-invariante h sur L à courbure positive au sens des courants. Alors

lct(h) = sup{c > 0; 2H + 2cP ⊂ cN (h) + 2Q},
où N (h) est le corps de Newton de h.

our otenir une expression de l9invrint αD nous déterminons quelles sont les métriques dont le seuil log nonique est le plus petitF sl s9git des métriques dont le potentiel onvexe est linéireF out ei donne le résultt suivntF Théorème 1.4. L'invariant α de L relatif à l'action de K × K est donné par la formule : α K×K (L) = sup{c > 0; c(P + (-P W )) ⊂ Q H}, où P W est l'ensemble des points W -invariants de P et est la soustraction de Minkowski entre deux ensembles convexes.

in prtiulierD si le groupe G est semisimpleD ou s9il y su0smment de symétries supplémentiresD il y une métrique dont le seuil log nonique est égl à l9invrint lph X elle dont le potentiel onvexe est l fontion nulleF Corollaire 1.3. Supposons que G soit un groupe semisimple. Alors

α K×K (L) = sup{c ≥ 0; cP ⊂ Q H}.
Il s'agit aussi du rayon inscrit de Q H par rapport à P . in(nD nous lulons l9invrint α pour quelques exemples de ompti(E tions de groupesF Corollaire 1.4. Soit X la compactication magnique de PSL n+1 (C), alors

α K×K (-K X ) = 1 1 + n 2 ( n 2 + 1)
.

IT

get exemple montre que de telles vriétés ne stisfont ps l ondition de in en générlF in fitD il n9est ps di0ile de se onvinre qu9il ne ser jmis stisfit pour les ompti(tions mgni(ques de groupes semisimples djoints sns fteur de rng unF xous devons don utiliser une utre méthodeF v formule que nous otenons s9pplique en prtiulier ux vriétés toriques et permet de retrouver le lul d9invrint α sur les vriétés toriquesD otenu préédemment dns helISF ve texte de et rtile est églement reproduit dns l9ppendie eF 1.2.4 Chapitre 6 ve hpitre T ontient le résultt prinipl de l thèseF xous déterminons dns e hpitre une ondition néessire et su0snte d9existene d9une métrique de uählerEiinstein sur une ompti(tion de groupeF gette ondition est de plus lulle numériquement sur les exemples en termes du polytope ssoié à l ompti(tionF oit G un groupe rédutif onnexeD X une ompti(tion lisse et pno de GD de polytope ssoié P F oit Φ le système de rines de GD Φ + un hoix de rines positivesF xotons P + l9intersetion de P ve l hmre de eyl posiE tiveD 2ρ l somme des rines positivesD et Ξ l9intérieur reltif du ône engendré pr les rines simplesF xous identi(ons a et son dul pr le hoix d9un produit slireD et notons dp l mesure de veesgue sur a normlisée pr le réseu des rtèresF Théorème 1.5. vorsque G est semisimpleD Ξ est le ône ouvert engendré pr les rines simples de ΦF vorsque G n9est ps semisimpleD l dimension de Ξ est stritement inférieure u rng r de GF in prtiulierD lorsque G est un toreD ρ est l9origine et Ξ = {0}D de sorte que l9on retrouve le ritère usuel pour les vriétés toriquesF in e'etD dns e s l mesure de huistermtErekmn est l mesure de veesgue sur P = P + D et le ritère signi(e que le ryentre de P est l9origineF

xous donnons ensuite quelques exemples de luls et otenons de nouE veux exemples de vriétés de uählerEiinstein pnoD mis ussi un exemple de ompti(tion d9un groupe semisimple qui n9dmet uune métrique de uählerEiinsteinF Exemple 1.5. ves ompti(tions mgni(ques des groupes semisimples irE rédutiles de rng deux dmettent des métriques de uählerEiinsteinF ges exemples sont de nouveux exemples de métriques de uählerEiinsteinF r exempleD l onnissne de l omposnte onnexe de l9identité de leur IU groupe d9utomorphismes @voir u hpitre RA permet de voir qu9ils ne sont ni homogènesD ni toriquesF v9exemple suivntD qui n9dmet ps de métrique de uählerEiinsteinD montre en prtiulier que l9nnultion de l9invrint de putki sur une ompti(tion de groupe ne su0t ps à ssurer l9existene d9une métrique de uählerEiinsteinD lors que 9étit le s pour les lsses de vriétés sphériques déjà étudiées @homogènesD toriques ou horosphériquesAF Exemple 1.6. v9élté de l ompti(tion mgni(que de Sp(4) en l9orite fermée est une ompti(tion lisse de pno qui n9dmet ps de métrique de uählerEiinsteinF our prouver le résulttD l première étpe est de trduire l9éqution de uählerEiinsteinD et plus générlement l9éqution de l méthode de ontinuitéD restreinte à l9orite ouverte omme une éqution de wongeEempère réelle sur un ône de R r F ve lul du wongeEempère omplexe e'etué u hpitre Q permet de le fireD et nous otenons que le potentiel onvexe u t d9une solution K × KEinvrinte de l méthode de ontinuité u temps t véri(e

MA R (u t )(x) α∈Φ + α(∇u t (x)) 2 = e -(tut+(1-t)u ref )(x) J(x)
(2P + ) -4ρ)(ξ) = (t ∞ -1)(v -4ρ)(ξ),
où t ∞ est l limite des t pour lesquels |x t | est (niD et v est l fontion support du polytope 2P F gei permet d9otenir l ondition su0snte de notre résulttF our l ondition néessireDnous l9otenons églement grâe à l9nnultion de l9intégrle préédenteD ppliquée à ν = u -ln J où u est le potentiel onvexe d9une métrique uählerEiinsteinF ppelons que dns le s toriqueD l ondition du théorème est équivlent à l9nnultion de l9invrint de putkiF hns le s où G n9est ps un toreD e n9est plus le sF in prtiulierD si X est l ompti(tion mgni(que d9un groupe semisimpleD l9nnultion de l9invrint de putki ne donne uune informtionF g9est pour rempler el que nous vons utilisé le ryentre du polytope P + pr rpport à l mesure de huistermtErekmnD qui est un invrint des ompti(tions de groupesF i G n9est ps un toreD l ondition n9est ps l9nnultion de et invrintD mis le fit que le ryentre soit dns une ertine zone du polytopeF he mnière équivlenteD omme l9nnultion du putki est équivlente à l9nnultion d9un nomre (ni d9intégrlesD iiD nous otenons un ensemle (ni d9inéglités et d9églités à stisfire pr des intégrlesF endnt l preuve de notre ritèreD nous lulons églement l plus grnde orne inférieure de ii de l vriété lorsqu9elle n9dmet ps de métrique de uählerEiinsteinF g9est le temps mximl d9existene d9une solution dns l méE thode de ontinuitéD mis ussi l orne supérieure des t < 1 tels qu9il existe une forme de uähler ω dns c 1 (X) ve Ric(ω) ≥ tω zéIIF Théorème 1.6. Supposons qu'il n'y a pas de métriques de Kähler-Einstein sur X et soit R(X) la plus grande borne inférieure de Ricci de X. Alors

R(X) = sup t < 1; t 1 -t (2ρ -bar DH (P + )) + 2ρ ∈ (P + + (-Ξ)) .
Introduction 2.1 Fano Kähler-Einstein metrics 2.1.1 Kähler-Einstein metrics e iemnnin metri g is iinstein if its ii urvture Ric(g) stis(es g = λRic(g) for some onstnt λF e uählerEiinstein metri on omplex mnifold X is iemnnin metri tht is oth uähler nd iinsteinF he dt of uähler metri g is equivlent to the dt of its ssoited uähler form ωF sn similr wyD the ii urvture of uähler metri n e onsidered s (1, 1)Eform tht we denote y Ric(ω)F vet us desrie this form lollyF e uähler form ω on X n e written on su0iently smll lol hrt s i∂∂ϕ for some smooth nd stritly plurisuhrE moni funtion ϕ on n open suset of C n F vet Hess C (ϕ) denote the omplex ressin of ϕD i.e. the mtrix whose oe0ients re the ∂ 2 ϕ ∂zi∂zj F ell tht funtion ϕ is stritly plurisuhrmoni @pshA if its omplex ressin is positive de(nite everywhereF he ii form of ω is then de(ned in this hrt y Ric(ω) = i∂∂ (-ln det Hess C (ϕ)) .

prom now on we will ssume tht the mnifold X is omptF he min oE servtion to do is tht for ny uähler metri ω on the ompt uähler mnifold XD the ii form of ω lies in (xed ohomology lss depending only on XD whih is the (rst ghern lss c 1 (X) of XF xow onsider the uählerEiinstein equtionD in terms of formsX

Ric(ω) = λω
for some onstnt λF essume tht ω is uähler formD solution to this equtionF sf λ is negtiveD tht mens tht there is uähler form -λω in the lss -c 1 (X) whih mens c 1 (X) < 0F sf λ is zeroD then tht mens tht 0 = Ric(ω) ∈ c 1 (X) nd thus tht the (rst ghern lss vnishesF wnifolds stisfying this re lled gliEu mnifoldsF pinlly if λ is positiveD tht mens tht there is uähler PH form in c 1 (X)D i.e. tht this lss is positiveD nd the mnifold X is then lled pnoF his disussion shows (rst tht the uählerEiinstein eqution nnot hve solution when the (rst ghern lss of the omplex mnifold X is not de(nite or zeroF purthermoreD it shows tht given mnifold X where there ould e uählerEiinstein metri with non zero onstnt λD one hs only to onsider uähler metris in the lss c 1 (X)D or -c 1 (X) depending on the sign of λF he onstnt λD when it is not zeroD n e normlized to either I or EIF essume now tht Ric(ω) nd λω re in the sme lssF e will explin how the uählerEiinstein eqution redues to hi on funtion thnks to the ∂∂ElemmF vet ω ref e (xed referene uähler metri in the lss of ωF fy the ∂∂Elemm we n writeD on one hnd till lollyD this is equivlent to

ω = ω ref + i∂∂ϕ
-∂∂ ln det Hess C (ϕ ref + ϕ) det Hess C (ϕ ref ) = -∂∂(f -λϕ).
fut we n write the left hnd side s -∂∂ ln ω n ω n ref where n is the dimension of XD nd then oth expressions re vlid glollyF o we hve ω n = e f +c-λϕ ω n ref for some onstnt c @ ∂∂Eext funtion on X whih is omptAF he onstnt c is determined y the volume of ω n D nd n e sored in f whih we de(ned only up to n dditive onstntF pinlly we n write

(ω ref + i∂∂ϕ) n = e f -λϕ ω n ref nd this
hi in φ is in ft equivlent to the uählerEiinstein equtionF he se when c 1 (X) < 0 ws solved y euin in euUTX there lwys exists uählerEiinstein metriD nd the se when c 1 (X) = 0 ws solved y uF sn ft the following more generl @gliEuA heoremD onjetured y gliD ws proved y u PI Theorem. [Yau78] Let [ω 0 ] be the class of a Kähler form on X and θ a form representing c 1 (X), then there exists a Kähler metric ω ∈ [ω 0 ] such that Ric(ω) = θ.

2.1.2 Fano Kähler-Einstein metrics sn the pno seD the uählerEiinstein eqution is not lwys solvleF Obstructions he (rst min ostrutionD proved y wtsushimD shows tht the onneted omponent ontining the identity of the utomorphism group of pno mniE fold dmitting uählerEiinstein metri must e redutive groupF st gives even more s it sys tht the group of holomorphi isometries of uählerEiinstein metri is s ig s possileF Theorem. [Mat57] Let X be a Fano manifold and assume that it admits a Kähler-Einstein metric. Then the connected component of the identity in the automorphism group Aut 0 (X) is a reductive complex algebraic group, and the group of holomorphic isometries of a Kähler-Einstein metric is a maximal compact subgroup of Aut 0 (X).

putki then found n integrl invrint putVQD now lled the Futaki invari-antD whih is vie hrter from the vie lger of holomorphi vetor (elds η(X) of X to RF putki proved tht it vnishes whenever X dmits uählerE iinstein metriF emrk tht the vie lger of holomorphi vetor (elds is the vie lger of the vie group Aut 0 (X)F he previous ostrution thus implies tht η(X) is redutive vie lgerF

The continuity method yn the positive sideD prts of the work tht hs een done for the resolution of the uählerEiinstein eqution in the se c 1 (X) ≤ 0 n still e used in the pno seF xmely these ses were solved using ontinuity method nd deriving priori estimtes on the solutionsF he ontinuity method onsists in onsidering insted of the uählerEiinstein eqution lone fmily indexed y t ∈ [0, 1] of equtionsD where the eqution t t = 1 is the uählerEiinstein equtionF he im is to prove tht the set of t suh tht there exists solution is nonEemptyD open nd losedF gonnexity then implies tht there is solution t time t = 1F he work of euin nd u n e used to show thtD in the ontinuity method tht we will next desrieD we hve opennessF he losedness is where there re ostrutionsD ut we n use their work to redue to determining priori C 0 estimtes on the solutionsF he fmily of equtions in the @usulA ontinuity method for the pno se is the followingX Tian's α-invariant in otined in iVU su0ient ondition to get C 0 estimtesF st inE volves n invrint of the mnifoldD α(X) D lled the αEinvrintD tht enodes the possile singulrities of singulr hermitin metris with non negtive urE vture on -K X F sf this invrint stis(es α(X) > n/(n + 1) then there exists uählerEiinstein metri on XF sn the se of mnifold whih dmits n tion of ompt groupD this invrint n e re(ned y onsidering only metris invrint under this group tionF he su0ient ondition for the existene of uählerEiinstein metri remin the smeF wore generllyD one n de(ne the lph invrint for ny mple line undle on omplex mnifold XF K-stability he iggest dvne in the pno uählerEiinstein prolem in reent yers ws the resolution of the uEinEhonldson onjeture y ghenEhonldsonEun ghISD ghISD ghIS @see lso iAF his result reltes the existene of uählerEiinstein metri on pno mnifold with ondition of lgeroE geometri stilityF st gives neessry nd su0ient ondition for the existeneF nfortuntely for the purpose of (nding exmples of uählerEiinstein metrisD this ondition is not esy to hek in prtie for most mnifoldsD nd is not yet very well understoodF 2.1.3 Examples of Fano Kähler-Einstein manifolds sn dimension oneD the only exmple of ompt uähler mnifold tht is pno is the projetive line P 1 D whih dmits s uählerEiinstein metri the puiniE tudy metriF por pno surfesD lso lled hel ezzo surfesD the uählerE iinstein prolem ws solved y in iWH who showed tht in this se the existene of uählerEiinstein metri is equivlent to the vnishing of the putki invrintD or even to the ft tht the vie lger of holomorphi vetor (elds is redutiveF his is no longer true in higher dimensions s in showed y n exmple in iWUF vet us desrie some of the lrgest fmilies of exmples of uählerEiinstein pno mnifoldsF he (rst exmple of pno uählerEiinstein mnifold in ny dimension is of ourse the projetive spe P n equipped with the puiniEtudy metriF wore generllyD ny pno ompt homogeneous mnifold dmits uählerEiinstein metriF PQ he su0ient ondition given y in in terms of αEinvrint is very powerful tool tht llowed to (nd mny fmilies of exmples of pno uählerE iinstein metris @see e.g. fegHPD iD üÿIQAF por exmpleD in introdued this invrint to prove tht permt hypersurfe of dimension n with degree greter thn n -1 dmits uählerEiinstein metriF sn the se of tori mnifoldsD i.e. mnifolds of dimension n equipped with n tion of the torus (C * ) n with n open nd dense oritD the (rst mjor existene result ws proved y ftyrev nd elivnov fWW using the αEinvrintF hey showed tht ll pno tori mnifolds tht were symmetricD i.e. for whih there exists ompt sugroup of utomorphisms leving no nonzero holomorphi vetor (eld invrintD dmitted uählerEiinstein metriF his did not solve however the uählerEiinstein prolem for tori mnifoldsF sndeedD for smll dimensions @n ≤ 6AD ny pno tori mnifolds either dmits no uählerEiinstein metri or is symmetriF trting from dimension UD there re nonEsymmetri uählerEiinstein tori mnifoldsF he exmples were found y xill nd 'enholz xII with omputer ssistneD y using the theorem of ng nd hu who ompletely solved the uählerEiinstein prolem for tori mnifoldsF Theorem. [WZ04] Let X be a Fano toric manifold. Then X admits a Kähler-Einstein metric if and only if the Futaki invariant of X vanishes.

(ω ref + i∂∂ϕ t ) n = e f -
es wuhi proved wVUD the putki invrint of pno tori mnifold is given y the ryenter of the polytope ssoited to the mnifoldF sn prtiulrD the putki invrint vnishes if nd only if the ryenter of the polytope is the originF he ryenter of polytope with integrl verties n e omputedD so the riterion n e heked in prtie one the polytope of tori mnifold is givenF xill nd 'enholz used lssi(tion of the orresponding polytopes up to dimension V to (nd n exmple of polytope of pno tori mnifold with ryenter the origin ut not symmetriF fy ong9s omputtion of the αEinvrints of pno tori mnifolds onHSD not only the exmples were nonE symmetri ut lso in9s riterion in terms of αEinvrint ould not e used to get existene of uählerEiinstein metriF ng nd hu9s method of proof is to get diretly C 0 estimtes long the ontinuity methodF ine our work follows the sme strtegy s they doD let us explin the strting point of their methodF yne min ingredient is to use wtE sushim9s theorem to derive tht uählerEiinstein metri on tori mnifold must e invrint under the tion of the ompt torus (S 1 ) n ⊂ (C * ) n F yne n further ssume thtD if we strt from ompt torus invrint referene metri in the ontinuity methodD ll the solutions for ny t re invrint under the ompt torusF hen restriting to the open (C * ) n oritD they trnslte the uählerEiinstein eqution s rel wongeEempère eqution on R n involving onvex rel funE tion with presried ehvior t in(nityF o see thisD the (rst remrk is tht the quotient of (C * ) n y (S 1 ) n is isomorphi to R n F eondlyD the uählerEiinstein eqution n e lolly expressed s omplex wongeEempère eqution in the PR lol potentil of the metriD whih is psh funtionF hirdlyD in n pproprite hoie of oordintes @logrithmi oordintesAD the omplex wongeEempère of (S 1 ) n Einvrint funtion is equl to the rel wongeEempère in the R n quoE tient oordintesD nd psh funtion eomes onvex funtion on the quotientF pinlly the onvex funtion otined on R n must stisfy some symptoti eE hvior onditions if it omes from metri tht extends from the open orit to the whole mnifoldF sn this settingD ng nd hu re then le to otin estimtes for the solutions of suh rel wongeEempère equtions on R n F en almost-homogeneous manifold is mnifold equipped with n tion of omplex vie group with dense oritF por exmpleD tori mnifolds re lmostEhomogeneous under the torus tionF he (rst exmples of non homoE geneous pno uählerEiinstein mnifolds were otined y kne nd uoiso uVTD uVV s P 1 Eundles over other pno uählerEiinstein mnifoldsF he exmples they exhiited were lmostEhomogeneousF es generliztion of the exmples given y uoiso nd kneD odest nd piro studied the existene of uählerEiinstein metris on homogeneous tori undles IHF hese re homogeneous undles over the )g mnifold of omplex semisimple groupD with (er tori vrietyF hey studied these vrieties to determine when they were pnoD nd to determine whenD in tht seD the putki invrint vnishedF emrk tht these vrieties re lmostE homogeneous for redutive groupD nd tht the quotient of the open orit under the mximl ompt sugroup is gin isomorphi to R r where here r is the dimension of the tori (erF es in the tori seD one n use the ompt group invrine to simplify the uählerEiinstein equtionD nd one still otins rel wongeEempère eqution on R r with the sme type of presried symptoti ehvior s in the tori seF epplying the work of ng nd hu with minor modi(tionsD odest nd piro onlude tht their mnifolds dmit uählerE iinstein metri if nd only if the putki invrint vnishesF o onlude this nonEexhustive list of exmplesD let us mention tht qun studied the existene of uählerEiinstein @nd more generlly nonilA metris on lmostEhomogeneous mnifolds of ohomogeneity oneD whih mens tht the mximl ompt sugroup ts with n hypersurfe oritF yn these mniE foldsD whih re hrd to ompletely lssifyD the uählerEiinstein eqution must trnslte to some ordinry di'erentil eqution thnks to the ompt group E tionF he ltest rtiles y qun on this sujet re quIID quIID quIID quIQF 2.1.4 Group compactications he pno mnifolds tht we study in this thesis re some group ompti(E tionsF vet G e onneted redutive liner omplex lgeri group @exmples inlude GL n (C)D SL n (C)D (C * ) n D etFAF e ompt tori vriety Z of dimension r is ompti(tion of the torus T = (C * ) r F he ft tht the torus ts on the whole mnifold Z nd tht T ⊂ Z is n orit for theD syD left multiplitionD mens tht Z is in ft T Eequivrint ompti(tion of T F sn other wordsD it is the dt of n PS emedding j of T in ompt mnifold Z suh tht Z dmits T tion nd j is T Eequivrint s mp i.e. j(t • z) = t • j(z)F iquivlentlyD we ould hve used the tion of T on itself y multiplition on the right y the inverseF ine T is elinD oth tions re equivlentF hen onsidering non elin redutive group GD the two tions re no longer equivlentF yne ould still onsider leftEGEequivrint ompti(tionsD or rightEGEequivrint ompti(tionsD ut we will onsider the stronger sE sumption tht the ompti(tion is equivrint under oth tionsF e thus onsider G × GEequivrint ompti(tions of omplex onneted redutive liner lgeri group GF por simpliity we will ll these ompti(tions of GD or group ompti(tions when the group is not (xedF yne min reson to onsider oth tions is tht suh vrieties re spherilF e vriety X is lled spheril if it is equipped with n tion of redutive group G suh tht forel sugroup B of G ts with n open orit on XF st implies tht G hs n open nd dense orit in XD whih is homogeneous spe under GF sn prtiulrD spheril vriety is lmost homogeneousF hey re the lmost homogeneous vrieties for whih the lssi(tion theory is the most dvned nd where mny of the prolems solved for tori vrieties hve hope to (nd similr resolutionF sn ftD most of the exmples of pno uählerEiinstein mnifolds mentioned previously re spheril vrietiesF st is esy to see tht ompt mnifolds homogeneous under redutive group re spherilD nd we hve seen tht tori vrieties re spherilF he exmples studied y odest nd piro elong to the fmily of horospheril vrietiesD nd ontins oth homogeneous mnifolds nd tori mnifoldsF es we hve seenD the tori vrieties re lso group ompti(tionsD for elin lgeri groupsF yn the oppositeD the 4lest elin4 redutive groups re the semisimple groupsF he most fmous exmples of ompti(tions of semisimple groups re the wonderful ompti(tions of djoint semisimple groups onstruted y he gonini nd roesi hgVQF sn shortD these re smooth nd pno mnifoldsD onstruted s the losure of the G × GEorit of the lss of the identity in P(End(V λ )D where V λ is the irredutile representtion of G of highest weight regulr nd dominnt weight λF honldson suggested in his survey honHV to study the uählerEiinstein exE istene prolemD nd more generlly the existene of extreml or onstnt slr urvture uähler metrisD for spheril vrietiesF hey (t in the (rst tegory of vrieties presented y honldson in honHVD etion R euse they re mulE tipliity freeF emrk tht the smller lss of group ompti(tions tht we onsider here in ft lso (ts in the seond tegory of mnifolds presented y honldson in honHVD etion RF PT 2.2 Results and organization of the work 2.2.1 Chapter 3 he im of ghpter Q is to give the tools neessry to work on redutive group GD nd thus on the open nd dense orit of ompti(tion of GF he (rst setion provides the de(nition of redutive groups nd rells mny usul tools to study themF he root system Φ ssoited to suh group will e very importnt s our results will lwys e stted in terms of this root systemD for the prt oming from the ig oritF he seond setion dels with the tion of mximl ompt group K on G oth on the left nd on the rightF por redutive groupD the orresponding quotient of G y K × K is losed one in R r where r is the rnk of the groupF e identify it with the losed positive eyl hmer a + in the grtn sulger a of the vie lger of mximl torus of GF e onsider funtions on G invrint under K × KF hese orrespond to funtions on the quotientD nd we rell how result of ezd nd voe llows to hrterize plurisuhrmoniity of funtion on G s onvexity of the funtion on the quotientF hen we rell the KAK integrtion formulD whih trnsltes the integrl of K × KEinvrint funtion on G with respet to rr mesure s n integrl on the quotient with respet to mesure on the quotient oneD solutely ontinuous with respet to veesgue mesureD with n expliit potentil denoted y JF pinlly we ompute the omplex ressin Hess C (ψ) of K × KEinvrint funtion ψ on G in hoie of lol oordintesF Theorem 2.1. Let ψ be a K × K invariant function on G, and f the associated function on a. Then in an appropriate choice of coordinates and for a ∈ a + , the complex Hessian matrix of ψ is diagonal by blocks, equal to:

Hess C (ψ)(exp(a)) =         1 4 Hess R (f )(a) 0 0 0 M α1 (a) 0 0 0 . . . . . . . . . . . . . . . 0 0 0 M αp (a)        
where Hess R (f ) is the real Hessian of f , the (α i ) i∈{1,...,p} run over the positive roots of Φ and M α is dened by:

M α (a) = 1 2 α(∇f (a)) coth(α(a)) i -i coth(α(a))
where ∇f is the gradient of f with respect to a xed scalar product on a extending the Killing form on the semisimple part.

es we hve seen in this introdutionD this omputtion will e very importnt to express the uählerEiinstein eqution on the ig orit of group ompti(E tionF wore preiselyD we will use the determinnt of this ressinD the omplex wongeEempèreD denoted y MA C (ψ)F PU Corollary 2.2. Let ψ be a K ×K invariant function on G, and f the associated function on a. Then in an appropriate choice of coordinates and at a ∈ a + , if r denotes the rank of G and p the number of positive roots, we have

MA C (ψ)(exp(a)) = 1 4 r+p MA R (f )(a) α∈Φ + α(∇f (a)) 2 sinh 2 (α(a))
.

Chapter 4

ghpter R is n introdution to group ompti(tions with fous on the polytope ssoited to polriztion of suh vrietyF he polytope ontins ll the informtion out the oundry divisorD nd out the hosen mple line undleF his is generliztion of the polytope ssoited to polrized tori vriety nd in ft we use this usul orrespondene to present the se of group ompti(tionsF sndeedD given ompti(tion X of GD onsider T mximl torus in G nd Z its losure in XF hen Z is tori vriety ontining ll the informtion out the ompti(tion @if G is (xedAF e give some exmples of smooth nd pno group ompti(tionsF here re mny suh exmplesX every djoint semisimple group dmits nonil group ompti(tionD lled its wonderful ompti(tionD nd it is smooth nd pnoF efter thtD we onsider hermitin metris with non negtive urvture on linerized mple line undles on group ompti(tionsF o suh metri h we ssoite plurisuhrmoni funtion on GF rovided h is K ×KEinvrintD this determinesD y ghpter QD onvex funtion on a + D lled the onvex potentil of hF e prove tht the ft tht h is metri on polriztion of X presries the symptoti ehvior of the onvex funtionD nd desrie this symptoti ehvior in terms of the ssoited polytopeF Theorem 2.3. Let (X, L) be a polarized compactication of G, with associated polytope P . The singular hermitian K × K-invariant metrics h on L with non negative current curvature are in bijection with the convex W -invariant functions ϕ : a -→ R satisfying the condition that there exists a

C 1 ∈ R such that ϕ(x) ≤ f P (x) + C 1
on a with f P the support function of the polytope 2P . Furthermore, h is locally bounded if and only if there exists in addition a constant C 2 such that

f P (x) + C 2 ≤ ϕ(x) ≤ f P (x) + C 1 .
The function ϕ associated to h is its convex potential.

et the end of the hpter we omine ll we hve presented up to here @inluding the omputtion of the omplex ressinA to give link etween the degree of n mple line undle L nd the volume of its polytope with respet to the huistermtErekmn mesureF PV Proposition 2.4. Let (X, L) be a smooth polarized compactication of G, corresponding to the polytope P . Then

deg(L) = C 2P + α∈Φ + (α(p)) 2 dp
for some constant C depending only on the group G.

his result is onsistent with the expliit formul otined y uzrnovskii uzVU nd frion friVWD ut we do not determine the onstnt C expliitly hereD euse we will not need it in the followingF 2.2.3 Chapter 5 sn ghpter S we ompute the αEinvrint of n mple line undle on pno group ompti(tionF wore preisely we ompute the αEinvrint with respet to the tion of mximl ompt sugroup K × K of G × GF e do this y omputing the log nonil thresholds of ny K ×KEinvrint non negtively urved hermitin metri h on L in terms of onvex ody sE soited to hF his onvex odyD tht we ll the xewton ody of hD is the domin of the vegendreEpenhel trnsform of the onvex potentil of hF he method to ompute the log nonil threshold is to redue to n integrility riterion on the open orit GD then on R r y the KAK integrtion formulF et this point we hve to determine when the exponentil of onve funtion is integrle with respet to the mesure with potentil J ginst the veesgue mesureF sing the integrility riterion derived y quenni s n nlyti proof of rowld9s omputtion of log nonil thresholds of monomil idelsD we n get suh riterion in terms of the xewton odyF o get n expression of the αEinvrintD we show whih re the metris with potentilly miniml log nonil thresholdF he omintion of these llows to prove the following theoremF Theorem 2.5. Let G be a connected complex reductive group, and X be a smooth Fano compactication of G. Let L be an ample G × G-linearized line bundle on X, whose associated polytope is P . Denote by Q the polytope associated to the anticanonical line bundle -K X . Then

α K×K (L) = sup{c > 0; c(P + (-P W )) ⊂ Q H},
where P W denotes the subset of W -invariant points of P , H is the convex hull of the images by W of the sum of positive roots, and is the Minkowski dierence of convex sets.

woreoverD if the group G is semisimpleD or if there re enough dditionl symmetriesD there is metri whose log nonil threshold is the αEinvrintF Corollary 2.6. Assume that G is a semisimple group. Then

α K×K (L) = sup{c ≥ 0; cP ⊂ Q H}.
It is also the inradius of Q H with respect to P . PW pinlly we ompute the αEinvrint for some exmples of group ompti(E tionsF Corollary 2.7. Let X be the wonderful compactication of PSL n+1 (C), then

α K×K (-K X ) = 1 1 + n 2 ( n 2 + 1)
.

prom this exmple we see tht suh mnifolds do not stisfy in9s su0ient ondition in generlF sn ft it is not hrd to onvine oneself tht it will not e stis(ed for most wonderful ompti(tionsF hus we need to use nother methodF he formul we otined pplies lso to tori mnifolds nd llows to reover our previous omputtion of the αEinvrint on tori mnifolds helISF he text of this rtile is lso reprodued in eppendix eF 2.2.4 Chapter 6 ghpter T ontins the min result of the thesisF xmely we determine neessry nd su0ient ondition for the existene of uählerEiinstein metri on group ompti(tionF his ondition is further numerilly omputle in terms of the polytope of the group ompti(tionF vet G e onneted omplex redutive groupD X smooth nd pno omE pti(tion of GD with ssoited polytope P F vet Φ e the root system of GD Φ + system of positive rootsF e denote y P + the intersetion of P with the positive eyl hmerD y 2ρ the sum of the positive rootsD nd y Ξ the reltive interior of the losed one generted y the simple rootsF pinllyD dp denotes the veesgue mesure normlized y the lttie of hrtersF Theorem 2.8. There exists a Kähler-Einstein metric on the smooth and Fano group compactication X if and only if the barycenter bar DH (P + ) :=

P + p α∈Φ + (α(p)) 2 dp P + α∈Φ + (α(p)) 2 dp -1 of P + with respect to the Duistermaat-Heckman measure is in 2ρ + Ξ.
hen G is semisimpleD Ξ is the open one generted y the simple roots of ΦF emrk tht when G is not semisimpleD the dimension of Ξ is stritly smller thn rF sn prtiulrD for G torusD ρ is the origin nd Ξ = {0}D so we reover the usul tori riterionF sndeedD the huistermtErekmn mesure then is just the veesgue mesure on P = P + D so the riterion is just tht the ryenter of P is the originF e provide some exmples of omputtions nd otin new exmples of uählerEiinstein pno mnifolds ut lso n exmple of ompti(tion of semisimple group tht dmits no uählerEiinstein metrisF QH Example 2.9. he wonderful ompti(tions of semisimple rnk two omE plex groups dmit uählerEiinstein metrisF hese exmples re new exmples of uählerEiinstein metrisF he knowlE edge of the onneted omponent of their utomorphism groups @see ghpter RA shows tht they re not homogeneousD nd not toriF he following non uählerEiinstein exmple shows in prtiulr tht the vnishing of the putki invrint on group ompti(tion is not enough to ensure the existene of uählerEiinstein metriD wheres it ws for the lsses of spheril vrieties previously studied @homogeneousD tori or horospherilAF Example 2.10. he low up of the wonderful ompti(tion of Sp 4 (C) t the losed orit is smooth pno group ompti(tion whih dmits no uählerE iinstein metrisF o prove the theoremD the (rst step is to trnslte the uählerEiinstein equE tion restrited to the dense orit s rel wongeEempère eqution on one in R r F hen we follow the sme strtegy s ng nd hu to get C 0 estimtesF roweverD here the eqution is di'erentD the funtions we study re di'erent nd one hs to tke extr re of wht hppens ner the wlls of the oneF purthermoreD ng nd hu used the vnishing of the putki invrint to get C 0 estimtesF sn the se of ompti(tion of semisimple groupD the vnishing of the putki invrint does not give ny informtionF o reple this we (nd di'erent integrl invrint for pno group ompti(tionsD whih is the ryenter of the polytope with respet to the huistermtErekmn mesureF he onditionD s seen in the theoremD is no longer the vnishing of this integrl invrintD ut the ft tht this ryenter is in ertin zone in the polytopeF iquivlentlyD s the vnishing of the putki invrint is equivlent to the vnishing of (nite numer of integrlsD here we otin set of inequlities tht must e stis(ed y some integrlsF huring the proofD we lso ompute the gretest ii lower ound when there re no uählerEiinstein metrisF his is the mximl time of existene of solution in the ontinuity methodD whih is how we ompute itD ut lso the supremum of ll t < 1 suh tht there exists uähler form ω in c 1 (X) with Ric(ω) ≥ tω zéIIF Theorem 2.11. Assume there are no Kähler-Einstein metrics on X and let R(X) be the greatest Ricci lower bound of X. Then

R(X) = sup t < 1; t 1 -t (2ρ -bar DH (P + )) + 2ρ ∈ (P + + (-Ξ)) .
Example 2.12. hen X is the mnifold from exmple PFIHD we ompute the ext vlue of the gretest ii lower oundD whih is R(X) = 1046175339 1236719713 0.8459 . . .

QI Chapter 3

Reductive groups and invariant functions sn this hpter we (rst introdue redutive groups nd rell some results out redutive nd semisimple groups tht will e used in the rest of the textF e very importnt omintoril dt ssoited to suh group is its root systemF wost of the (rst setion of this hpter dels with the root systemD the orresponding root deomposition of the vie lger of the groupD nd the lssi(tion of redutive groups tht follows from itF his lssi(tion will llow to understnd some exmples of group ompti(tions tht we will lter onsiderF he seond setion of this hpter dels with the tion of the mximl ompt sugroup of redutive group y left nd right multiplitionF his tion is extremely importnt in the setting of uählerEiinstein metrisD nd we use lssil redutive group deomposition to ompute the omplex ressin nd omplex wongeEempère of funtion on the group invrint under oth tions of the ompt groupF 3.1 Reductive groups 3.1.1 Denition and maximal compact subgroup here re severl de(nitions for redutive groupsD tht re equivlent over the (eld CF vet G e omplex onneted liner lgeri group iFeF onneted lgeri sugroup of some GL N (C)F he (rst de(nition we give is in terms of the unipotent rdil of the group GF hen we will give other hrteriztions tht re equivlentF vet us (rst rell some usul notions in group theoryF e use the ooks prWVD forWI s referenes for this setionF he derived subgroup D(G) of group G is the sugroup generted y ll ommuttors of elements of GF qiven group GD one n onsider its derived QP series

G D(G) D(D(G)) • • •
e group G is solvable if its derived series eventully rehes the trivil sugroup {e} of GF he radical R(G) of n lgeri group G is the mximl onnetedD norml nd solvle sugroup of GF en lgeri group G is sid unipotent if it is isomorphi to losed sugroup of some U T n D where U T n denotes the group of n × n upper tringulr mtries with ll digonl oe0ients equl to oneF he unipotent radical R u (G) of n lgeri group G is the mximl onnetedD norml nd unipotent sugroup of GF his llows to de(ne redutive nd semisimple groupsF Example 3.9. he vie lger of SL n (C) is denoted y sl n (C) nd onsists of the mtries whose tre is zeroF vet us desrie the uilling form on sl 2 (C)F e usul sis for sl 2 (C) is the sis (h 2 , g 2 , g -2 ) where

Denition 3.1. en lgeri group G is reductive if its unipotent rdil R u (G) is trivilF st is semisimple if its rdil R(G) is trivilF Example 3.2. e torus (C * ) n is redutive group tht is not semisimpleD nd unipotent group is not redutiveF Remark 3.3. emrk tht R u (G) ⊂ R(G)D so
h 2 = 1 0 0 -1 , g 2 = 0 1 0 0 , g -2 = 0 0 1 0 .
hen we n express the uilling formX

B(a 1 h 2 + a 2 g 2 + a 3 g -2 , b 1 h 2 + b 2 g 2 + b 3 g -2 ) = 8a 1 b 1 + 4(a 2 b 3 + a 3 b 2 ).
e semisimple Lie algebra is vie lger tht is diret sum of simple vie lgersF he vie lger of semisimple vie group is semisimpleF e fundmentl result out the uilling form is tht vie lger g is semiE simple if nd only if its uilling form is non degenerteF purthermore it is then negtive de(nite on the ompt rel form k of gF st is ler on the other hnd tht the uilling form on n elin vie lger vnishes everywhereF sn the se of the vie lger g of redutive groupD roposition QFR shows tht we n deompose the vie lger s g = Z(g) ⊕ [g, g] where [g, g] is the vie lger of D(G) nd s suh is semisimpleD nd Z(g) is elin nd is the enter of gF he uilling form B on g is nondegenerte on [g, g] nd zero on Z(g)F e n lso oserve tht these two sulgers re orthogonl with respet to the uilling formF e n hoose ny nondegenerte iliner symmetri form on Z(g) to extend B| [g,g] to nondegenerte iliner symmetri form on the whole of gF e (x suh hoie in the followingD nd denote it y •, • F 3.1.3 Roots e egin y de(ning the root system ssoited to semisimple vie lgerD then relte this to semisimple groupsF his will lso e pplied to redutive groups through roposition QFRF QR Semisimple Lie algebras and root systems vet g e omplex semisimple vie lgerF ell tht g ts linerly on itself through the djoint tionF ghoose t mximl elin sulger of g onsisting of elements h suh tht ad(h) is digonlisleF estrited to tD whih is elinD the djoint tion is simultneously digonlisle nd the orresponding eigenspe deomposition of g is lled the root decompositionF sing the nottion

g α = {x ∈ g; ad(h)(x) = α(h)x ∀h ∈ t} for ny α ∈ t * D nd denoting y Φ the set of nonzero α ∈ t * suh tht g α = {0}D the root deomposition redsX g = g 0 ⊕ α∈Φ g α .
purthermoreD we hve g 0 = t rumUPD VFPD nd ny two grtn sulgers t re onjugte y n utomorphism of gF he set Φ is lled the root sytem of gD its elements the roots of gF he uilling form B llows to ssoite to eh root α ∈ Φ the unique element

h α of t suh tht α(h) = B(h α , h) for ll h ∈ tF vet us introdue lso the nottion (α, β) := B(h α , h β )F
he root system Φ stis(es the following onditionsX ! Φ spns rel suspe E of t * D of rel dimension equl to the omplex dimension of tD on whih (•, •) extends to positive de(nite formY ! if α ∈ ΦD there re extly two multiples of α in Φ whih re α nd -αY ! if α, β ∈ ΦD then the imge β -2(β,α) (α,α) α of β y the re)etion determined y α is in ΦY ! if α, β ∈ ΦD then 2(β,α) (α,α) ∈ ZF sn ftD these onditions re the xioms de(ning n abstract @reduedA root systems rumUPD WFPD nd the following lssil theorem sttes tht omplex semisimple vie lgers re lssi(ed y root systemsF Theorem 3.10. [Hum72,18.4] For any abstract root system Φ, there exists a unique (up to isomorphism) complex semisimple Lie algebra g whose root system is Φ.

sn dditionD the root systems of simple vie lgers re omintorilly lssi(edD nd the root system of diret sum is the produt of the root systemsD so ll semisimple omplex vie lgers re omintorilly lssi(ed in this wyF vet us lso reord some properties of the root deompositionF Proposition 3.11. [Hum72,8.3 and 8.4] Let α, β ∈ Φ, then the root space g α is of complex dimension one;

if α + β ∈ Φ then [g α , g β ] = g α+β ; the subspace [g α , g -α ] is in t and one dimensional; more precisely, if x ∈ g α and y ∈ g -α , then [x, y] = B(x, y)h α . QS wore generllyD if α, β ∈ a * D then [g α , g β ] ⊂ g α+β D so if 0 = α + β / ∈ Φ then [g α , g β ] = {0}F
o semisimple group G one n ssoite the root system Φ of its vie lger gF roweverD two non isomorphi semisimple groups n hve the sme vie lgerD for exmple this is the se with SL n nd PGL n F o distinguish two suh groups one needs extr dtF his will e disussed in the next setionF Root system of a reductive group vet now G e redutiveD with mximl ompt group KF ghoose S mximl torus of KD nd let T e its omplexi(tion in GF hen T is lso mximl torus of GF Denition 3.12. he omplex dimension rk(G) := r of T D whih is lso the rel dimension of SD is lled the rank of GF vet Φ e the root system of (G, T )D iFeF the root system of the semisimple prt [g, g] of g with the hoie of mximl elin sulger t ss the semisimple prt of the vie lger of T F he grtn involution θ indues n involution on t * ss whih preserves the rootsD nd we still denote y θ the orresponding involution of ΦF purthermoreD θ sends α ∈ Φ to -αF e will denote y a the vetor spe is ⊂ t where s = t ∩ k is the vie lger of SF e denote y A the imge of a in G y the exponentil mpF

Weights Semisimple case

vet Φ e the root system of semisimple vie lger gD let E e the suspe of t * generted y the roots of ΦF he(ne M sc to e the set of ll m ∈ E suh tht 2(m,α) (α,α) ∈ Z for ll α ∈ ΦD nd ll its elements the weights of ΦF he set M sc is lttie in EF e Weyl wall in E is hyperplne de(ned y n eqution of the form (x, α) = 0 for some root α ∈ ΦF gll weight m regular if (m, α) = 0 for ll α ∈ ΦD i.e. m is not on ny eyl wllF e suset ∆ of Φ is set of simple roots if it is sis of ED nd ny root in Φ hs either ll of its oordintes positive in this sis or ll of its oordintes negtiveF he set of roots with positive oordintes is then denoted y Φ + nd its elements re lled the positive roots of ΦF ghoose set of simple roots ∆ in ΦF e weight m is lled dominant if it stis(es (m, α) ≥ 0 for ll α ∈ ∆F his is equivlent to (m, α) ≥ 0 for ll α ∈ Φ + F he sis of E formed y the m i suh tht 2(mi,αj ) (αj ,αj ) = δ ij is lso sis of the lttie M sc F sts elements re lled the fundamental weights of ΦD nd the dominnt weights re the elements with positive oordintes in this sisF he losed one generted y the fundmentl weights

E + := { r i=1 x i m i ; x i ≥ 0}
QT is lled the positive closed Weyl chamber of EF e will sy tht its interior E + is the @openA positive Weyl chamber of EF he positive eyl hmer is onneted omponent of the omplement of the union of eyl wllsF he other omponents re lled the Weyl chambers of EF ih would e the positive eyl hmer for n pproprite hoie of simple rootsF sn ft the Weyl group W of ΦD de(ned s the (nite group generted y the re)etions m → m -2(m,α) (α,α) α for α ∈ ΦD ts trnsitively on the set of eyl hmersD nd the losed positive eyl hmer is fundmentl domin for the tion of W on EF vet G e semisimple group nd T e mximl torus in GF he(ne the lttie of weights M of G s the lttie of hrters of T F e will see tht it llows to distinguish the groups with the sme vie lgerF qiven root system ΦD let M ad e the lttie generted y the roots of ΦD nd rell tht M sc the lttie generted y the fundmentl weights of ΦF Theorem 3.13. [FH91, Theorem 23.16] Given a root system Φ and a lattice M between M ad and M sc , there exists a unique (up to isomorphism) semisimple group G whose root system is Φ and whose lattice of weights is M .

he semisimple group G ad with weight lttie M ad is lled adjointD nd the semisimple group G sc with weight lttie M sc is lled simply connectedF sn ft if M 1 ⊂ M 2 re the weight ltties of two semisimple groups G 1 nd G 2 with the sme root system then G 1 is isomorphi to the quotient of G 2 y (nite groupF Reductive case vet us now onsider redutive group GF e n still de(ne the lttie of weights M of G y hoosing mximl torus T nd onsidering its lttie of hrtersF purthermore we ssoite to G the root system Φ of its derived sugroup D(G) s eforeF hen the lssi(tion of redutive groups follows from roposition QFR nd heorem QFIQF emrk lso tht if t denotes the vie lger of the hosen mximl torusD we still hve the root deomposition with the properties of roposition QFIIX

g = t ⊕ α∈Φ g α .
vet lso N e the lttie of omplex one prmeter sugroups of T F e my lso ll N the lttie of coweights of GF foth M nd N re free elin groups of rnk r nturlly dul to eh otherF sdentify a with N ⊗ R nd a * with M ⊗ RF emrk tht if G is semisimpleD a * is extly EF ghoose set of positive roots Φ + F his llows to de(ne losed Weyl chamber a + in aD y x ∈ a + if nd only if α(x) ≥ 0 for ll α ∈ Φ + F henote y A + the suset of A whih is the imge y the exponentil mp of the losed eyl hmer a + F he open Weyl chamber a + is de(ned s the interior of the QU losed eyl hmerF sn the se when G is semisimple this is equivlently de(ned y x ∈ a + if nd only if α(x) > 0D ut not in the se of redutive groupsF he (nite group W = N G (T )/T is lled the Weyl group of G with respet to the mximl torus T F st is lso the eyl group of the root system ΦF st ts on T nd indues n tion on aF he closed eyl hmer a + is fundmentl domin for the tion of W on aF

Examples

Rank one here re only two semisimple groups of rnk oneF hey re PGL 2 (C) whih is djoint nd SL 2 (C) whih is simply onnetedF heir ommon vie lger is sl 2 (C) whih hs only two roots in its root systemF o there is only one root system of rnk one tht is denoted y A 1 F sdentifying one root with 2 ∈ R gives n identi(tion of a * with RF sf we hoose this root s the positive root then the positive eyl hmer is R + F he root lttieD generted y the rootsD is 2Z nd the weight lttie of sl 2 (C) is ZF he weight lttie of PGL 2 (C) is thus 2Z nd the weight lttie of SL 2 (C) is ZF sdentifying a with its dul we get lso tht the oweight lttie of

PGL 2 (C) is Z nd the oweight lttie of SL 2 (C) is 2ZF
Rank two here re four root systems of rnk PD up to isomorphismF yne is otined s produt of two opies of A 1 F his is for exmple the root system of SL 2 (C) × SL 2 (C)F he other three re irreduileF hey re denoted y A 2 D B 2 nd G 2 F pigures QFI DQFP nd QFQ give the representtions of these root systems in the usul euliden plne R 2 F he lk irles represent the rootsD the white irles represent the rest of the root lttie nd the rosses represent the points of the weight lttie tht re not in the root lttieF prom this nd heorem QFIQ we see tht there re two semisimple groups with root system A 2 D the djoint one whih is PGL 3 (C) nd the simply onneted one whih is SL 3 (C)F por type B 2 there is gin one djoint group SO 5 (C) nd one simply onneted group Sp 4 (C)F pinlly for type G 2 there is only one groupD denoted gin G 2 tht is oth djoint nd simply onnetedF yn the (gures re lso represented the eyl hmersD whih oinide for a nd a * in the identi(tions we mdeF 3.1.6 A basis of g e will now (x onneted redutive group G nd give sis of gD tking into ount the ft tht G is the omplexi(tion of ompt group KF his sis will give rise to lol omplex oordintes on G in whih we will e le to ompute the omplex ressin mtrix of K × KEinvrint funtion on GF ell tht we hve two deompositions of gX

QV pigure QFIX oot system A 2 pigure QFPX oot system B 2 QW pigure QFQX oot system G 2
! the grtn deomposition g = k ⊕ ikD ! nd the root deomposition g = t ⊕ α∈Φ g α F e wnt to omine these two to otin sis suitle for omputtionsF he(ne the following rel suspes of kX

k α := {x ∈ k; ad(h) 2 (x) = α(h) 2 x ∀h ∈ g 0 }, for α ∈ ΦF emrk tht k α = k -α F
st is linked with the root deomposition y the equlityX

g α ⊕ g -α = k α ⊕ ik α .
emrk tht if θ denotes the grtn involution then θ(g α ) = g -α F ih g α is of omplex dimension oneD so eh k α is of rel dimension twoF his gives deomposition of kX

k = s ⊕ α∈Φ + k α .
uppose for the moment tht g is semisimpleFhere is speil omplex sis of g dpted to the root deomposition relUVD ghpter sD vemm QFIF vet h α ∈ a = is e the unique element suh tht Example 3.14. sn the se of sl 2 (C)D we desried the uilling form previouslyD nd the grtn involution sends mtrix to the opposite of the onjugte of its trnsposeD so we get

B(h, h α ) = α(h) for ll h ∈ tF emrk tht h -α = -h α F
h 2 = 1/4 0 0 -1/4 , e 2 = 0 1/2 0 0 , e -2 = 0 0 1/2 0 ,
nd so the sis of isu 2 otined is formed y the mtries

e 2 -e -2 = 0 1/2 -1/2 0 , ie 2 + ie -2 = 0 i/2 i/2 0 . 3.2 K × K-invariant functions on G
sn this setion we will (rst rell the lssil KAK deomposition of redutive groupF his is the deomposition tht enodes the K × K orits of GD or equivlently tht desries the quotient under this tionF st will e used in the following to mnipulte K × KEinvrint funtions on GF e rell or prove here the min tools for thisF he (rst is the trnsltion of smoothness or positivity propertiesD the seE ond is n integrtion formul dpted to the KAK deompositionD nd the lstD whih is proved hereD is omputtion of the omplex ressin of K × KE invrint funtion in the sis de(ned in etion QFIFTF his gives lso n exE pression of the omplex wongeEempèreD whih is wht we will minly use in the next hptersF 3.2.1 KAK decomposition and invariant functions he KAK deomposition n e stted s the following deomposition of n element of GF Proposition 3.15. [Kna02, Theorem 7.39] Let g ∈ G be any element, then we can write g = k 1 tk 2 , with k 1 , k 2 ∈ K, and t ∈ A. Furthermore, in this decomposition, t is uniquely determined up to the action of the Weyl group W .

enother wy to stte this result is y sying tht ny g ∈ G n e written g = k 1 exp(a)k 2 D where k 1 , k 2 ∈ K nd a ∈ a + uniquely determined y gF sndeedD we hve seen efore tht a + is fundmentl domin for the tion of W on aF st mens lso tht the quotient of G y the tion of oth left nd right K tion n e identi(ed with A + D or a + F he KAK deomposition implies tht K × K invrint funtion ψ on G only depends on its vlues t points in AF iquivlentlyD one n onsider the funtion f de(ned on a y f (a) = ψ(exp(a))F purthermoreD the funtion f is W EinvrintF e reord here how some properties on ψ trnslte to properties on f F he (rst one is out smoothnessF RI Proposition 3.16. [FJ78, Theorem 4.1] The correspondence ψ → f gives a bijection between K×K-invariant functions on G (resp. smooth K×K-invariant functions on G) and W -invariant functions on a (resp. smooth W -invariant functions on a).

xextD the following result of ezd nd voe tells us tht plurisuhrmoniity trnsltes to onvexityF his is to e relted to the se of funtions on (C * ) n invrint under (S 1 ) n D whih is hevily used in tori geometryF sn tht seD psh funtion on (C * ) n orresponds to onvex funtion on R n F sn ftD this esy result is suse of ezd nd voe9s resultD with G = (C * ) n F sndeedD for the redutive group (C * ) n D the ompt torus (S 1 ) n is mximl ompt sugroupD nd the KAK deomposition reds

(C * ) n = (S 1 ) n (R * + ) n (S 1 ) n .
roweverD sine it is elinD only one (S 1 ) n ftors ountsD nd the deompoE sition is given y tking on eh ftor the ngle nd the modulusF he set A ⊂ G is thus (R * + ) n in this se nd it is the imge y the exponentil of R n F e smooth funtion ψ from n open suset of C n to R is plurisubharmonic @we will sy pshA if its omplex ressin is non negtiveF st is strictly psh if its omplex ressin is positiveF his is still de(ned forD syD lolly integrle funtionsD in the sense of distriutionsF Proposition 3.17. [AL92, Theorem 1] The correspondence ψ → f gives a bijection between K × K-invariant psh functions on G (resp. smooth strictly psh K × K-invariant functions on G) and W -invariant convex functions on a (resp. smooth strictly convex W -invariant functions on a).

Haar measure and KAK integration formula

vet G e redutive groupF hen there exists unique leftEinvrint positive smooth mesure on GD up to positive onstntF uh mesure is lled Haar measureD nd will e denoted y dgF e restrit here to redutive groups ut of ourse rr mesures exist on more generl groupsF rr mesures on redutive groups stisfy stronger propertyX they re lso invrint under the right tion of G @we sy tht redutive group is unimodular AF e will integrte K × KEinvrint funtions on G with respet to the rr mesureD nd we wnt to express suh n integrl in terms of the restrition to A of the funtionF sing the KAK deompositionD it is possile to mke vrile hnge nd get the formul in the following theoremD y omputing toinF his omputtion ws originlly done y rrishEghndr nd n e found in ook of relgsonF vet J denote the funtion on a de(ned y

J(x) = α∈Φ + sinh 2 (α(x)).

RP

Theorem 3.18. [Kna02, Proposition 5.28] (see also [Hel84, Theorem 5.8]) Let dg denotes a Haar measure on G, and dx a Lebesgue measure on a + , then there exists a constant C > 0 such that for all

K × K-invariant positive function ψ on G, G ψ(g)dg = C a + J(x)ψ(exp(x))dx.
st is esy to (nd rr mesure on vie groupF pirst hoose ny sis of the otngent spe t the neutrl element eF y here we hoose omplex sis dz 1 , . . . , dz n of T * e GD nd uild the top exterior produt

i n dz 1 ∧ dz 1 ∧ • • • ∧ dz n ∧ dz n .
e multiplied y i n to get rel formF hen trnsport this 2nEform y the tion of G on itself y right trnsltionF his gives smooth volume form on G invrint under the right tionD so rr mesure on GF nimodulrity implies tht the volume form uilt this wy is lso G × GEinvrintF

Complex Hessian matrix on G

e will ompute in this setion the omplex ressin of K × KEinvrint funtion on G in terms of the rel wongeEempère of the ssoited W Einvrint funtion on aF his omputtion is vlid in n pproprite hoie of omplex oordintes on GX t the identity element e ∈ GD hoose the sis of T e G = g given in etion QFIFTF his gives omplex sis of T e GD nd y the tion of G y multiplition on the rightD we n trnsport this to omplex sis of ny T g GF hese lso de(ne lol omplex oordintes ner every element of GF sndeedD the exponentil eing iholomorphism from neighorhood of 0 ∈ g to neighorhood of the neutrl element e in GD we get holomorphi oordintes ner eF henD omposing with the multiplition on the right y g ∈ GD this de(nes holomorphi oordintes on neighorhood of gF wore preiselyD if (l j ) n j=1 denotes the hosen sis of gD the mp orresponding to the lol oordintes is the mp

C n → G de(ned y (z 1 , . . . , z n ) → exp(z 1 l 1 + • • • z n l n )g.
e will ompute the omplex ressin with respet to these oordintesF sf ψ is funtion on G we denote y Hess C (ψ)(g) the omplex ressin nd y MA C (ψ)(g) the determinnt of the omplex ressin of ψ t gD lled the omplex wongeEempèreD everything with respet to the oordintes given oveF sf f is funtion on aD then we denote y MA R (f )(x) the determinnt of its rel ressin t xF e denote y ∇f the grdient of f with respet to the slr produt •, • on aF Theorem 3.19. Let ψ be a K ×K invariant function on G, and f the associated function on a. Then in the coordinates above and for a ∈ a + , the complex RQ Hessian matrix of ψ is diagonal by blocks, equal to:

Hess C (ψ)(exp(a)) =         1 4 Hess R (f )(a) 0 0 0 M α1 (a) 0 0 0 . . . . . . . . . . . . . . . 0 0 0 M αp (a)        
where the (α i ) i∈{1,...,p} run over the positive roots of Φ and M α is dened by:

M α (a) = 1 2 α(∇f (a)) coth(α(a)) i -i coth(α(a)
) .

Corollary 3.20. Let ψ be a K×K invariant function on G, and f the associated function on a. Then in the coordinates above and at a ∈ a + , if r denotes the rank of G and p the number of positive roots, we have

MA C (ψ)(exp(a)) = 1 4 r+p MA R (f )(a) α∈Φ + α(∇f (a)) 2 sinh 2 (α(a)) . Proof. ine MA R (f )(a) = det(Hess R (f )(a))D we just hve to ompute the deE terminnt of M α F his is det(M α ) = ( 1 2 α(∇f (a))) 2 (coth(α(a)) 2 -1) = ( 1 2 α(∇f (a))) 2 cosh(α(a)) 2 -sinh(α(a)) 2 sinh(α(a)) 2 = ( 1 2 α(∇f (a))) 2 1 sinh(α(a)) 2
Remark 3.21. enother wy to write this is s

MA C (ψ)(exp(a)) = 1 4 r+p MA R (f )(a) 1 J(a) α∈Φ + α(∇f (a)) 2
where J is the funtion involved in the KAK integrtion formulF Example 3.22. gonsider the se G = PSL 2 (C)F hen a + R * + D nd there is only one positive root tht we n identify with the identity on RF hen

Hess C (ψ)(exp(a)) = 1 2   f (a)/2 0 0 0 f (a)coth(a) if (a) 0 -if (a) f (a)coth(a)   nd the omplex wongeEempère redsX MA C (ψ)(exp(a)) = 1 16 f (a)(f (a)) 2 1 sinh 2 (a)
.

RR

Remark 3.23. enormlizing orretly the sis we nD nd willD ssume thtX

MA C (ψ)(exp(a)) = MA R (f )(a) 1 J(a) α∈Φ + α(∇f (a)) 2 .
Corollary 3.24. By taking, this time, the trace of the complex Hessian, we recover the expression of the laplacian applied to a K × K-invariant function on G, also called the rdil lplin:

∆ r (ψ)(a) :=Tr(Hess C (ψ)(exp(a))) = 1 4 Tr(Hess R (f )(a)) + α∈Φ + α(∇f (a))coth(α(a)).
he rest of the setion is devoted to the proof of the theoremF he tehnique of the proof is sed on the work of fielwski fieHRF sn prtiulrD the ide to use the deomposition in vemm QFPT nd the fkerEgmpellErusdor' formul ppers in this rtileF e egin y introduing these two toolsF

The Baker-Campbell-Hausdor formula es forml series in the vriles x nd yD the logrithm of exp(x) exp(y) is well de(nedF e denote this y BCH(x, y)F he fkerEgmpellErusdor' formul is the followingF Proposition 3.25. There exists a neighborhood U of 0 in g such that for all x and y in U , BCH(x, y) is convergent and denes an element of g, and we have exp(x) exp(y) = exp(BCH(x, y)).

purthermore we know expliitly the (rst terms of BCH(x, y)F e will only use the followingX

BCH(x, y) = x + y + 1 2 [x, y] + O
where O denotes terms of order higher thn P in x nd yF

l = A + B + C; if l ∈ α∈Φ g α then B = 0. if l ∈ k α ,
if l = e α + θ(e α ) then l = ie α -iθ(e α ); if l = ie α -iθ(e α ) then l = -e α -θ(e α ).
sn the sttementD the result is more nd more preise s we know more preisely the element onsideredF sn the proof we will egin y the very preise se of the elements of the sis nd work our wy up y linerityF hen y linerity the (rst point holds true for ny l ∈ i α∈Φ + k α D with B = 0F fut we hve

α∈Φ g α = α∈Φ + k α ⊕ i α∈Φ + k α ,
so we hve the deomposition for ny l ∈ α∈Φ g α D with B = 0F pinlly for l ∈ tD it su0es to deompose l long t = s ⊕ aF fy linerity nd the root deompositionD we otin the proposition for ny l ∈ gF RT Using the Baker-Campbell-Hausdor formula e wnt to ompute the omplex ressin of ψ in the hosen system of oordintesD t point exp(a) for a in the open eyl hmer a + F sf l 1 nd l 2 re two vetors in the hosen sis of kD we thus wnt to omputeX

H l1,l2 (a) := ∂ 2 ∂z 1 ∂z 2 z1,z2=0 ψ(exp(z 1 l 1 + z 2 l 2 ) exp(a)).
here re di'erent sesD ording to the suspes where l 1 nd l 2 lieF e will (rst desrie the prt of the rgument tht is used in ll sesD whih relies on the fkerEgmpellErusdor' formulD nd then del with eh se seprtelyF sing the deomposition from vemm QFPT on z 1 l 1 + z 2 l 2 we n write

z 1 l 1 + z 2 l 2 = A 1 + B 1 + C 1 with A 1 ∈ kD B 1 ∈ a nd C 1 ∈ Ad(exp(a))(k)D nd ll re of homogeneous degree one in z 1 nd z 2 F vet D 1 = 1 2 ([B 1 , A 1 ] + [C 1 , A 1 ] + [C 1 , B 1 ]),
it is of order two in z 1 nd z 2 F vet us now use gin vemm QFPT to get

D 1 = A 2 + B 2 + C 2 .
with A 2 ∈ kD B 2 ∈ a nd C 2 ∈ Ad(exp(a))(k)D nd ll re of homogeneous degree two in z 1 nd z 2 F hen the fkerEgmpellErusdor' formul llows to prove the following lemmF his n e seen s n expliit in(nitesiml KAK deompositionF xote tht to lighten the nottions we do not write the dependene on z 1 D z 2 D ut ll the terms de(ned ove A j D B j D C j nd D 1 re in ft funtions of these two omplex vrilesF Lemma 3.27. We can write

exp(z 1 l 1 + z 2 l 2 ) exp(a) = k 1 exp(B 1 + B 2 + a + O)k 2
where O denotes terms of order greater than two in z 1 and z 2 .

Proof. e egin y pplying roposition QFPS to exp(-A 1 ) exp(A 1 + B 1 + C 1 )D nd get tht this is equl to

exp B 1 + C 1 + 1 2 [-A 1 , B 1 + C 1 ] + O 1 ,
where O 1 denotes terms of order greter thn P in z 1 nd z 2 F RU hen we multiply on the right y exp(-C 1 ) nd getD with roposition QFPS ginD

exp B 1 + 1 2 [-A 1 , B 1 + C 1 ] + 1 2 [B 1 , -C 1 ] + O 2 ,
where O 2 denotes terms of order greter thn P in z 1 nd z 2 F fy de(nition of D 1 D we hve proved

exp(-A 1 ) exp(z 1 l 1 + z 2 l 2 ) exp(-C 1 ) = exp (B 1 + D 1 + O 2 ) . ell tht D 1 = A 2 + B 2 + C 2 D nd
tht ll of these re of degree two in z 1 nd z 2 F e pply nother time the roposition QFPSD to exp(-A 2 ) exp(B 1 + D 1 + O 2 )D ut here we only need to use the (rst term in the development of BCHF e might sy tht A 2 ommutes up to order two with elements of degree greter or equl to one in

z 1 D z 2 F e get exp(-A 2 ) exp(B 1 + D 1 + O 2 ) = exp(B 1 + B 2 + C 2 + O 3 ),
where O 3 denotes terms of order greter thn P in z 1 nd z 2 F yne further use of the fkerEgmpellErusdor' formul yields

exp(-A 2 ) exp(B 1 + D 1 + O 2 ) exp(-C 2 ) = exp(B 1 + B 2 + O 4 ),
where O 4 denotes terms of order greter thn P in

z 1 nd z 2 F gonsider now exp(C 2 ) exp(C 1 )F ine C 1 , C 2 ∈ Ad(exp(a))(k)D we hve exp(C 2 ) exp(C 1 ) = exp(a)k 2 exp(-a)
for some k 2 ∈ KF yn the other hndD we hve k 1 := exp(A 1 ) exp(A 2 ) ∈ KF umming up we hve proved tht

exp(z 1 l 1 + z 2 l 2 ) = k 1 exp(B 1 + B 2 + O 4 ) exp(a)k 2 exp(-a).
fut then

exp(z 1 l 1 + z 2 l 2 ) exp(a) = k 1 exp(B 1 + B 2 + O 4 ) exp(a)k 2 ,
nd one lst pplition of roposition QFPS gives the lemmD euse

B 1 , B 2 nd a ommuteX exp(z 1 l 1 + z 2 l 2 ) exp(a) = k 1 exp(B 1 + B 2 + a + O)k 2
where O denotes terms of order greter thn P in z 1 nd z 2 F Lemma 3.28. We have

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 2 + B 1 ).

RV

Proof. e (rst use K × KEinvrine of ψ nd vemm QFPU to write

ψ(exp(z 1 l 1 + z 2 l 2 ) exp(a)) = ψ(exp(a + B 1 + B 2 + O)).
hen

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 z1,z2=0 ψ(exp(z 1 l 1 + z 2 l 2 ) exp(a)) = ∂ 2 ∂z 1 ∂z 2 0 ψ(exp(a + B 1 + B 2 + O))
euse O is of order greter thn twoD this eomes

= ∂ 2 ∂z 1 ∂z 2 0 ψ(exp(a + B 1 + B 2 )) sine a + B 1 + B 2 ∈ aD this is = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 1 + B 2 )
st remins to determine B 1 + B 2 for ll oe0ients of the ressinD nd then to ompute the oe0ientF por thtD sine we wnt to redue to rel oordintesD we rell tht if

z 1 = x 1 + iy 1 nd z 2 = x 2 + iy 2 then ∂ 2 ∂z 1 ∂z 2 = 1 4 ∂ 2 ∂x 1 ∂x 2 + ∂ 2 ∂y 1 ∂y 2 + i 4 ∂ 2 ∂x 1 ∂y 2 - ∂ 2 ∂y 1 ∂x 2 .
Determining H l1,l2 (a) Lemma 3.29. Assume l 1 , l 2 ∈ s. Then H l1,l2 (a) is the corresponding coecient of 1 4 Hess R (f )(a) :

H l1,l2 (a) = 1 4 ∂ 2 ∂y 1 ∂y 2 0 f (a + y 1 il 1 + y 2 il 2 ).
Proof. sn this se we hve

z 1 l 1 +z 2 l 2 = A 1 +B 1 +0 with A 1 = x 1 l 1 +x 2 +l 2 ∈ s nd B 1 = y 1 l 1 + y 2 l 2 ∈ aD nd A 1 nd B 1 ommuteD so D 1 = 0 nd B 2 = 0F hen y vemm QFPVD H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + y 1 il 1 + y 2 il 2 ) = 1 4 ∂ 2 ∂y 1 ∂y 2 0 f (a + y 1 il 1 + y 2 il 2 ) RW Lemma 3.30. Assume l 1 ∈ k α and l 2 ∈ s, then H l1,l2 (a) = 0.
Proof. vet us (rst determine the

A 1 , B 1 , C 1 suh tht z 1 l 1 +z 2 l 2 = A 1 +B 1 +C 1 F sing vemm QFPTD write il 1 = -coth(α(a))l 1 + 1 sinh(α(a)) (Ad(exp(a))(l 1 )
with l 1 = 1 α(a) ad(a)(il)F hen we hve

A 1 = x 1 l 1 -y 1 coth(α(a))l 1 + x 2 l 2 B 1 = y 2 il 2 C 1 = y 1 sinh(α(a)) (Ad(exp(a))(l 1 ) = y 1 il 1 + y 1 cosh(α(a))l 1 e must now ompute D 1 = 1 2 ([B 1 , A 1 ] + [C 1 , A 1 ] + [C 1 , B 1 ])F sn ft we must only determine B 2 whih is the prt of D 1 tht lies in aF e hve [B 1 , A 1 ] = [y 2 il 2 , x 1 l 1 -y 1 coth(α(a))l 1 + x 2 l 2 ] = -y 1 y 2 coth(α(a))[il 2 , l 1 ] + x 1 y 2 [il 2 , l 1 ] xow il 2 ∈ a nd l 1 , l 1 ∈ k α ⊂ g α ⊕ g -α so [il 2 , l 1 ], [il 2 , l 1 ] ∈ g α ⊕ g -α D
nd the third point of vemm QFPT pplies to show tht the a omponent of [B 1 , A 1 ] is zeroF por the seond prtD write

[C 1 , A 1 ] =x 1 y 1 cosh(α(a))[l 1 , l 1 ] -y 2 1 coth(α(a))[il 1 , l 1 ] + x 2 y 1 [il 1 , l 2 ] + x 2 y 1 cosh(α(a))[l 1 , l 2 ] e hve here [l 1 , l 1 ], [l 1 , l 2 ] ∈ kD nd [il 1 , l 2 ] ∈ g α ⊕ g -α s oveD so only [il 1 , l 1 ]
mttersF fy the properties of the root deompositionD

[il 1 , l 1 ] ∈ (g -2α ⊕ g 0 ⊕ g 2α ) ∩ ik nd g -2α = g 2α = {0}F o [il 1 , l 1 ] ∈
aF fut in ft we do not need to determine it more expliitly euse it ppers s term in y 2 1 nd these re ignored in the omputtion of ∂∂F por the third prtD

[C 1 , B 1 ] = y 1 y 2 [il 1 , il 2 ] + y 1 y 2 cosh(α(a))[l 1 , il 2 ] with [il 1 , il 2 ] ∈ k nd [l 1 , il 2 ] ∈ g α ⊕ g -α so
there is no ontriution to B 2 F e hve thus proved tht

B 2 = - 1 2 y 2 1 coth(α(a))[il 1 , l 1 ].

SH

vemm QFPV now gives

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 2 + B 1 ) = ∂ 2 ∂z 1 ∂z 2 0 f (a + y 2 il 2 - 1 2 y 2 1 coth(α(a))[il 1 , l 1 ]) = 0 Lemma 3.31. Assume l 1 ∈ k α1 and l 2 ∈ k α2 , with α 1 = α 2 positive roots. Then H l1,l2 (a) = 0.
Proof. sing vemm QFPTD we write

il 1 = -coth(α 1 (a))l 1 + 1 sinh(α 1 (a)) (Ad(exp(a))(l 1 ) il 2 = -coth(α 2 (a))l 2 + 1 sinh(α 2 (a)) (Ad(exp(a))(l 2 ) with l 1 = 1 α1(a) ad(a)(il 1 ) nd l 2 = 1 α2(a) ad(a)(il 2 )F hen we hve A 1 = x 1 l 1 + x 2 l 2 -y 1 coth(α 1 (a))l 1 -y 2 coth(α 2 (a))l 2 B 1 = 0 C 1 = y 1 1 sinh(α 1 (a)) (Ad(exp(a))(l 1 ) + y 2 1 sinh(α 2 (a)) (Ad(exp(a))(l 2 ) nd D 1 = 1 2 [C 1 , A 1 ] = 1 2 [y 1 il 1 + y 2 il 2 + y 1 cosh(α 1 (a))l 1 + y 2 cosh(α 2 (a))l 2 , A 1 ] e hve y 1 cosh(α 1 (a))l 1 + y 2 cosh(α 2 (a))l 2 nd A 1 in kD so their rket remins in k nd does not pper in B 2 F e ompute [y 1 il 1 + y 2 il 2 , A 1 ] whih is equl to x 2 y 1 [il 1 , l 2 ] -y 2 1 coth(α 1 (a))[il 1 , l 1 ] -y 1 y 2 coth(α 1 (a))[il 2 , l 1 ] + x 1 y 2 [il 2 , l 1 ] -y 2 2 coth(α 2 (a))[il 2 , l 2 ] -y 1 y 2 coth(α 2 (a))[il 1 , l 2 ].
egin the properties of the root deomposition tell us tht

[il 1 , l 2 ]D [il 1 , l 2 ]D [il 2 , l 1 ], nd [il 2 , l 1 ] re in α∈Φ g α D so the orresponding terms do not onE triute to B 2 F es eforeD [il 1 , l 1 ] nd [il 2 , l 2 ] re in aD so we get B 2 = 1 2 (-y 2 1 coth(α 1 (a))[il 1 , l 1 ] -y 2 2 coth(α 2 (a))[il 2 , l 2 ]).

SI

epplying vemm QFPVD we get

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 2 + B 1 ) = ∂ 2 ∂z 1 ∂z 2 0 f (a - 1 2 (y 2 1 coth(α 1 (a))[il 1 , l 1 ] + y 2 2 coth(α 2 (a))[il 2 , l 2 ])) = 0
uppose now tht α 1 = α 2 = αF he suspe k α is two dimensionlD nd we hve hosen sis formed y the vetors e α + θ(e α ) nd ie α -iθ(e α )F pirst we del with the se when l 1 = l 2 F Lemma 3.32. Suppose l 1 = e α + θ(e α ) and l 2 = ie α -iθ(e α ). Then

H l1,l2 (a) = i 2 α(∇f (a)),
and

H l2,l1 (a) = - i 2 α(∇f (a)).
Proof. sing vemm QFPTD we writeD just s in the previous proof

il 1 = -coth(α(a))l 1 + 1 sinh(α(a)) (Ad(exp(a))(l 1 ) il 2 = -coth(α(a))l 2 + 1 sinh(α(a)) (Ad(exp(a))(l 2 )
with

l 1 = l 2 l 2 = -l 1 hen we hve A 1 = (x 1 + y 2 coth(α(a)))l 1 + (x 2 -y 1 coth(α(a)))l 2 B 1 = 0 C 1 = y 1 1 sinh(α(a)) (Ad(exp(a))(l 1 ) + y 2 1 sinh(α(a)) (Ad(exp(a))(l 2 ) nd D 1 = 1 2 [C 1 , A 1 ] = 1 2 [y 1 il 1 + y 2 il 2 + y 1 cosh(α(a))l 2 -y 2 cosh(α(a))l 1 , A 1 ]
SP yne gin the rket of y 1 cosh(α(a))l 2 -y 2 cosh(α(a))l 1 with A 1 yields only terms in k so we ompute [y 1 il 1 + y 2 il 2 , A 1 ]D whih is equl to

y 2 (x 1 + y 2 coth(α(a)))[il 2 , l 1 ] + y 1 (x 2 -y 1 coth(α(a)))[il 1 , l 2 ].
sing the expliit hoies of l 1 nd l 2 we hve

-[il 2 , l 1 ] = [il 1 , l 2 ] = [i(e α + θ(e α )), ie α -iθ(e α )] = [ie α , -iθ(e α )] + [iθ(e α ), ie α ] = 2[e α , θ(e α )] = -2[e α , e -α ] = -2h α .
pinlly we hve

B 2 = (y 2 x 1 + y 2 2 coth(α(a)) -y 1 x 2 + y 2 1 coth(α(a)))h α .
epplying vemm QFPVD we get

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 2 + B 1 ) = i 4 ∂ 2 ∂x 1 ∂y 2 - ∂ 2 ∂y 1 ∂x 2 0 f (a + (y 2 x 1 -y 1 x 2 )h α ) = i 2 (Df ) a (h α )
where (Df ) a denotes the di'erentil of f t aD so

H l1,l2 (a) = i 2 h α , ∇f (a)
y de(nition of h α D this is

H l1,l2 (a) = i 2 α(∇f (a)).
Lemma 3.33. Suppose now that l 1 = l 2 = e α + θ(e α ), then

H l1,l2 (a) = 1 2 α(∇f (a))coth(α(a)).
Proof. sing vemm QFPTD we write

il 2 = il 1 = -coth(α(a))l 1 + 1 sinh(α(a)) (Ad(exp(a))(l 1 ) SQ with l 2 = l 1 = ie α -iθ(e α )
hen we hve

A 1 = (x 1 + x 2 )l 1 -(y 1 coth(α(a)) + y 2 coth(α(a)))l 1 B 1 = 0 C 1 = y 1 1 sinh(α(a)) (Ad(exp(a))(l 1 ) + y 2 1 sinh(α(a)) (Ad(exp(a))(l 2 ) nd D 1 = 1 2 [C 1 , A 1 ] = 1 2 [(y 1 + y 2 )il 1 + (y 1 cosh(α(a)) + y 2 cosh(α(a)))l 1 , A 1 ]
yne gin the rket of (y 1 cosh(α(a)) + y 2 cosh(α(a)))l 1 with A 1 yields only terms in kD so we just ompute

[(y 1 + y 2 )il 1 , A 1 ] = -(y 1 + y 2 ) 2 coth(α(a))[il 1 , l 1 ].
sing the expliit hoies of l 1 we hve

[il 1 , l 1 ] = [i(e α + θ(e α )), ie α -iθ(e α )] = -2h α .
pinlly we hve B 2 = (y 1 + y 2 ) 2 coth(α(a))h α .

epplying vemm QFPVD we get

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 2 + B 1 ) = 1 4 ∂ 2 ∂y 1 ∂y 2 0 f (a + (y 2 1 + 2y 1 y 2 + y 2 2 )coth(α(a))h α ) = coth(α(a)) 2 (Df ) a (h α ) = coth(α(a)) 2 h α , ∇f (a) = coth(α(a)) 2 α(∇f (a)).
he lst step is to ompute the oe0ient of the ressin with l 1 = l 2 = ie α -iθ(e α )D nd the result is extly the sme s in the previous seX SR Lemma 3.34. Assume that l 1 = l 2 = ie α -iθ(e α ), then

H l1,l2 (a) = 1 2 α(∇f (a))coth(α(a)).
Proof. sing vemm QFPTD we write

il 2 = il 1 = -coth(α(a))l 1 + 1 sinh(α(a)) (Ad(exp(a))(l 1 )
with

l 2 = l 1 = -e α -θ(e α )
he eginning of the omputtion does not hngeX we hve

A 1 = (x 1 + x 2 )l 1 -(y 1 coth(α(a)) + y 2 coth(α(a)))l 1 B 1 = 0 C 1 = y 1 1 sinh(α(a)) (Ad(exp(a))(l 1 ) + y 2 1 sinh(α(a)) (Ad(exp(a))(l 2 ) D 1 = 1 2 [(y 1 + y 2 )il 1 + (y 1 cosh(α(a)) + y 2 cosh(α(a)))l 1 , A 1 ]
yne gin the rket of (y 1 cosh(α(a)) + y 2 cosh(α(a)))l 1 with A 1 yields only terms in kD so we just ompute

[(y 1 + y 2 )il 1 , A 1 ] = -(y 1 + y 2 ) 2 coth(α(a))[il 1 , l 1 ].
xow the expression of l 1 hs hngedD ut we hve

[il 1 , l 1 ] = [i(ie α -iθ(e α )), -e α -θ(e α )] = 2[e α , θ(e α )] = -2h α .
sn other words we hve gin

B 2 = (y 1 + y 2 ) 2 coth(α(a))h α ,
nd pplying vemm QFPVD we get gin

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 2 + B 1 ) = coth(α(a)) 2 α(∇f (a)).

SS

Chapter 4

Group compactications and metrics sn this hpter we de(ne the group ompti(tionsD nd derive from the theory lssifying suh vrieties the informtion tht we will use to get our resultsF o eh group ompti(tionD equipped with polriztionD i.e. n mple linerized line undle on itD is ssoited polytope tht ontins ll the informtion out the ompti(tionF sn prtiulrD this polytope determines the symptoti ehvior of the potentils of metris on the ig oritF e egin y short setion desriing line undles on redutive group G tht re linerized y G × GF his llows to introdue the notion of linerized line undle nd will e used to de(ne the potentils of metris on the ig orit G of ompti(tion of GF hen we give rief overview of the theory of tori vrietiesD whih is used then to study group ompti(tionF sndeedD group ompti(tion dmits tori suvriety tht ontins ll the informtion out the ompti(tionF sf we hve polriztionD it restrits to polriztion on the tori suvrietyD nd thus trnsltes s the dt of n integrl polytope y the lssil theory of projetive tori vrietiesF e lso provide desription of exmples of group ompti(tionD the most importnt lss of suh eing the wonderful ompti(tionsF hey turn out to e pno mnifolds nd we desrie the polytope ssoited to their ntinonil polriztionF e then introdue the di'erent notions of potentil of hermitin metri on line undleD nd use the polytopes to desrie the symptoti ehvior of potentils of metris on the group oritF gomining the symptoti desriptionD the KAK integrtion formul nd the omputtion of the omplex wongeE empère from the (rst hpterD we see how we n reoverD up to onstntD the formul for the degree of n mple line undle on group ompti(tion whih ws omputed y uzrnovskii uzVU nd frion friVWF

G × G de(ned y (g 1 , g 2 ) • (g, t) = (g 1 gg -1 2 , -χ(g 2 )t
). he onstrution of s ove gives s(g) = (g, 1)D nd we reover the hrter χ s the hrter of the tion of diag(G) on L e F wore generlly one n get similr result for homogeneous spesD see uuvVWF e n lso determine if ll line undles on G n e G × GE linerizedF his is the se if G is simply onnetedD nd in generl if G sc is the simply onneted group ove GD then every line undle on G is G sc × G sc E linerizedF his is lso explined in uuVWD uuvVWF

σ v = {n ∈ N ⊗ R; m(n) ≥ 0 ∀m ∈ C v }.
he norml fn Σ of P is the olletion of ones whih onsists of the ones σ v nd their fesF yf ourse some of these ones hve ommon fes tht must e ounted only oneF his olletion of ones stis(es the following two onditions gvIID heorem PFQFPX ! for ny σ ∈ ΣD the fes of σ re in ΣD nd ! the intersetion of σ with nother one in Σ is union of fes of σF sn generl olletion of onvex rtionl polyhedrl ones stisfying these two onditions is lled fan nd orresponds to tori vrietyD not neessrily projetiveF yne of the mjor reltionship etween tori vriety nd its fn is the oritE one orrespondeneD whih we rell here long with the polytope versionF fy onventionD polytope with nonEempty interior in M ⊗ R hs unique fe of dimension rD whih is itselfF Proposition 4.5. There is a bijective correspondence between the following: the T -orbits in X of complex dimension k; the cones in Σ of real odimension k in N ⊗ R; SV the faces of P of real dimension k in M ⊗ R. Furthermore, in each case we can dene a partial order by saying that an orbit (resp. cone, face) is smaller than another one if it is in its closure. Then the correspondence between orbits and faces is order-preserving, and it reverses the order between orbits and cones.

he polytope dds the informtion out the mple line undle to the fnF wore generlly we desrie the omintoril dt ssoited to linerized line undle on tori vriety @ssumed to e projetive hereAF 4.2.2 Line bundles vet X e tori vriety s oveF vet L e T Elinerized line undle on XF por ny T E(xed point x on XD T ts linerly on the (er of L t xF henote y v x the opposite of the hrter of this tionF e de(ne the support function g L of the line undle L s the pieewise liner funtion on N ⊗RD whih tkes the vlue v x (n) t point n in the losure of the one of dimension r orresponding to the (xed point x y the oritEone orrespondeneF e n lso ssoite to L the polytope P L de(ned y

P L = {m ∈ M ⊗ R; g L (n) ≤ m(n) ∀n ∈ N ⊗ R}.
he lttie points of P L determine sis of the spe H 0 (X, L) of lgeri setions of L gvIID roposition RFQFQF wore preiselyD if s 0 denotes T E equivrint setion of LD then the setions s m de(ned y s m (t) = m(t)s 0 (t) on T for m ∈ P L ∩ M extend to X nd form sis of lgeri setions of LF yne n hrterize the mpleness of L in terms of its support funtion g L F Proposition 4.6. [CLS11, Lemma 6.1.13] The line bundle L is ample if and only if g L is concave, and v x = v y for any two dierent xed points x and y in X.

here re severl properties of the line undles @nd more generlly divisorsA on the tori vriety tht n e red o' from the ssoited polytope or support funtionD see gvIIF he polytope ssoited to polrized tori vriety (X, L) is P L F his explins one diretion of the orrespondeneF e ttrt the reder9s ttention to the ft tht the support funtion of the line undle is not the support funtion of the polytope P L F sn ft the support funtion of the polytope P L will e more importnt to usF st is the onvex funtion

v L : N ⊗ R → R de(ned y v(x) = sup{m(x); m ∈ P L }.
sn the se when L is nefD we hve v(x) = -g L (-x) so the dt of v L is equivlent to the dt of g L or LF purthermoreD this funtion is pieewise liner with respet to the opposite of the fn ΣF SW pigure RFIX olytope nd fn of P 2 Example 4.7. gonsider the omplex projetive line P 1 F st is tori pno mnifoldF he polytope ssoited to the ntinonil line undle is [-1, 1] whih hs verties in the lttie ZF he support funtion of -K P 1 is x → -|x| nd the support funtion of the polytope [-1, 1] is x → |x|F Example 4.8. pigure RFI gives the fn of P 2 nd the polytope orresponding to the ntinonil line undleF he support funtion of the ntinonil line undle is liner on eh one of the fnD equl to -x -y when x, y ≥ 0D to 2x -y when x ≤ 0D y ≥ xD nd to 2y -x when y ≤ 0D x ≥ yF he support funtion of the polytope is liner on the opposite of these onesX it is -x -y when x, y ≤ 0D 2x -y when x ≥ 0D y ≤ xD nd 2y -x when y ≥ 0D x ≤ yF 4.2.3 Smoothness criterion he smoothness of tori vriety is n informtion tht n e esily seen on the fnD or on the ssoited polytope for polrized vrietyF vet us (rst rell the de(nition of helznt polytope efore stting the riterionF Denition 4.9. e fullEdimensionl onvex lttie polytope P ⊂ M ⊗R is lled Delzant if the slopes of the edges t eh vertex form sis of M F e will lso ll one smooth if it is generted y prt of sis of N F st is ler tht polytope is helznt if nd only if ll the full dimensionl ones of its norml fn re smoothD nd this implies tht ll ones of the norml fn re smoothF Proposition 4.10. [CLS11, Theorem 3.1.19] Given a polarized toric variety (X, L) with associated polytope P , the following are equivalent:

X is smooth ; the polytope P is Delzant ; all the cones of the normal fan of P are smooth. e use s referene the sixth hpter in fuHSD whih is onvenient refE erene for the results on group ompti(tions we will useF he referenes to the originl ppers n e found in this ookF 4.3.2 Group compactications and polytopes e n now give the generliztion of the orrespondene etween polrized vrieties nd polytopes to the setting of group ompti(tionsF vet G e redutive groupD nd X ompti(tion of GF ghoose T mximl torus in G nd denote y Z the losure of T in XF e hve seen tht Z is tori vrietyF his tori suvriety dmits in ddition n tion of the eyl group W of GF vet L e n mple G × GElinerized line undle on XF he restrition of L to Z is line undleD linerized y the normlizer N G×G (T )F Theorem 4.14. [[AK05, Theorem 2.4], based on [AB04a, AB04b]] The restriction of L to Z is an ample line bundle, and this gives a W -invariant lattice polytope P associated to the polarized group compactication (X, L). Conversely, given a W -invariant full-dimensional lattice polytope P , there exists a polarized G × G-equivariant compactication of G whose associated polytope is P .

his omes with kind of oritEfe orrespondene ginF TI Proposition 4.15. Let (X, L) be a polarized compactication of G, with associated polytope P . The G × G orbits in X are in an order preserving bijective correspondence with the W -orbits of faces of the polytope P .

vet us now explin one wy to reover group ompti(tion from its polytopeD remrk tht this works in prtiulr for tori vrietiesD for whih we did not explin this diretion yet eitherF his generl onstrution is gin tken from efHRD efHRD euHSF ell (rst tht the lger of regulr funtions on G is desried s G × GErepresenttion in imIID heorem PFIS yX

C[G] λ∈M ∩(a * ) + End(E λ )
where E λ is the (nite dimensionl irreduile representtion of G of highest weight the dominnt weight λD nd End(

E λ ) = E * λ ⊗ E λ is the spe of endoE morphisms of E λ D nd is n irreduile G × GErepresenttionF wore preiselyD let us desrie how this isomorphism is relizedF en element σ ⊗ x ∈ E * λ ⊗ E λ de(nes mtrix oe0ient f σ⊗x of the representtion E λ X f σ⊗x (g) = σ(g • x) whih is n element of C[G]F
qiven W Einvrint full dimensionl lttie polytope P in M ⊗ RD de(ne P + to e the prt of P lying in the positive eyl hmer of a * F vet C e the one over (1, P + ) in Z ⊕ M R F he vetor spe

R P := µ∈C∩(Z⊕M ) End(F µ ) ⊂ C[C * × G]
hs nturl struture of sulgerD nd is (nitely genertedF e n thus de(ne X := Proj(R P ) nd oherent shef L = O(1)F his X is in ft ompti(tion of G nd L is G × GElinerized mple line undle on X whose ssoited polytope is P F he polytope P of (X, L) lso enodes the struture of the spe of setions of L s G × G representtionF xmelyD we hveX

H 0 (X, L) = λ∈P + ∩M End(V λ ).

Smoothness criterion

vet us now give the prtil smoothness riterion otined y elexeev nd utzrkovD where we gin restrit to the se of group ompti(tionsF Proposition 4.16. [AK05, Proposition 2.5] If X is smooth then the associated polytope is Delzant. If P is Delzant, and no vertex of P lies in a Weyl wall, then X is smooth. TP Remark 4.17. sn the seond seD the dded ondition tht no vertex of P lies in eyl wll ensures tht X is toroidl s spheril vrietyD i.e. tht no B×BE stleD not G × GEstle divisor of X ontins losed G × GEoritF oroidlD smooth ompti(tions of groups re lso lled regulr ompti(tions of groupsF por toroidl ompti(tions of groupsD it ws lredy known tht the smoothness of the group ompti(tion nd of its tori suvriety were equivlentF e turn now to some exmples of group ompti(tionsF e lredy reE viewed the tori vrietiesD whih re ompti(tions of groups with no semiE simple prtF yn the opposite endD the most known fmily of suh vrieties onsists of the wonderful ompti(tions of semisimple djoint groupsF e will present these in the following setionD inluding desription of the line undles on themF e do not desrie in generl ll line undles on group ompti(tions to void lengthening the text too muhD ut the ird group of ny spheril vriety ws desried y wihel frion in friVWD nd desription of the line undles on regulr ompti(tions even more similr to the one given for tori vrieties is possileD see fifWHF 4.4 Wonderful compactications 4.4.1 Denition and existence e wonderful ompti(tion n e de(ned y some of its remrkle propE ertiesF Denition 4.18. e G × GEequivrint ompti(tion X of G is lled wonE derful if it stis(es the onditionsX ! X is smoothY ! X \ G is the union of smooth norml rossing prime divisorsD with nonE empty intersetionsY ! the G × GEorits in X re preisely the intersetions of fmilies of these divisorsF he existene of suh ompti(tion for semisimple djoint group ws proved y de gonini nd roesi in hgVQF sn ftD they onsidered ompti(tions of symmetri spes under semisimple djoint groupD ut we fous here on group ompti(tionsF Theorem 4.19. [DCP83] If G is a semisimple adjoint group then G admits a unique wonderful compactication.

here exists severl onstrutions of wonderful ompti(tionsF qiven ny regulr nd dominnt weight λ in M D if P denotes the onvex hull of the imges of λ y W D then the ompti(tion of G ssoited to P is the wonderful ompti(tion of GF e will see tht this gives ll polriztionsD nd whih one orresponds to the ntinonil line undleF TQ here is in ft simpler wy to desrie the wonderful ompti(tion given regulr nd dominnt weight λF vet E λ e the irreduile representtion of G with highest weight λF gonsider the G × G projetive representtion P(End(E λ ))F hen the losure in P(End(E λ )) of the orit of the identity in End(E λ ) is the wonderful ompti(tion of GF

Line bundles

vet G = G ad e semisimple djoint group nd X the wonderful omptE i(tion of GF vet M sc e the lttie generted y the fundmentl weights of the root system Φ of gF e priori M ⊂ M sc is di'erent from M sc F sndeedD for n djoint group GD the lttie of weights @or hrters of T ⊂ GA is generted y the roots of ΦF vet G sc e the orresponding simply onneted groupF he vriety X is lso spheril under the G sc × G sc Etion indued y the G ad × G ad EtionF he only losed orit is isomorphi to the full )g vriety

(G sc × G sc )/(B sc × B sc ) = G sc /B sc × G sc /B sc
for G sc × G sc F he restrition mp from the ird group of X to the ird group of the losed orit is injetiveF ell tht to hrter of B sc is ssoited G sc Elinerized line undle on G sc /B sc D in the following wyX if V λ denotes the oneEdimensionl representtion of B sc ssoited to the hrter λD one gets line undle on G sc /B sc y onsidering the (er produt G sc × Bsc V -λ F eny G sc ×G sc Elinerized line undle on G sc × G sc /B sc × B sc is of the form L(λ, µ) where λ nd µ re two hrters of B sc nd L(λ, µ) is the tensor produt of the pullks y the two projetions to G sc /B sc of the orresponding line undles on G sc /B sc F he imge of the ird group of X onsists of the line undles of the form L(-w 0 λ, λ) where w 0 is the element of the eyl group of G sending the positive eyl hmer to the negtive oneF purthermoreD properties of the line undles re enoded in the orresponding hrter λF his is summrized in the following propositionF Theorem 4.20. sn ftD ll line undles on X n e G sc × G sc ElinerizedD where G sc is the simply onneted semisimple group over GF hese mnifolds re espeilly interesting for our purposes euse they re pnoF he following result gives the polytope ssoited with the ntinonilly polrized wonderful ompti(tionF ell tht given root system Φ with positive roots Φ + D ρ denotes the sum of the fundmentl weights of ΦD nd is lso equl to hlf the sum of the positive rootsF sn prtiulrD 2ρ ∈ M F TR Proposition 4.21. [BK05, Proposition 6.1.11] Let G be a semisimple adjoint group, and X its wonderful compactication. Then the fan corresponding to the toric subvariety Z ⊂ X is given by the subdivision of a induced by the Weyl chambers and their faces. The anticanonical line bundle of X is associated with the weight 2ρ + r i=1 α i where α i for 1 ≤ i ≤ r are the simple roots of Φ + . In other words, -K X = L(2ρ + r i=1 α i ). This implies that X is Fano, and that the polytope associated to the anticanonically polarized G × Gequivariant compactication (X, -K X ) is the convex hull of the images by W of 2ρ + r i=1 α i .

Remark 4.22. he ntinonil line undle on G/B is ssoited to the hrter 2ρF yn the other hndD the hrter of the tion of T on the (xed point of the 0ne tori vriety de(ne y the eyl hmer is the sum of the simple roots @whih re the genertors of the dul one of the eyl hmerAF he proposition shows tht the hrter orresponding to the ntinonil line undle on X is the sum of these twoF his losed orit is the egre emedding of P 1 × P 1 in P 3 D nd it turns out tht this identi(tion is lso equivrint under PGL 2 (C) × PGL 2 (C) one we remrk tht P 1 is the )g mnifold of PGL 2 (C)F st is then ler tht P 3 is the wonderful ompti(tion of PGL 2 (C)F he group PGL 2 (C) is of rnk oneD nd we n hoose s mximl torus the set T := {[a : 0 : 0 : d]; ad = 0} formed y the lsses of digonl mtriesF he losure of

T in P 3 is {[a : b : c : d]; b = c = 0}D nd is isomorphi to P 1 F

Rank two examples

por eh root sytem of rnk twoD there is orresponding djoint semisimple group of rnk twoF por A 1 × A 1 D the djoint group is PGL 2 (C) × PGL 2 (C) nd the orresponding wonderful ompti(tion is the produt P 3 × P 3 D or y the TS previous exmple the produt of two opies of the wonderful ompti(tion of PGL 2 (C)F gonsider now the root system A 2 F he orresponding djoint semisimple group is PGL 3 (C)F he polytope orresponding to its wonderful ompti(E tion is given in pigure RFPF he tori suvriety in this se is the low up of P relevnt prt for us is the onneted omponent of the identityF vet us rell the resultF vet G ad e semisimple djoint group nd G the orresponding simply onneted groupF vet X e the wonderful ompti(tion of G ad F rite G = (SL 2 (C)) n × G where G ontins no diret ftor isomorphi to SL 2 F hen Aut 0 (X) (PSL 4 ) n ×(G ad ×G ad ) where G ad is the djoint group orresponding to G F sn prtiulrD when G ontins no SL 2 ftorD we hve Aut 0 (X) G ad ×G ad F his result shows tht when there is no SL 2 ftorD the wonderful ompti(E tion of the djoint group is neither homogeneous nor toriF st is not homogeneous euse Aut 0 (X) leves the oundry X \ G ad invrintF st is not tori euse if Z is tori vrietyD every mximl torus of Aut 0 (Z) is of the dimension of ZF rereD remrk tht dim(G) = rk(G) + Card(Φ) ≥ 3rk(G) ut the mximl torus of G ad × G ad is of dimension 2rk(G)D so X nnot e toriF sn the se of the wonderful ompti(tion X of Sp 2n (C)D whih is not djointD ezzini proved in ezHW tht the onneted utomorphism group Aut 0 (X) is the imge of Sp rnk is less thn three n e found in uzIPF sn prtiulr if we fous on toroidl ompti(tions of simple groupsD we see in uzIPD le U tht we re only missing two suh mnifoldsF he (rst is the low up of the wonderful ompti(tion of PGL 3 t the losed orit whose polytope is represented in pigure RFTD nd the seond is the low up of the wonderful ompti(tion of Sp 4 (C) t the losed orit whose polytope is represented in pigure RFUF o otin the polytopes for these exmplesD whih re not wonderfulD one n use the generl desription of the ntinonil divisor given y frion in friVWF sn the se of toroidl ompti(tionD the desription is simpli(ed y uzzi @see uzIPD pge PRT or his hh thesis uzAF st turns out tht the support funtion v of the polytope of -K X n e desried s v = v G + v Z where v G is de(ned y v(x) = 2ρ(x) on the positive eyl hmer nd is W E invrintD nd v Z is de(ned s -g -K Z (-x) where Z is the tori suvriety in X nd g -K Z is the support funtion of the line undle -K Z on ZF por oth of these mnifoldsD the onneted group of utomorphisms Aut 0 (X) is the imge of G × GF st is ler tht it ontins this imgeD nd we hve seen tht for the orresponding wonderful ompti(tionsD the onneted utoE morphism group is preisely this imgeF fut sine our mnifolds re lowups of these wonderful mnifoldsD flnhrd9s lemm @flSTD roposition sFID see lso fIQD roposition RFPFIA gives n inlusion of the onneted utomorphism groups of the low ups in the onneted utomorphism groups of the wonderful onesF o we get our sttementF 4.6 Hermitian metrics on line bundles e will desrie in this setion how to see the potentil of K × KEinvrint @singulrA hermitin metri h on L s W Einvrint funtion on aF TW pigure RFUX xon wonderfulD pno toroidl ompti(tion of Sp 4 (C) epplying this to nonEnegtively urved singulr metrisD we get orresponE dene etween the K ×KEinvrint singulrD nonEnegtively urved metris on L nd W Einvrint onvex funtions on a stisfying symptoti onditionsF hese onditions re given in terms of the polytope P D nd re the onditions orreE sponding to tori metris on the restrition of L| Z to the tori sumnifold Z of XF

Potentials and quasi psh functions

Local potentials vet X e ompt uähler mnifold nd L line undle on XF e hermitian metric h on L is the dtD for eh x ∈ X of hermitin form on the (er L x over XF qiven lol triviliztion of LD sy sD on n open suset U ⊂ XD we hve hoie of sis for eh spe L x D x ∈ U F o the dt of hermitin form on L x in this sis is just omplex numer |s(x)| 2 h D the norm of s(x) with respet to the hermitin formF e n de(ne funtion ϕ on U D y

x → -ln(|s(x)| 2 h )
tht we ll the lol potentil of h with respet to sF e hermitin metri h is determined y ll its lol potentilsF he hermitin metri is lled smooth @respF continuousA if ll of its lol potentils re smooth @respF ontinuousAF e will onsider lso singulr hermitin metris whih re those for whih the lol potentils re L 1 loc F pinlly hermitin metri h on L is sid to e locally bounded if ll its lol potentils on su0iently smll open sets re oundedF UH o smooth hermitin metri on L is ssoited @IDIAEform ω h D lled its curvatureF yne wy to de(ne it is lollyX if ϕ is lol potentil of h on U D then it is lso lol ∂∂ potentil of ω iFeF ω h = i∂∂ϕ on U F purthermoreD ω h lies in the (rst hern lss c 1 (L)F he urvture is still well de(ned s urrent for singulr hermitin metris thnks to the ssumption on the potentilsF Global potential here is nother notion of potentil for hermitin metri hD this time glolD given referene metri h 0 F he(ne the potentil of h with respet to h 0 to e the funtion ψ suh tht for ξ ∈ L x D

|ξ| 2 h = e -ψ(x) |ξ| 2 h0 .
emrk tht oth urvture forms lie in c 1 (L) so y the ∂∂Elemm there exists funtion ψ suh tht

ω h0 = i∂∂ψ + ω h .
he potentil of h with respet to h 0 is suh funtionF Positivity e smooth hermitin metri on L is sid to hve positive curvature if its urvture ω h is uähler metriF emrk tht the existene of suh metri is equivlent to the mpleness of the line undle LF wore generllyD singulr metri h is sid to hve non-negative curvature if ω h ≥ 0 s urrentF et the level of lol potentils this trnsltes in the following wyX singuE lr hermitin metri hs nonEnegtive urvture if its lol potentils re psh funtionsD nd it is smooth nd hs positive urvture if nd only if its lol potentils re smooth nd stritly psh funtionsF vet h 0 e ontinuousD non negtively urved metri on LD nd let ω h0 e its urvture urrentF he(ne the ω h0 Epsh functions on X s the upper semionE tinuous funtions ϕ on X suh tht ω + i∂∂ϕ ≥ 0F

he ω h0 Epsh funtions prmetrize ll non negtively urved metris on LD s the potentils of suh metris with respet to h 0 F 4.

UI

emrk thtD restrited to T D ψ is the potentil of the hermitin metri on L| T indued y h with respet to the restrition of

s 0 F Proposition 4.23. Assume that h is K × K-invariant, then ψ is also K × K- invariant.
Proof. vet k 1 , k 2 ∈ K nd z ∈ GF fy using the triviliztion s 0 D we n idenE tify L| G with G × CD with the tion of (g 1 , g 2 ) ∈ G × G sending (z, t) to (g 1 zg 2 , χ(g 2 )t) for some hrter χ of GF hen

ψ(k 1 zk 2 ) = -ln(|s 0 (k 1 zk 2 )| 2 h ) = -ln(|(k 1 zk 2 , 1)| 2 h ) = -ln(|χ(k 2 )| -2 |(k 1 zk 2 , χ(k 2 ))| 2 h ) = -ln(|χ(k 2 )| -2 |(k 1 , k 2 ) • (z, 1)| 2 h ) emrk tht sine K is omptD |χ(k 2 )| = 1D nd y K × KEinvrine of hD we get ψ(k 1 zk 2 ) = -ln(|s 0 (z)| 2 h ) = ψ(z).
vet ϕ(x) = ψ(exp(x)) e the funtion indued y ψ on the vie lger aF ell from roposition QFIT tht ψ is ompletely determined y ϕD whih is W EinvrintF uppose now tht h is non negtively urvedF hen ψ is pshD nd so ϕ is onvex y roposition QFIUF Denition 4.24. e will ll ϕ the convex potential of hF

Asymptotic behavior of the convex potential

A special metric fefore stting the theorem giving the symptoti ehvior of non negtively urved metris on LD we need to introdue speil ontinuous metriD tht we will denote y h L F his will reple the ftyrevEshinkel metri @see for exmE ple eppendix e or wiHHA de(ned in the tori seF sn ft it is onstruted from this metri on the tori sumnifoldF ell tht we denote y P the polytope of the polriztion (X, L)D nd tht it is lso the polytope ssoited to the polrized tori mnifold (Z, L| Z )F vet g P e the support funtion ssoited to the line undle L| Z F hen there exists ontinuous hermitin metri h P on L| Z D toriD W E invrint nd non negtively urvedD lled the ftyrevEshinkel metri nd whose onvex potentil is the funtion f P : x → -2g P (-x)F emrk tht if we de(ne the support funtion v of polytope Q s v(x) = sup{ x, q ; q ∈ Q} then f P is the support funtion of the polytope 2P F UP Remark 4.25. e onsider here the support funtion of 2P euse in the de(nition of the potentil we took ||s 0 || 2 h nd not just ||s 0 || h F e n extend h P to ontinuous hermitin metri h L on LD sine h P is W EinvrintF sndeedD the stilizer in K × K of point x of Z ts linerly on the (er L x D whih is omplex lineD nd so the metri h P is invrint under this tionF he onvex potentil of h L is still f P F Asymptotic behavior Theorem 4.26. The singular hermitian K × K-invariant metrics h with non negative current curvature are in bijection with the convex W -invariant functions ϕ : a -→ R satisfying the condition that there exists a C 1 ∈ R such that ϕ(x) ≤ f P (x) + C 1 on a, and ϕ is then the convex potential of h. Furthermore, h is locally bounded if and only if there exists in addition a constant C 2 such that

f P (x) + C 2 ≤ ϕ(x) ≤ f P (x) + C 1 .
Proof. vet h e singulr hermitin K × KEinvrint metri with non negtive urrent urvture on LF vet ϕ e its onvex potentilF ell tht h L denotes the metri onstruted oveD nd let ω L e the urvture urrent of h L F rite v the potentil of h with respet to h L F st is n ω L Epsh metri on XF sn prtiulrD v is ounded from ove on XF henote y u the funtion on a ssoited to the K × KEinvrint funtion v| G F hen we see tht the funtion ϕ -f L is equl to u nd thus ounded from oveF sf furthermore h is lolly ounded then sine h L is lso lolly oundedD the funtion v is ounded on XF o u = ϕ -f L is ounded on aF gonverselyD let ϕ e onvex W Einvrint funtion suh tht ϕ(x) ≤ f P (x)+ CF e hoose ny referene metri h 0 on L tht is smoothD positively urved nd K × KEinvrintF hen y the (rst diretion there exist onstnts C 1 nd C 2 suh tht if ϕ 0 is the potentil of h 0 we hve

f P (x) + C 2 ≤ ϕ 0 (x) ≤ f P (x) + C 1 .
vet ω 0 e the urvture form of h 0 F gonsider the funtion u := ϕ -ϕ 0 F st will e enough to show tht the funtion v on G orresponding to u extends to n ω 0 Epsh funtion on XF pirst remrk tht v = ψ -ψ 0 D nd y roposition QFIUD ψ is psh on GF he ssumption on ϕ implies tht uD nd thus vD re ounded from oveF sndeedD we hve

u = ϕ -ϕ 0 ≤ f P + C -ϕ 0 ≤ C -C 2 .
e lssil result on psh funtions is tht psh funtion extends over n nlyti suset if nd only if it is lolly ounded oveF rere pplying tht UQ with v llows to extend v to n ω 0 Epsh funtion on XF he orresponding singulr hermitin metri h hs non negtive urvtureD is K × KEinvrintD nd hs onvex potentil ϕF por lolly ounded metrisD one just needs to use the re(nement tht if psh funtion is lolly ounded then it extends to ounded psh funtionF Smooth metrics sn the se of polrized tori mnifoldsD quillemin quiWR found neessry nd su0ient ondition for smooth stritly onvex funtion to e the onvex potentil of smooth positively urved tori hermitin metriF his ondition is tht the vegendre trnsform u of the onvex funtion is of the form

u(p) = i l i (p) ln(l i (p)) + v(p)
where v is smooth funtion on 2P D nd the l i re the liner forms de(ning 2P F elexeev nd utzrkov stte tht the ondition still holds on smooth polrized group ompti(tions euHSD roposition QFPF e will not use this ondition hereF here re simpler onsequenes of metri eing smoothD whih we will use thoroughly in the following nd in ghpter TF Proposition 4.27. Let h be a smooth K × K-invariant hermitian metric with positive curvature on L, and let ϕ be its convex potential. Then the gradient ∇ϕ of ϕ denes a dieomorphism from a to the interior of 2P , identifying a with a * by the scalar product •, • . Furthermore, the restriction of ∇ϕ to a + is a dieomorphism to the interior of 2P + .

Proof. ine h is smooth nd positively urvedD ϕ is smooth nd stritly onvex funtion on aF o ∇ϕ is di'eomorphismF st remins to determine the imgeF he smoothness of h implies tht it is lolly oundedF o y heorem RFPTD we hve

f P (x) + C 1 ≤ ϕ(x) ≤ f P (x) + C 2
where f P is the support funtion of the polytope 2P F his implies tht ∇ϕ(a) = Int(2P )F fy W EinvrineD we lso hve ∇ϕ(a + ) = Int(2P + )F 4.7 Volume forms and the Duistermaat-Heckman measure 4.7.1 Moment map and Duistermaat-Heckman measure vet (X, L) e smooth polrized ompti(tion of GD orresponding to the polytope P F vet ω e K × KEinvrint uähler form in c 1 (L)F gonsider the moment map µ of (X, ω) with respet to the tion of K × KF he intersetion of the imge of µ with the positive eyl hmer in (a ⊕ a) * UR @regrded s suspe of (k ⊕ k) * A is onvex polytopeD lled the @uirwnA moment polytopeF st follows from the work of frion friVU tht this moment polytope n e identi(ed with P + the intersetion of P ⊂ M R with the positive eyl hmer in a * D where we identify a with its ntidigonl emedding in a ⊕ aF he Duistermaat-Heckman mesure dσ is the pushforwrd of the viouville mesure ω n /n! under the moment mp µ : X → P F frion friVW found n expliit expression for the huistermtErekmn mesure in this situtionF vet dq e the veesgue mesure on P normlized to give unit volume to the fundmentl domin of the lttie in P F vet Ψ deE note the root system of G × GD whih is the disjoint union of two opies of ΦD nd Ψ + hoie of positive rootsD omptile with the hoie of Φ + F vet lso ρ G×G denote the hlf sum of the positive roots of G × GF hen the density of dσ with respet to dq is

ν DH (q) = β∈Ψ + (β, q) (β, ρ G×G ) .

Degree of an ample line bundle

Proposition 4.28. Let (X, L) be a smooth polarized compactication of G, corresponding to the polytope P . Then

deg(L) = C 2P + α∈Φ + (α(p)) 2 dp
for some constant C depending only on the group G. Furthermore, if u is the convex potential of a smooth positively curved K × K-invariant metric on L, then

deg(L) = C a + α∈Φ + (α(∇u(a))) 2 MA R (u)(a)da.
Proof. vet h e smooth positively urved K × KEinvrint hermitin metri on LD with urvture the uähler form ωF vet s e G × {e}Eequivrint setion of LD nd ϕ the potentil of h with respet to sF e thus hve ω = i∂∂ϕ on GF vet dg denote the rr mesure otined on G y the hoie of the sis of g mde in etion QFIFTF sf z 1 , . . . , z n denote the lol omplex oordintes in whih we omputed the omplex ressinD lolly we n write

ω n = i n MA C (ϕ)dz 1 ∧ . . . ∧ dz n = MA C (ϕ)dg
= C a + α∈Φ + α(∇u(a)) 2 MA R (u)(a)da
e use the vegendre trnsform to trnsport this integrl to n integrl on P + F imply putD sine u is smooth nd stritly onvexD we n use the vrile hnge p = ∇u(a)F hen it is ler tht dp = MA R (u)(a)daF he imge y ∇u of a + is the interior of 2P + D y roposition RFPU nd identifying a with a * y the slr produt •, • D so pplying the vegendre trnsform yields

X ω n = C 2P + α∈Φ + α(p) 2 dp.
Remark 4.29. his is in ftD up to multiplitive onstntD the integrl with respet to the huistermtErekmn mesureF sndeedD we hve

ν DH (q) = β∈Ψ + (β, q) (β, ρ G×G ) nd for q = (p, -p) in a ⊕ aD so if β = (α, 0) or β = (0, -α) for α ∈ ΦD we hve (β, q) = 2α(p). hus for some onstnt CD ν DH (p, -p) = C α∈Φ + α(p) 2 dp.

UT

Remark 4.30. he sme proof would giveD for g ny ontinuous funtion on

P + D P + g(p) α∈Φ + (α(p)) 2 dp = a + g(∇u(a)) α∈Φ + (α(∇u(a))) 2 MA R (u)(a)da.
Remark 4.31. he onstnts tht pper in our sttements ove ould e determined expliitly y studying preisely whih rr mesure ppers with our hoie of sis in ghpter QF fut for our purposes we will never need to determine these onstnts expliitlyF UU Chapter 5

Alpha invariants of polarized group compactications sn this hpter we ompute the lph invrint of ny linerized mple line undle L on group ompti(tion X of G with respet to the tion of K × K mximl ompt sugroup of G × GF his is done y omputing the log nonil thresholds of non negtively urved singulr K × KEinvrint hermitin metri on L in terms of onvex ody ssoited to itD tht we ll the xewton ody of the metriF o this end we (rst trnslte the log nonil threshold of metri to n integrility ondition on the glol potentil of the metri with respet to (xed referene metriF henD restriting to the dense orit GD nd using the KAK integrtion formulD this eomes n integrility sttement for onvex potentils of metrisF trting from the nlyti version of the omputtion of log nonil threshE olds of monomil idelsD we otin n integrility riterion in our situtionD involving the xewton odies previously mentionedF sing this riterion nd the eyl group tion we otin n expression of the lph invrintD tht is prtiulrly simple in the se of the ompti(tion of semisimple groupF e then ompute the lph invrint of the ntinonil line undle for some exmples of group ompti(tionsF he su0ient riterion of existene of uählerEiinstein metris in terms of lph invrint is unfortuntely never stis(edD despite the ft thtD t lest for most wonderful ompti(tionsD the group K × K is mximl ompt sugroup of Aut 0 (X)F

Log canonical thresholds on compact manifolds

sn this (rst setion we onsider X ompt omplex mnifold tht is not neessrily group ompti(tionD nd L line undle on XF UV Denition 5.1. vet x e point in XD nd h hermitin metri on LF he complex singularity exponent @or local log canonical threshold A of h t xD whih we denote y lct(h, x) is the supremum of ll c > 0 suh tht e -cϕ is inteE grle with respet to veesgue mesure in neighorhood of xD where ϕ is the potentil of h with respet to triviliztion s of L in neighorhood of xX ϕ(z) := -ln(|s(z)| 2 h ).

Remark 5.2. sf h is lolly ounded metri then on su0iently smll neighE orhood of ny pointD the potentil ϕ is ounded funtionD so it is integrleF st mens tht for ny suh metriD lct(h, x) = ∞ t ny point xF Denition 5.3. vet h e hermitin metri on LD then the log canonical

threshold of h is de(ned s lct(h) = inf x∈X (lct(h, x)).
Proposition 5.4. Let h be a singular hermitian metric on L, h 0 a locally bounded hermitian metric on L, and ψ the potential of h with respect to h 0 . Let also dV be any smooth volume form on X. Then we have

lct(h) = sup c > 0; X e -cψ dV < ∞ .
Proof. vet x e ny point in XD nd s triviliztion of L on neighorhood U of xF p to shrinking U D we n ssume tht the lol potentil ϕ 0 of h 0 with respet to s is oundedF vet ϕ e the lol potentil of h with respet to s nd ψ the potentil of h with respet to h 0 F hen y de(nition of ψD we hve ψ = ϕ -ϕ 0 on U D nd sine ϕ 0 is oundedD the integrility of e -cϕ with respet to veesgue mesure on neighorhood of x is equivlent to the integrility of e -cψ on the sme neighorhoodF purthermoreD in the neighorhood of ny point x in XD the integrility with respet to veesgue mesure is equivlent to integrility with respet to smooth volume formF he funtion ψ is de(ned everywhere on XD e -cψ is positiveD nd X is omptD so e -cψ is integrle with respet to dV in the neighorhood of ny point in X if nd only if X e -cψ dV < ∞F ke 0 < c < lct(h)D then c < lct(h, x) for ll x ∈ XD so X e -cψ dV < ∞F his mens tht

lct(h) ≤ sup c > 0; X e -cψ dV < ∞ . gonverselyD if c > lct(h) then there exists x ∈ X suh tht c > lct(h, x) ut then X e -cψ dV = ∞D so c ≥ sup c > 0; X e -cψ dV < ∞ F king the in(mum gives the other inequlityX lct(h) ≥ sup c > 0; X e -cψ dV < ∞ .
his proves the propositionF UW 5.2 Newton body of a hermitian metric sn this setion we introdue onvex ody ssoited to ny non negtively urved singulr K × KEinvrint hermitin metri h on n mple linerized undle L on group ompti(tion XF e (rst de(ne onvex set ssoited to ny funtionD whih is nturl set to onsider in the se of onvex funtionsF epplying this onstrution to the onvex potentil of hermitin metri yields onvex ody tht is ontined in the polytope of LD tht will e used to ompute the log nonil threshold of hF Then

Newton set of a function

N σ (cf ) = cN σ (f ) N σ (f + c) = N σ (f ) if f ≤ g then N σ (f ) ≤ N σ (g).
In particular, if for some constants c 1 and c 2 ,

g + c 1 ≤ f ≤ g + c 2 on σ, then N σ (f ) = N σ (g).
sf now the one is hnging insted of the funtionD we hve gin some esy properties of the xewton setF Proposition 5.8. Let f : a → R be any function.

If σ is covered by other cones σ

i iFeF σ = ∪σ i , then N σ (f ) = N σi (f ). If σ 1 ⊂ σ 2 then N σ2 (f ) ⊂ N σ1 (f ).

VH

vet us now prove less trivil result whih will e used in the omputtion of log nonil thresholdsF Proposition 5.9. Let v : a → R be a onvexD pieewise liner function along a decomposition σ = ∪σ i of a cone in cones of full dimension. Denote by v σi the element of a * such that v

(x) = v σi (x) on σ i . Then N σ (v) = Conv{v σi }+(-σ ∨ ). Proof. pirstD it is ler tht N σi (v) = v σi + (-σ ∨ i ) for ll iD nd thusD tht N σ (v) = i v σi + (-σ ∨ i )F sn prtiulrD we hve the esy inlusion N σ (v) ⊂ Conv{v σi } + (-σ ∨ ). o prove the other inlusionD it is enough to show tht for ll iD jD v σi ∈ v σj + (-σ ∨ j )F e use the onvexity of vF vet x ∈ Int(σ i ) nd y ∈ Int(σ j )F e hveD for ny 0 ≤ t ≤ 1D v(ty + (1 -t)x) ≤ tv σj (y) + (1 -t)v σi (x).
hen t is lose to HD ty + (1 -t)x is still in σ i D so we get

tv σi (y) + (1 -t)v σi (x) ≤ tv σj (y) + (1 -t)v σi (x). his implies tht (v σj -v σi )(y) ≥ 0.
his is true for ll y ∈ Int(σ j )D so for ll y ∈ σ j D so this mens tht v σj -v σi ∈ σ ∨ j D or in nother order v σi ∈ v σj + (-σ ∨ j )F

Newton set of convex functions

por this prgrph onlyD we will llow onvex funtions to tke the vlue +∞F sf f is suh funtion we de(ne its domain y

dom(f ) := {x ∈ a; f (x) < ∞}.
e impose however tht ll funtions onsidered hve non empty dominF sn the rest of the textD we ssume dom(f ) = aF he (rst remrk to e mde is tht the xewton set of funtion f on the whole of a is the domin of its vegendreEpenhel trnsform @or onvex onjugteA

f * de(nedD for m ∈ a * D y f * (m) := sup{m(x) -f (x); x ∈ a}.
vet σ e onvex oneD nd de(ne the onvex funtion δ σ s the inditor funtion of σD i.e.

δ σ (x) = 0 if x ∈ σ nd δ σ (x) = ∞ otherwiseF hen it is not hrd to hek tht N σ (f ) = N a (f + δ σ )F sn other words N σ (f ) is the domin of the onvex onjugte of f + δ σ F
e will rell lssil result on onvex funtionsD whih llows to express the xewton set of sum s the winkowski sum of the xewton sets of the summndsF pirst rell the de(nition of in(ml onvolutionX VI Denition 5.10. vet f nd g e two onvex funtionF he inmal convolution of f nd g is the funtion f g de(nedD for x ∈ aD y f g(x) = inf{f (x -y) + g(y); y ∈ a}.

Theorem 5.11. [Roc97,Theorem 16.4] Let f and g be two convex functions on a, such that the relative interiors of the domains of f and g have a point in common. Then

(f + g) * (m) = f * g * .
Proposition 5.12. Let σ be a convex cone, and f a convex function with

dom(f ) = a, then N σ (f ) = N a (f ) + (-σ ∨ ).
Proof. e hve seen tht N σ (f ) is the domin of the onvex onjugte of f +δ σ D ut y heorem SFIID this is lso the domin of the funtion f * δ * σ F e n pply the heorem euse the intersetion of the domins of f nd δ σ is σF he domin of n in(ml onvolution is the winkowski sum of the domins of the two funtions involvedD so we just need to ompute the domin of δ * σ F fy de(nition we hek tht this is -σ ∨ D nd otin the sttementF Proposition 5.13. Let f and g be two convex functions, both with domain a, and σ a convex cone.

Then N σ (f + g) = N σ (f ) + N σ (g).
Proof. e hveD y the previous propositionD

N σ (f + g) = N a (f + g) + (-σ ∨ ).
fut y the sme proofD

N a (f + g) = N a (f ) + N a (g), so N σ (f + g) = N a (f ) + N a (g) + (-σ ∨ ) = N σ (f ) + N σ (g)

Newton body of a metric

vet X e ompti(tion of GD polrized y LF vet h e K ×KEinvrint hermitin metri with non negtive urvture on LD nd ϕ its onvex potentil with respet to (xed leftEinvrint triviliztion of L on GD whih is funtion on aF Denition 5.14. e will ll Newton body of h the set N (h) := N a (ϕ)F vet P L e the polytope orresponding to the polriztion LF VP Example 5.15. vet h L e the metri onstruted in etion RFTFQF sts onvex potentil is the support funtion of 2P L D so N (h L ) = 2P L D whih n e heked y roposition SFWF emrk tht the onvex potentil of h L is pieewise liner with respet to the opposite of the fn of the tori suvrietyF Proposition 5.16. The Newton body of h is stable under the action of the Weyl group W .

Proof. vet ϕ e the onvex potentil of hD nd let m ∈ a * F uppose tht

ϕ(x) -m(x) ≥ C
for some onstnt C nd for ll x ∈ aF vet w ∈ W F fy W Einvrine of ϕD the inequlity is equivlent to

C ≤ ϕ(w • x) -m(x) ≤ ϕ(w • x) -w -1 • m(w • x) ine w indues ijetion of aD we get tht for ll w ∈ W D m ∈ N (h) if nd only if w • m ∈ N (h)D whih mens tht N (h) is W EinvrintF Proposition 5.17. Let h be a K × K-invariant hermitian metric with non negative curvature on L. Then N (h) ⊂ 2P L . If in addition h is locally bounded, then N (h) = 2P L .
Proof. henote y ϕ L the onvex potentil of the metri onstruted in eE tion RFTFQF ell from the sme setion tht the onvex potentil ϕ of K × KEinvrint hermitin metri h with non negtive urvture on L stisE (es ϕ ≤ ϕ L + C 2 on a for some onstnt C 2 D nd tht if h is lolly ounded then we hve in ddition ϕ L + C 1 ≤ ϕ for some onstnt C 1 F xow the result esily follows from roposition SFU nd ixmple SFISF 5.3 Integrability criterions 5.3.1 Integrability criterion on a cone e will use the following propositionD proved y quenni in queIPF st is n nlyti proof nd generliztion of the omputtion y rowld of the log nonil thresholds of monomil idelsF he sttement given here is slightly di'erent thn the sttement in queIPD ut is in ft equivlent @see eppendix e for detilsAF Proposition 5.18. [Gue12] Let f be a convex function on a. Assume that σ is a smooth polyhedral cone in a = N R . Then e -f is integrable on a translate (equivalently on all translates) of σ if and only if 0 is in the interior of the Newton body of f : 0 ∈ Int(N σ (f )). VQ 5.3.2 Integrability with respect to J pix G redutive groupD let Φ e its root systemD Φ + hoie of positive rootsF ell tht J is the funtion de(ned on a y

J(x) = α∈Φ + sinh 2 (α(x)).
he hlf sum of positive roots is denoted y ρF e wnt to prove the following integrility riterionD with respet to the mesure J(x)dxF Proposition 5.19. Assume that a + = i σ i where each σ i is a smooth polyhe- dral cone of full dimension r. Let l be a function on a, convex on each cone σ i . Then

a + e -l(x) J(x)dx < +∞ if and only if 4ρ ∈ Int(N a + (l)).
Lemma 5.20. Let σ be a smooth full dimensional polyhedral cone in a + , l be a convex function on a, then the following are equivalent:

σ e -l(x) J(x)dx < ∞; σ e -l(x)+4ρ(x) dx < ∞; 4ρ ∈ Int(N σ (l)).
Proof. riting

sinh(α(x)) = e α(x) -e -α(x) 2 = 1 2 e α(x) (1 -e -2α(x) ),
we get tht

J(x) = 1 2 2Card(Φ + ) e 2 α∈Φ + α(x) α∈Φ + (1 -e -2α(x) ) 2 . por ny x ∈ a + nd α ∈ Φ + D α(x) > 0D so 0 ≤ e -2α(x) < 1F his implies 0 < α∈Φ + (1 -e -2α(x) ) 2 ≤ 1D so 0 < J(x) ≤ 1 2 2Card(Φ + ) e 4ρ(x) .
his (rst inequlity llows to sy tht if σ e -l(x)+4ρ(x) dx < ∞ then σ e -l(x) J(x)dx < ∞.

vet us now prove the onverseF ghoose γ point in the interior of σF essume tht e -l+4ρ is not integrle on σF hen y the usul integrility riterion @roposition SFIVA e -l+4ρ is lso non integrle on γ + σF x) .

VR fut nowD for x ∈ γ + a + nd α ∈ Φ + D we hve α(x) ≥ c = min β∈Φ + β(γ) > 0D so 0 ≤ e -2α(x) ≤ e -2c < 1D nd this implies 1 -e -2c 2 2Card(Φ + ) e 4ρ(x) ≤ J(x) ≤ 1 2 2Card(Φ + ) e 4ρ(
his gives tht σ e -l(x) J(x)dx ≥ γ+σ e -l(x) J(x)dx ≥ γ+σ e -l+4ρ dx ≥ ∞
o we hve shown the equivlene of the two (rst points in the lemmF fy the usul riterion the seond point is lso equivlent to

0 ∈ Int(N σ (l -4ρ)) = -4ρ + Int(N σ (l)).
vetting 4ρ go to the leftD we onlude the proofF xow we n prove the propositionD just y gluing the prtsF Proof. tust remrk tht sine the funtion e -l(x) J(x) is positive nd the ones re full dimensionlD a + e -l(x) J(x)dx < +∞ if nd only if σi e -l(x) J(x)dx < +∞ for ll iF por eh of these integrls we n use the lemmD so the neessry nd su0ient ondition eomes 4ρ ∈ Int(N σi (l)) for ll iD or equivlently 4ρ ∈ Int( i N σi (l))F o onludeD oserve tht N a + (l) = i N σi (l) y emrk SFVF

Log canonical thresholds on group compactications

vet G e redutive groupF vet X e smooth pno G × GEequivrint ompti(tion of GF vet L e n mple line undle on XD whose ssoited polytope is P F henote y Q the polytope ssoited to the ntinonil undle -K X F vet lso H denote the onvex hull of ll imges of 2ρ y the eyl group W F e will onsider only K × KEinvrint metris on LD nd use the KAK integrtion formul tht we rell hereX Proposition 5.21. [Kna02] If f is a K×K invariant function on G, dg denotes a Haar measure on G, and dx a Lebesgue measure on a + , then

G f (g)dg = C a + J(x)f (exp(x))dx
for some constant C > 0 independent of f . VS e wnt to prove the following Theorem 5.22. Let h be a K ×K-invariant hermitian metric with non negative curvature on L, then

lct(h) = sup{c > 0; 2H + 2cP ⊂ cN (h) + 2Q} e (rst introdue some nottionsF vet us (x s 0 left G equivrint triviliztion of L on G nd s 1 left G equivrint triviliztion of -K X on GF
vet u e the onvex potentil of h with respet to the setion s 0 F vet lso u 0 e the support funtion of P nd h 0 e the orresponding metriF st hs lolly ounded potentilsF henote y ψ the potentil of h with respet to h 0 F ine X is pnoD we n hoose h 1 smooth metri on -K X with positive urvtureD nd let u 1 e its onvex potentil with respet to s 1 F his hoie determines smooth volume form on XD whih writesD on GD dV = e -u1 dg where dg is the rr mesure s -1

1 ∧ s -1 1 F
Remark 5.23. sn prtiulrD the integrl of this volume form is (niteD so pE plying the KAK integrtion formul this mens tht a + e -u1 Jdx < ∞ fy the integrility riterion @roposition SFIWAD this implies tht

4ρ ∈ Int(N (h 1 )) = Int(2Q).
enother wy to sy tht is

H ⊂ Int(Q)F
Proof. sing roposition SFRD then restriting to the dense oritD we getX

lct(h) = sup c > 0; X e -cψ dV < ∞ = sup c > 0; G e -cψ dV < ∞ ine ψ(exp(x)) = u(x) -u 0 (x)D we n now use the KAK integrtion formul to writeX lct(h) = sup c > 0; a + e -c(u-u0) e -u1 Jdx < ∞ .
hen roposition SFIW givesX

lct(h) = sup {c > 0; 4ρ ∈ Int(N a + (cu -cu 0 + u 1 ))} = sup{c > 0; 4ρ ∈ N a + (cu -cu 0 + u 1 )}
VT vet σ i e the losures of the ones of full dimension in the fn sudivision of a + orresponding to XF hen u 0 is liner on eh -σ i F e write u i 0 the orresponding element of a * F e hve

lct(h) = sup{c > 0; ∀i, 4ρ ∈ N -σi (cu -cu 0 + u 1 )} = sup{c > 0; ∀i, 4ρ + cu i 0 ∈ N -σi (cu + u 1 )} ell from roposition SFW tht P = N a (u 0 ) ⊂ u i 0 + σ ∨ i D so tht lct(h) = sup{c > 0; ∀i, 4ρ + cP ∈ N -σi (cu + u 1 )} = sup{c > 0; 4ρ + cP ∈ N a + (cu + u 1 )} = sup{c > 0; 2H + 2cP ⊂ N a (cu + u 1 )} y W EinvrineF o onlude it remins to remrk tht oth u nd u 1 re onvexD so y roposition SFIQD N a (cu + u 1 ) = cN a (u) + N a (u 1 ) = cN (h) + 2Q.

Alpha invariant on group compactications

Denition 5.24. vet X e ompt omplex mnifoldD K ompt suE group of the utomorphisms group of XD nd L KElinerized line undle on XF he alpha invariant of L reltive to the group KD denoted y α K (L) is the in(mum of the log nonil thresholds of ll KEinvrint singulr hermitin metris on L with non negtive urvtureF fefore stting the min resultD let us introdue two notions out onvex odiesF vet P nd Q e two onvex odies in a * F ell the de(nition of the Minkowski dierenceX

Q P = {x|x + P ⊂ Q}.
enother expression of the winkowski di'erene is the followingD whih shows tht it is onvex if Q is onvexX

Q P = p∈P (-p + Q). sf P 1 D P 2 nd Q re three onvex odiesD then P 1 + Q ⊂ P 2 if nd only if P 1 ⊂ P 2 QF
Denition 5.25. he inradius of Q with respet to P is the numerX

inr(P, Q) := sup{c ≥ 0|∃x x + cP ⊂ Q}.
VU he lph invrint of n mple line undle on smooth pno ompti(E tion of semisimple group n e esily expressed in terms of the polytope ssoited to L s n inrdius etween two onvex odiesF e stte (rst the result for generl redutive group ompti(tionsD nd then we will see how the sttement is simpli(ed in the se when the group is semisimpleF Theorem 5.26. Let G be a reductive group, and X be a smooth Fano group compactication of G. Let L be an ample G × G-linearized line bundle on X, whose associated polytope is P . Denote by Q the polytope associated to the anticanonical line bundle -K X . Then

α K×K (L) = sup{c > 0; c(P + (-P W )) ⊂ Q H},
where P W denotes the subset of W -invariant points of P .

Proof. vet h e ny K × KEinvrint metri on L with non negtive urvtureF he xewton ody of h is onvex nd W EstleF sn prtiulr it ontins W Einvrint point pD for exmple the ryenter of the orit of ny point in N (h)F henote y h p the K × KEinvrint metri on L with non negtive urvture whose onvex potentil is the funtion x → p(x)F hen {p} = N (h p ) ⊂ N (h)D so y the expression of the log nonil thresholds from heorem SFPPD lct(h) ≥ lct(h p )F ine ll suh h p for p ∈ 2P W de(ne singulr hermitin metri with nonE negtive urvtureD this remrk llows to write the lph invrint s

α K×K (L) = inf p∈2P W lct(h p ).
xow from the expression of the log nonil threshold we get

lct(h p ) = sup{c > 0; 2H + 2cP ⊂ cN (h p ) + 2Q} = sup{c > 0; -cp + 2cP ⊂ 2Q 2H}
hen the expression of the lph invrint further simpli(es s

α K×K (L) = inf p∈2P W sup{c > 0; -cp + 2cP ⊂ 2Q 2H} = sup{c > 0; ∀p ∈ 2P W , -cp + 2cP ⊂ 2Q 2H} = sup{c > 0; 2cP + (-2cP W ) ⊂ 2Q 2H}
hividing y two yields = sup{c > 0; c(P + (-P W )) ⊂ Q H} whih is the expression in the sttement of the heoremF Remark 5.27. sn the tori seD we reover our previous omputtion from eppendix eX α (S 1 ) n (L) = sup{c > 0; c(P + (-P )) ⊂ Q}.

VV

Corollary 5.28. Assume that G is a semisimple group. Then

α K×K (L) = inr(P, Q H).
Proof. sf G is semisimpleD we hve P W = {0}F sn ftD the metri h 0 whose onvex potentil is the zero funtion stis(es

α K×K (L) = lct(h 0 ) = sup{c > 0; cP ⊂ Q H}.
end this is equl to the inrdius inr(P, Q H)F sndeedD one inequlity is trivilX

inr(P, Q H) ≥ α K×K (L)F gonverselyD ssume c ≤ inr(P, Q H)D iFeF there exists n x ∈ a * suh tht x + cP ⊂ Q H. hen sine P nd Q H re stle under W EtionD we lso hve ∀w ∈ W, w • x + cP ⊂ Q H.
gonvexity nd the ft tht the ryenter of the W Eorit of x is 0 imply tht cP ⊂ Q HD so c ≤ α K×K (L)F e hve thus proved the other inequlity inr(P, Q H) ≤ α K×K (L)F Remark 5.29. sn the se of redutive groupsD the lph invrint is not n inrdiusD ut we n ound it from ove y n inrdiusX α K×K (L) ≤ inr(P + (-P ) W , Q H).

Additional symmetries

sf the polytopes P nd Q dmit dditionl ommon symmetriesD then the vlue of the lph invrint n e improvedF sndeedD the symmetries of Q trnslte to (nite sugroup O of the utomorphisms group of the vriety XD nd if P is stle under these symmetriesD then it is linerized y OF e n thus onsider the lph invrint with respet to the igger group generted y K × K nd OD tht we denote K O F e then hveD dpting the proof of heorem SFPTD

α K O (L) = sup{c > 0; c(P + (-P W,O )) ⊂ Q H}.
sn prtiulrD if the only (xed point under W, O is the originD then just s in the semisimple seD we get

α K O (L) = inr(P, Q H).
VW

Examples

por wonderful ompti(tions of semisimple djoint groupsD the polytope of the ntinonil line undle Q is determined y the root systemF sndeedD rell tht it is the onvex hull of the imges y W of the weight 2ρ + r i=1 α i where the α i re the simple roots of Φ + nd 2ρ is the sum of the positive rootsF sn prtiulrD when G = (PSL 2 (C)) n D for ny n ≥ 1D the simple roots re the sme s the positive rootsD so Q = 2HF Corollary 5.30. Let X be the wonderful compactication of (PSL 2 (C)) n , then

α K×K (-K X ) = 1 2 .
Proof. epplying gorollry SFPV gives

α K×K (-K X ) = inr(2H, H) = 1 2 .
wore generlly for type A n D hoosing n pproprite ordering of the simple roots α 1 , . . . , α n D we n write the positive roots s

α i + α i+1 + • • • + α j
for eh pir (i, j) with 1 ≤ i ≤ j ≤ nF e see then tht the oe0ient of α k in the sum of positive roots n l=1 α l is equl to the rdinl of the set {(i, j); 1 ≤ i ≤ k ≤ j ≤ n}F his is k(n -k + 1)F edding the sum of simple rootsD we see tht the k th Eoordinte of the vertex de(ning the polytope of the wonderful ompti(tion of PSL n+1 (C) in the sis of simple roots is 1 + k(n -k + 1)F hen from our resultD the lph invrint is esily seen to e the mximum of ll c > 0 suh tht for eh kD c(1+k(n-k +1)) ≤ 1F e dedue the following vlue for the lph invrintF Corollary 5.31. Let X be the wonderful compactication of PSL n+1 (C), then

α K×K (-K X ) = 1 1 + n 2 ( n 2 + 1)
.

WH

Chapter 6

Existence of Kähler-Einstein metrics on group compactications vet X e smooth pno ompti(tion of onneted redutive group GF henote y P the polytope ssoited to the ntinonil polriztion of XF ell tht the root system Φ of G is the root system of the derived vie lger [g, g]D nd 2ρ denotes the sum of positive rootsD in a * F vet •, • denote the slr produt on a introdued in ghpter Q @it extends the uilling form on the semisimple prtAF e use it to identify a with a * F sn prtiulrD P is identi(ed with polytope in aD nd P + with its intersetion with a + euse the positive eyl hmers in a nd a * orrespondF ell tht in the redutive seD if a z denotes the tori prt nd a ss the semisimple prt of aD then the positive eyl hmer a + ss is de(ned s usul s the open eyl hmer generted y the fundmentl weightsD nd a + is the produt

a + = a z × a + ss ⊂ a.
vet Ξ e the reltive interior of the losed one generted y the simple rootsF enother de(nition of Ξ isX

Ξ = {p ∈ a ss ; p, x > 0 ∀x ∈ a + ss }.
he results re the followingF Theorem 6.1. There exists a Kähler-Einstein metric on X if and only if the barycenter bar DH (P + ) of P + with respect to the Duistermaat-Heckman measure is in 2ρ + Ξ.

en expression of the ryenter bar DH (P + ) is the followingD where Φ + deE WI notes the positive roots of the root system Φ of GX bar DH (P + ) =

P + p α∈Φ + (α(p)) 2 dp P + α∈Φ + (α(p)) 2 dp -1
.

yserve tht when G is semisimpleD the one Ξ is the open one generted y the simple roots of ΦF hen G is not semisimpleD the dimension of Ξ is stritly smller thn rF sn prtiulrD for G torusD ρ is the origin nd Ξ = {0}D so we reover the usul tori riterionF sndeedD the huistermtErekmn mesure then is the veesgue mesure on P = P + D so the riterion is just tht the ryenter of P is the originF hen there re no uählerEiinstein metrisD we n see how fr we n go in the ontinuity methodD nd thus we hve the vlue of the gretest ii lower oundF sndeed this invrint R(X) of pno mnifold XD de(ned s R(X) := sup{t; ∃ω ∈ c 1 (X), Ric(ω) ≥ tω} ws shown to oinide with the supremum of ll times t whih there exists solution in the ontinuity method zéIIF Theorem 6.2. Assume there are no Kähler-Einstein metrics on X and let R(X) be the greatest Ricci lower bound of X. Then R(X) = sup t < 1; t 1 -t (2ρ -bar DH (P + )) + 2ρ ∈ (P + + (-Ξ)) .

his hpter is devoted to the proof of these resultsF he neessry prt of the ondition is otined in roposition TFPI nd the su0ient prt in heE orem TFQHF por the gretest ii lower oundD we (rst prove tht it is lower thn the quntity in the theorem in roposition TFPPD nd end the proof with heorem TFQIF 6.1 Continuity method 6.1.1 In general vet X e pno mnifoldF pix referene uähler form ω ref in the lss 2πc 1 (X)F he uähler forms in 2πc 1 (X) n ll e written s ω ref + i∂∂ψ with ψ smooth nd ω ref Estritly psh funtion on XD iFeF suh tht ω ref + i∂∂ψ > 0F he uählerEiinstein eqution Ric(ω) = ω on X trnsltesD in terms of ω ref E psh funtionsD s the wongeEempère eqution

(ω ref + i∂∂ψ) n = e f ref -ψ ω n ref , @TFIA
where f ref is the normalized Ricci potential of ω ref de(ned s the ω ref Epsh funE tion tht stis(es

ω ref + i∂∂f ref = Ric(ω ref ) nd X e f ref ω n ref = X ω n ref F
WP vet h ref e smooth hermitin metri on -K X with urvture form ω ref F hen it determines volume form dV on X de(ned in lol triviliztion s of -K X y dV = |s| 2 h ref s -1 ∧ s -1 F hen up to onstntD the ii potentil f ref is equl to the logrithm of the potentil of dV with respet to ω n ref F e hoose h ref @y multiplying y slrA suh tht f ref is indeed equl to thtF he following fmily of equtions is the one used in the usul ontinuity method for the uählerEiinstein equtionX

(ω ref + ∂∂ψ t ) n = e f ref -tψt ω n ref . @TFPA
o show the existene of uählerEiinstein metri on XD it is enough to show tht the set I of 0 ≤ t ≤ 1 suh tht this eqution dmits solution is extly [0, 1]F fy the work of euin euUT nd u uUVD 0 ∈ ID nd I is openF purthermoreD it is enough to know uniform priori estimtes on the C 0 norm of ψ t D to ensure the losure of ID nd thus the existene of solution t t = 1D iFeF uählerEiinstein metriF e rell tht y C 0 estimtesD we menD s in most of the litertureD uniform ontrol on sup x∈X |ψ t (x)|F sn ftD we n even onentrte only on uniform upper ound on ψ t @see iuVVD roposition PFI or iVUD pges PQS nd PQTAF Notation 6.3. vet us (x some 0 < t 0 ∈ ID whih exists sine 0 ∈ I nd I is openF vet us summrize the onsequene of wht we hve relled in this setionF Proposition 6.4. Assume that [t 0 , t 1 [⊂ I, that ψ t denotes the solution at t ∈ [t 0 , t 1 [, and that there exists a constant C such that ψ t ≤ C ∀t ∈ [t 0 , t 1 [. Then t 1 ∈ I.

Reduction to the open orbit

he estimtes were otined y ng nd hu HR in the tori seD y restriting to the open dense torus nd using onvex nlysisF e follow the sme generl frmeworkD ut severl modi(tions re neessryF uppose now tht X is G × GEequivrint smooth nd pno omptiE (tion of GF vet P e the polytope ssoited to the ntinonil undle -K X F fy the tion of K × KD if we hoose h ref K × KEinvrintD we n ssume tht the funtions ψ t in eqution @TFPA re K × KEinvrintF e usul wy to do this is to onsider only the times t for whih there exists K × KEinvrint solution nd prove openness in this situtionD ut this is not enough to get n ostrution to the existene of uählerEiinstein metris or n upper ound on R(X)F o otin this we use the stronger result tht in ft solution t time t if it exists is unique nd thus neessrily K × KEinvrint if h ref isF his follows from the uniqueness result for twisted @or generlizedA uählerEiinstein metris IRD gorollry IFRF WQ fy ontinuity of the solutions ψ t D it is enough to prove uniform upper ound on the restritions of ψ t to the open nd dense orit G ⊂ XF vet ϕ t denote the funtion on a indued y ψ t F st is enough to give n upper ound for ϕ t F e lso denote y h t the hermitin metri on -K X whose potentil with respet to h ref is ψ t F vet u ref e the onvex potentil of h ref D de(ned on aD denote y u t the onvex funtion u ref + ϕ t whih is the onvex potentil of the metri h t F pinllyD we denote y w t the funtion

tu t + (1 -t)u ref F Proposition 6.5. Suppose ψ t is a K × K-invariant solution of equation @TFPA. Then for x ∈ a, MA R (u t )(x) α∈Φ + α(∇u t (x))) 2 = e -wt(x) J(x).
@TFQA ell tht J(x) = α∈Φ + sinh 2 (α(x))D where Φ is the root system of GF

Proof. e introdued in etion QFPFQ left GEinvrint setion s of the ntiE nonil undle -K G on GF st gives rise lso to rr volume form s -1 ∧ s -1 on GF purthermoreD we n express the potentil of (i∂∂ψ) n with respet to s -1 ∧ s -1 D for smooth funtion ψ on GD s

(i∂∂ψ) n = MA C (ψ)s -1 ∧ s -1 ,
where MA C (ψ) is the omplex wongeEempère of ψ is the lol oordintes given in etion QFPFQF vet ψ ref e the potentil of the referene metri h ref with respet to the setion s nd pply this to the funtion ψ ref + ψ t F st gives thtD on GD

(ω ref + ∂∂ψ t )| n G = (i∂∂ψ ref + ψ t ) n = MA C (ψ ref + ψ t )s -1 ∧ s -1
xow the omputtion of the omplex wongeEempère in lol oordintes from etion QFPFQ gives

(ω ref + ∂∂ψ t ) n (exp(x)) = MA R (u t )(x) α∈Φ + α(∇u t (x))) 2 1 J(x) s -1 ∧ s -1
for x ∈ a + F yn the other hndD the de(nition of the normlized ii potentil

f ref imply tht e f ref ω n ref = e -ψ ref s -1 0 ∧ s -1
0 , whih llows to write the right hnd side of eqution @TFPA s

e f ref -tψt ω n ref = e -tψt-ψ ref s -1 0 ∧ s -1 0 .
WR por x ∈ aD we hve

-tψ t -ψ ref (exp(x)) = -tϕ t (x) -u ref (x) = -tu t (x) -(1 -t)u ref (x) = -w t (x)
sn onlusionD t point exp(x) for x ∈ a + D eqution @TFPA reds

MA R (u t )(x) α∈Φ + α(∇u t (x))) 2 1 J(x) s -1 ∧ s -1 = e -wt(x) s -1 ∧ s -1 .
st is equivlent to the equlity of the potentils with respet to s -1 ∧ s -1 F purthermoreD y multiplying oth sides y J(x)D we otin the eqution of the sttementD tht is well de(ned on the whole of aD nd it is stis(ed y W E invrine nd smoothnessF 6.1.3 Strategy o (nd uniform upper ound for ϕ t we will introdue nother funtion ν t D nd study this funtionD following the strtegy used y ng nd hu in the tori seF wore preiselyD let j e the funtion on the open eyl hmer a + de(ned y j(x) = -ln(J(x))F e onsider the funtion ν t = w t + j de(ned on a + F e will show tht it is stritly onvex funtion on a + F st is proper in the following senseX es x goes to in(nityD or x goes to wll of a + D ν t (x) goes to in(nityF hese two properties of ν t imply tht it dmits unique minimumF vet m t e the minimum of ν t nd x t e the point of a + where this minimum is ttinedF e will otin estimtes on oth the vlue m t of the minimum nd on the distne from the origin |x t | of the point where it is ttinedF hen we need to relte these estimtes with the funtion tht we wnt to ontrolF xmely we will go from ν t to w t then u t nd (nlly ϕ t F o summrizeD the strtegy to prove estimtes on ϕ t is in three stepsX ! redue to estimtes on |m t | nd |x t |D ! (nd uniform estimtes |m t | ≤ CD ! get uniform ontrol |x t | ≤ C of x t F e will lso hve to prove the neessity of the ondition nd the upper ound on R(X)F fefore thtD we gther some preliminry resultsF 6.2 Preliminaries 6.2.1 Potentials of metrics in c 1 (X) e ollet some informtion out the potentils of smooth hermitin metE ris on -K X with positive urvture tht will e used severl times in the proofF WS he polytope ssoited to the ntinonil polriztion of X is denoted y P D P + is its intersetion with the positive eyl hmer nd v is the support funtion of 2P F Proposition 6.6. Let f : a -→ R be the convex potential of a smooth K × Kinvariant hermitian metric on -K X with positive curvature. Then

1. f is W -invariant, 2. ∇f (a) = Int(2P ) and ∇f (a + ) = int(2P + ), 3. |∇f | ≤ d for some constant d independent of f , 4. f (x) ≤ v(x -x 0 ) + f (x 0 )
for any x 0 ∈ a, and 5. f (x) ≥ v(x) + C 1 for some constant C 1 depending on f . Proof. ine smooth metri hs lolly ounded potentilsD roposition RFPT implies tht f is W Einvrint nd there exists onstnts

C 1 nd C 2 depending on f suh tht v(x) + C 1 ≤ f (x) ≤ v(x) + C 2 .
e wnt to prove the fourth pointF vet x 0 ∈ aF por ny 0 = y ∈ aD onsider the slope f (x0+ty)-f (x0) t D with t > 0F fy onvexity nd the two inequlities given y roposition RFPTD we see tht this slope inreses nd onverges to v(y) s t tends to in(nityF his shows tht for ny x = x 0 + y ∈ a \ {x 0 }D we hve

f (x) ≤ v(x -x 0 ) + f (x 0 ).
his inequlity is oviously lso stis(ed t x 0 D so the fourth point is provedF he seond point is extly the onlusion of roposition RFPUD nd it implies the thirdD euse the polytope P is oundedF ine P ontins the origin @y W EinvrineAD we n tke for exmple d equl the dimeter of 2P F Remark 6.7. his proposition in prtiulr pplies to the funtions u ref D u t D nd w t F 6.2.2 The functions j = -ln(J) and ν t he im of this setion is to study the funtions j nd ν t to show the existene of the minimum of ν t nd otin some prtil estimtes on themF e hoose n ritrry sis {e i } of aD nd orresponding oordintes {x i } when neessryF Lemma 6.8. The function j is strictly convex on a + .

Proof. e ompute the ressin of j nd hek tht it is positive de(niteF ell tht J(x) = α∈Φ + sinh 2 (α(x))F hen j is the following sumX

j(x) = -2 α∈Φ + ln(sinh(α(x)).
WT en esy omputtion shows tht ∂ 2 ∂x j ∂x i (-ln(sinh(α(x))) = α(e i )α(e j ) 1 sinh 2 (α(x))

.

o the ressin of j is the sum of semipositive mtriesD nd it is esy to hek tht the whole sum is de(niteD so the ressin of j is positive de(niteF Lemma 6.9. There exists a constant c such that for any x ∈ a + , we have

j(x) ≥ -4ρ(x) + c. Proof. rite sinh(α(x)) = e α(x) 1 -e -2α(x) 2 ≤ e α(x) 2 for x ∈ a + F hen j(x) = -2 α∈Φ + ln(sinh(α(x))) ≥ -2 α∈Φ + α(x) + c,
where c = 2 ln(2)Card(Φ + )F

e n now prove the existene of the minimum m t t x t F Lemma 6.10. The function ν t admits a unique minimum.

Proof. e know from vemm TFV tht ν t is stritly onvexF e lredy reE mrked tht ν t (x) tends to +∞ when x pprohes eyl wllF o prove the existene of minimum x t it remins to explin why ν t goes to in(nity t in(nityF roposition TFT implies tht w t (x) ≥ v(x) + C 1 for some onstnt C 1 D where v is the support funtion of the polytope 2P D so ν t (x) ≥ v(x) + j(x) + C 1 F hen ν t (x) ≥ v(x) -4ρ(x) + c + C 1 y vemm TFWF pinllyD the ft tht X is pno impliesD y emrk SFPQ tht 4ρ ∈ Int(P )D so ν t is indeed properF he hlf sum of positive roots ρ is in the interior of a + D so α(ρ) > 0 for ll α ∈ Φ + F e will use this s referene to ontrol the distne to the wllsF pirstD we n sy tht x t is not too lose to the wllsX Lemma 6.11. There exists a constant b 1 > 0 independent of t such that

x t ∈ b 1 ρ + a +

WU

Proof. fy de(nition of x t D the derivtive of ν t t x t vnishesF sn prtiulrD the diretionl derivtive of ν t in the diretion ρ is zeroX

(Dν t ) xt (ρ) = ∇ν t (x t ), ρ =: ∂ν t ∂ρ (x t ) = 0.
ell tht ν t = w t + jD nd tht the derivtives of w t re ounded y roposition TFTF sn prtiulr we get ound

∂w t ∂ρ (x t ) ≤ C.
yn the other hndD we n ompute the diretionl derivtive of jX ∂j ∂ρ

(x t ) = -2 α∈Φ + α(ρ)coth(α(x t )).
o we hve

2 α∈Φ + α(ρ)coth(α(x t )) ≤ C
ut sine ll the terms of the sum re positive nd ll the α(ρ) re stritly positiveD this implies tht for ll α ∈ Φ + D coth(α(x t )) ≤ CF yserve tht the funtion coth tends to +∞ t HD so we otin α(x t ) ≥ C for ll α for onstnt C > 0 independent of tF o onludeD oserve tht the intersetion of the hlf spes de(ned y α(x) ≥ C is ontined in trnslte b 1 ρ + a + for some b 1 > 0 su0iently smllD independent of tF e will lso need to ontrol the derivtives of j wy from the wllsF his is hieved y the following lemmF Lemma 6.12. For any b > 0, there exists a constant C such that for any

x ∈ bρ + a + , |∇(j)(x)| ≤ C.
Proof. ell tht

∂j ∂x i (x) = -2 α∈Φ +
α(e i )coth(α(x))

por x ∈ bρ + a + D we hve 1 < coth(α(x)) < coth(bα(ρ))D so for ny iD ∂j ∂x i (x) ≤ 2 α∈Φ + |α|coth(bα(ρ))
.

WV e will lso need to ontrol from elow the vlue of ν t ner the wllsF his will e hieved y the following tehnil propositionF por now we nnot ontrol ν t uniformly lose to the wllsD ut we will s soon s we ontrol m t F e will use twie the propositionD (rst to otin lower ound on m t D then to ensure e -νt is su0iently smll ner the wllsF emrk lso tht this proposition n e seen s preise sttement of wht we lled the properness of ν t ner the wllsF Proposition 6.13. For any M > 0, there exists a constant b > 0 independent of t such that for any x ∈ a + satisfying α(x) < bα(ρ) for some root α ∈ Φ + dening a wall of a + , we have

ν t (x) ≥ m t + M.
ell tht the roots de(ning the wlls re lso the simple roots of Φ + F Proof. vet x ∈ a + e suh tht α(x) < b 1 α(ρ) for some simple root α ∈ Φ + F gonsider the ry {x + sρ, s ≥ 0} strting from xF st meets the oundry ∂(b 1 ρ + a + ) of b 1 ρ + a + t unique point y = x + s 0 ρF purthermore y is in b 1 ρ + α ⊥ for simple root αF e n then write x = y -s 0 ρD nd s 0 stis(es

0 < b 1 -s 0 < bF gonsider α ∈ Φ + simple rootD nd y ∈ (b 1 ρ + α ⊥ ) ∩ ∂(b 1 ρ + a +
)F e will show tht there exists onstnt b > 0 independent of t suh tht ν t (y -sρ) ≥ m t + M for ll s suh tht 0 < b 1 -s < bD nd tht this b n e hosen independent of y nd αF his is enough to prove the proposition euse ny x s in the sttement is of the form ove for some αD y nd s s shown t the eginningF gonsider the funtion g(s) = ν t (y-sρ) on [0, b 1 [F e hve g(0) = ν t (y) ≥ m t y de(nition of m t F e onsider now the derivtive of gF ememer tht the derivtives of w t re uniformly oundedD y dD in solute vlue y roposition TFTF hen

g (s) ≥ -d + 2 β∈Φ + β(ρ)coth(β(y -sρ)).
ine ll the terms in the sum re positiveD we hve in prtiulr g (s) ≥ -d + 2α(ρ)coth(α(y -sρ)).

prom the ssumptionsD we ompute

α(y -sρ) = b 1 α(ρ) -sα(ρ) = (b 1 -s)α(ρ).
yserve tht the positive funtion coth is not integrle ner 0 + D so Proof. ell tht it is enough to otin uniform upper ound on u t -u ref whih is funtion de(ned on aF e hveD y roposition TFT with x 0 = x t D tht

u t (x) ≤ v(x -x t ) + u t (x t )
where v is the support funtion of 2P F sing the other inequlity for u ref we hve

u ref (x) ≥ v(x) + C 1 ≥ v(x -x t ) + C 1 -d|x t |.
gomining these two gives

(u t -u ref )(x) ≤ v(x -x t ) + u t (x t ) -v(x -x t ) -C 1 + d|x t | ≤ u t (x t ) -C 1 + d|x t | ≤ u t (x t ) -C 1 + dC x so we just hve to ontrol u t (x t )F e hve |m t | = |ν t (x t )| ≤ C m D i.e |tu t (x t ) + (1 -t)u ref (x t ) + j(x t )| ≤ C m . xow we hveX ! t ≥ t 0 > 0D ! |j(x t )| ≤ C 2 for some onstnt C 2 euse x t ∈ b 1 ρ + a + D ! nd u ref (x t ) ≤ sup{u ref (y); y ∈ B(0, C x )} =: C 3 F o u t (x t ) ≤ C m + C 2 + C 3 t 0 .
pinlly we hve proved the uniform upper ound

(u t -u ref )(x) ≤ C 4 := C m + C 2 + C 3 t 0 -C 1 + dC x .

Estimates on |m t |

e onsider the set

A t := {x ∈ a + ; m t ≤ ν t (x) ≤ m t + 1} ⊂ a + .
e will otin upper nd lower ound for the volume of A t F he upper ound will depend on m t F he key is to otin n upper ound tht is smll enough to give informtionD nmely its logrithm hs to e stritly dominted y m t F sn the ourse of the proof we will use the following property of A t F IHH Proposition 6.15. The set A t is a bounded and convex set.

Proof. ine m t is the minimum of ν t D A t is sulevel set of ν t whih is onvexD so A t is onvexF purthermoreD y the properness of ν t D A t is ounded setF Lemma 6.16. There is an upper bound on the volume of A t :

Vol(A t ) ≤ Ce mt/2
where the constant C > 0 does not depend on t ≥ t 0 .

Proof. pritz tohn proved in tohRVD heorem sss tht for ny onvex nd ounded suset A of R r D there exists n ellipsoid E suh tht

1 r E ⊂ A ⊂ E
where 1 r E is the diltion of E of ftor 1 r entered t the enter of the ellipsoid EF fy roposition TFIS we n (nd suh n ellipsoid E t for A t F vet T e liner trnsformtionD of determinnt oneD suh tht T (E) = B(y, δ) is llF e will otin n upper ound on δD thus getting n upper ound for the volume of Vol(A t ) euse

Vol(A t ) ≤ Vol(E) = Vol(T (E)) = Cδ r .
vet ν t e the funtion de(ned y ν t (x) = ν t (T -1 (x))F e wnt to use omprison priniple on B(y, δ r )F por tht we (rst show tht MA R (ν t )(x) ≥ Ce -mt on T (A t )F his is equivlent to showing tht MA R (ν t )(x) ≥ Ce -mt on A t F pirst remrk tht sine the ressin Hess R ν t of ν t stis(esX

Hess R ν t = tHess R u t + (1 -t)Hess R u ref + Hess R j, we hve det(Hess R ν t ) ≥ det(tHess R u t ), iFeF MA R (ν t )(x) ≥ t r MA R (u t )(x).
sing roposition TFS we dedue tht

MA R (ν t )(x) ≥ t r J(x)e -wt(x) α∈Φ + 1 α(∇u t (x)) 2 ≥ t r e -νt(x) α∈Φ + 1 α(∇u t (x)) 2 .
e tret the ftors seprtelyX ! e hve t ≥ t 0 > 0 for t 0 de(ned in xottion TFQF ! fy de(nition of A t D we hve e -νt(x) ≥ e -mt-1 on A t F IHI ! por ny x ∈ aD we hve ∇u t (x) ∈ 2P D so for ny α ∈ Φ + D α(∇u t (x)) is ounded ove independently of tF his implies tht

α∈Φ + 1 α(∇u t (x)) 2 ≥ c
for some positive onstnt cF sn onlusionD we indeed hve n inequlity MA R (ν t )(x) ≥ Ce -mt on A t D with C positive onstnt independent of t ≥ t 0 F xow we use the omprison priniple on B(y, δ r ) for rel wongeEempère equtionsX let g e the uxiliry funtion de(ned y

g(x) = C 1/r e -mt/r (|x -y| 2 - δ 2 r 2 ) + m t + 1. hen we hve ! g(x) = m t + 1 ≥ ν t (x) for x ∈ ∂B(y, δ r )D nd ! MA R (g)(x) = Ce -mt ≤ MA R (ν t )(x) on B(y, δ r )F o the omprison priniple gives tht ν t (x) ≤ g(x) on B(y, δ r )F sn prtiulrD we hve m t ≤ ν t (T -1 (y)) ≤ ν t (y) ≤ g(y)
≤ C 1/r e -mt/r (-δ 2 r 2 ) + m t + 1. e dedue from tht the following upper ound for δX δ ≤ 1 C 1/r re mt/2r . utting everything togetherD we otin Vol(A t ) ≤ Vol(B(y, δ) ≤ C e mt/2 . e turn now to lower ound on Vol(A t )F Lemma 6.17. There exists a constant c > 0 independent of t such that

Vol(A t ) ≥ c.
Proof. here exists onstnt b 2 independent of t suh tht 0 < b 2 < b 1 nd

A t ⊂ b 2 ρ + a + .
his is orollry of roposition TFIQD tking b 2 orresponding to M = 1F sndeedD y vemm TFIP nd roposition TFTD on b 2 ρ + a + D |∇(ν t )| is ounded independently of tD sy y M F hen it is ler tht the ll B(x t , M ) is ontined in A t F o Vol(A t ) ≥ c for some c > 0 independent of tF IHP Proposition 6.18. The following integral is independent of t:

a + e -νt(x) dx = V = 2P + α∈Φ + α(p) 2 dp.
Proof. epplying roposition RFPV with the mple line undle -K X D we hveD for some onstnt C depending only on GD nd for ny onvex potentil u of smooth K × KEinvrint positively urved hermitin metri on

-K X D deg(-K X ) = C a + α∈Φ + (α(∇u(x))) 2 MA R (u)(x)dx = C 2P + α∈Φ + α(p) 2 dp
e pply this to the onvex potentil u t D whih y roposition TFS stis(es

e -νt(x) = α∈Φ + (α(∇u(x))) 2 MA R (u)(x)
nd otin the sttementD with V = deg(-K X )/CF e n now prove the min result of this susetionF Proposition 6.19. There exists a constant C independent of t, such that

|m t | ≤ C.
There exist a constant κ > 0 and a constant C, both independent of t, such that for x ∈ a + ,

ν t (x) ≥ κ|x -x t | -C.
Proof. pollowing here honldson honHV rther thn ng nd huD we write ∂ν ∂ξ e -ν dx = 0.

fefore we get to the proofD let us remrk tht the funtion onsidered is integrleF wore generllyD we n remrk (rst tht for ny potentil u 0 D nd ny vetor ξD the funtion ∂u0 ∂ξ e -ν is integrle on a + F his is the se euse ∇u 0 ∈ 2P D nd e -ν ≤ Ce -(v-4ρ)+C @y roposition TFT nd vemm TFWA is oviously integrleF eondlyD we hve to show tht the funtion ∂j ∂ξ e -ν = ∂j ∂ξ Je -u is integrleF rite

∂j ∂ξ (x)J(x) = -2 α∈Φ + α(ξ)coth(α(x)) β∈Φ + sinh 2 (β(x)) = -2 α∈Φ + α(ξ)cosh(α(x))sinh(α(x)) β =α sinh 2 (β(x))
hen y omputtion similr to vemm TFWD we hve

|e 4ρ ∂j ∂ξ (x)J(x)| ≤ Ce 4ρ ,
so gin ∂j ∂ξ e -ν is integrleF Proof. ghoose sis (e i ) i=1..s of the semisimple prt a ss whih generte the eyl hmer s oneD nd sis (f j ) j=1..r-s of the entrl prt a z F gonsider the sets

Q( , M ) :=    i x i e i + j y j f j ; ∀i ≤ x i ≤ M, ∀j -M ≤ y j ≤ M    for 0 ≤ < M F vet S 1 ( , M ) = { i x i e i + j y j f j ∈ Q( , M ); ∃i x i = } nd S 2 ( , M ) = { i x i e i + j y j f j ∈ Q( , M ); ∃i x i = M or ∃j |y j | = M }F emrk tht ∂Q( , M ) = S 1 ( , M ) ∪ S 2 ( , M )F
emrk tht ∂ν ∂ξ e -ν = -∂e -ν ∂ξ F hen y the divergene formul pplied to e -ν we hve for > 0D

(x) ≤ v(ξ)D so ∂u ref ∂ξ e -νt V ≤ v(ξ).
yn the other hndD we n use here lso the ft tht ∂j ∂ξ ≤ -4ρ(ξ) for ξ ∈ a + to get

1 t -1 ∂j ∂ξ e -νt V ≥ 1 t -1 (-4ρ(ξ)). e thus hve v(ξ) ≥ t t -1 ∂u t ∂ξ e -νt V + 1 t -1 ∂j ∂ξ e -νt V ≥ t t -1 ξ, bar DH (2P + ) - 1 t -1 4ρ(ξ) ≥ ξ, 4ρ + t t -1 (bar DH (2P + ) -4ρ) he ft tht this is true ∀ξ ∈ a + mensD sine v is the support funtion of 2P D 4ρ + t t -1 (bar DH (2P + ) -4ρ) ∈ 2P + + (-Ξ).
6.5 Absence of estimates on |x t | e will ssume now tht there re no uählerEiinstein metris on XF e will denote y t ∞ := R(X) the gretest ii lower oundF yur ssumption implies tht |x t | is unounded s t tends to t ∞ F sndeed if it ws not the seD then we would hve estimtes on |x t | nd so y ll wht we hve done hereD there would e solution t time t ∞ nd y openness for times greter thn t ∞ F his is ontrditionF e n (nd sequene t i suh tht t i → t ∞ nd |x ti | → ∞F vet ξ t = xt |xt| ∈ a + F p to tking susequeneD we n lso ssume tht ξ t dmits limit ξ ∞ ∈ a + s t i → t ∞ F e will onsider n integrl equlity involving ν t nd onsider the limit s t i → t ∞ F he integrl equlity follows from the vnishing result lredy proved @roposition TFPHA pplied to w t X Lemma 6.23. We have Proof. ell tht

e -νt = α∈Φ + (α(∇(u t ))) 2 MA R (u t ).
sing the hnge of vriles given y ∇(u t )D we get he result follows y oserving tht V = 2P + α∈Φ + (α(p)) 2 dp y the sme hnge of vrilesD nd tht

2P + p α∈Φ + (α(p)) 2 dp = bar DH (2P + )V. sn prtiulrD the limit s t i → t ∞ is ξ ∞ , bar DH (2P + ) V.
por the other terms we need more work to ompute the limitsF e will prove the two following propositionsF Proposition 6.25. We have

lim ti→t∞ a + ∂j ∂ξ t e -νt = -4ρ(ξ ∞ )V.

IHW

Proposition 6.26. We have

lim ti→t∞ a + ∂u ref ∂ξ t e -νt = v(ξ ∞ )V.
e will (rst (nd domin D( ) of the form B(x t , δ)∩(bρ+a + ) where e -νt dx puts ll the mss up to 2 > 0F hen we write B(x t , δ)D we in generl men B(x t , δ) ∩ a + F Lemma 6.27. For any > 0, there exists a constant δ = δ( ) independent of t such that a + \B(xt,δ) e -νt dx < and e -κδ+C σ r δ r-1 < , where σ r is the area of a sphere of radius 1 in R r .

Proof. ell from roposition TFIW tht ν t (x) ≥ κ|x-x t |-CD for some κ > 0, C independent of tF yserve tht the funtion e -κ|x-xt|+C is well de(ned on aD positive nd integrleF o there exists δ > 0 suh tht a\B(xt,δ)

e -κ|x-xt|+C dx < .

fut then we lso hve

a + \B(xt,δ) e -νt dx ≤ a + \B(xt,δ)
e -κ|x-xt|+C < .

yf ourseD sine e -κy+C dereses exponentilly with respet to yD we n inrese δ so s to hve the seond onditionX e -κδ+C σ r δ r-1 < . Lemma 6.28. For any > 0, let δ = δ( ) be given by Lemma 6.27. There exists a constant b = b( ) > 0 such that if we denote by D = D( ) the domain B(x t , δ) ∩ (bρ + a + ), we have where dσ is the area measure of ∂D, which is piecewise smooth.

IIH

Proof. rere we wnt to use the roposition TFIQF xow tht we know tht m t is uniformly oundedD we n hoose M nd the orresponding b so thtX ∀x ∈ a + \ (bρ + a + ), e -νt(x) < max σ r δ r-1 , δ r ω r where ω r is the volume of the ll of rdius I in R r F vet us prove tht e -νt < σrδ r-1 on ∂DF e point x ∈ ∂D is either on the sphere of rdius δ entered t x t D or on ∂(bρ + a + )F sn the (rst seD we hveD y roposition TFIWD e -νt(x) ≤ e -κ|x-xt|+C ≤ e -κδ+C < σ r δ r-1 y the seond onsequene of vemm TFPUF sn the seond seD x ∈ ∂(bρ + a + )D so y the hoie of b oveD using the (rst term in the mximumD we hve e -νt(x) < σ r δ r-1 .

yviously the volume of ∂D is ≤ σ r δ r-1 D so ∂D e -νt dσ < ∂D σ r δ r-1 dσ < .

por the other prtD we use the ft tht e -νt(x) < δ r ωr on B(x t , δ) \ D ⊂ a + \ (bρ + a + )D whih implies tht

B(xt,δ)\D e -νt dx < B(xt,δ)\D δ r ω r dx <
using the ft tht the volume of B(x t , δ) \ D is ≤ δ r ω r F Lemma 6.29. Let > 0 and D = D( ) be the domain given by Lemma 6.28.

We have

e -νt + 4ρ(ξ ∞ )V ≤ D ∂j ∂ξ t e -νt + 4ρ(ξ t )V + |4ρ(ξ ∞ -ξ t )V |
he seond term tends to zero so there exists n i 0 suh tht for ll i

≥ i 0 D |4ρ(ξ ∞ -ξ t )V | ≤ 3
(x) = -2 α∈Φ + α(ξ t )coth(α(x)) ≤ -2 α∈Φ + α(ξ t ) = -4ρ(ξ t ) ell tht D(θ) ⊂ b(θ)ρ + a + for some b(θ) > 0D nd more preisely tht D(θ) = B(x t , δ(θ)) ∩ (b(θ)ρ + a + )F ghoose b 0 > 0 suh tht B(b 0 ρ, δ(θ)) ⊂ b(θ)ρ + a + F e n write ξ t = x t |x t | = x t -b 0 ρ |x t | + b 0 ρ |x t | = |x t -b 0 ρ| |x t | ξ 1 + |b 0 ρ| |x t | ξ 2 where ξ 1 = xt-b0ρ |xt-b0ρ| nd ξ 2 = b0ρ |b0ρ| F st gives ∂ ∂ξ t = |x t -b 0 ρ| |x t | ∂ ∂ξ 1 + |b 0 ρ| |x t | ∂ ∂ξ 2 .
IIQ vet x = x t + y ∈ DD we onsider the restrition of j to the line strting from b 0 ρ + y nd of diretion ξ 1 D whih ontins xF fy onvexityD we hve

∂j ∂ξ 1 (x) ≥ j(x) -j(y + b 0 ρ) |x t -b 0 ρ| . ell from vemm TFW tht j(x) ≥ -2 α∈Φ + α(x) + C = -4ρ(x) + C
on a + D for some onstnt CF epplying this gives

∂j ∂ξ 1 (x) ≥ -4ρ(x) + C -j(y + b 0 ρ) |x t -b 0 ρ| . xow going k to ∂j ∂ξt (x)D we hve ∂j ∂ξ t (x) = |x t -b 0 ρ| |x t | ∂j ∂ξ 1 (x) + |b 0 ρ| |x t | ∂j ∂ξ 2 (x) ≥ |x t -b 0 ρ| |x t | -4ρ(x) + C -j(y + b 0 ρ) |x t -b 0 ρ| + |b 0 ρ| |x t | ∂j ∂ξ 2 (x) ≥ -4ρ(x t + y) + C -j(y + b 0 ρ) |x t | + |b 0 ρ| |x t | ∂j ∂ξ 2 (x) so 0 ≥ ∂j ∂ξ t (x) + 4ρ x t |x t | ≥ -4ρ(y) + C -j(y + b 0 ρ) + |b 0 ρ| ∂j ∂ξ2 (x) |x t | xow y ∈ B(b 0 ρ, δ(θ)) is oundedD j is ounded on b(θ)ρ + a + D
(x) ≥ u ref (x) -u ref (x -x t ) |x t | ≥ v(x) + C |x t |
for some onstnt C independent of x in B(x t , δ)D y roposition TFT nd euse u ref is ounded on B(0, δ)F hen we n write

∂u ref ∂ξ t (x) ≥ v(ξ t + x -x t |x t | ) + C |x t | ≥ v(ξ t ) + C |x t | he lst step holds euse v is vipshitzF por i > i 0 for some i 0 D we thus hveD for x ∈ B(x t , δ)D ∂u ref ∂ξ t (x) -v(ξ t ) < 3 V.
sntegrting on the ll gives

B(xt,δ) ∂u ref ∂ξ t e -νt - B(xt,δ) v(ξ t )e -νt < 3 .
epplying vemm TFPU gin gives

B(xt,δ) v(ξ t )e -νt - a + v(ξ t )e -νt < dθ, with a + v(ξ t )e -νt = v(ξ t )V. IIS pinllyD sine ξ t onverges to ξ ∞ D there exists i 1 suh tht for i > i 1 D |v(ξ t )V -v(ξ ∞ )V | < 3 .
e hve proved tht for i > i 0 , i 1 D we hve

a + ∂u ref ∂ξ t e -νt -v(ξ ∞ )V < 2 3 + 2dθ =

Conclusion

e n now prove tht our ondition is su0ient for the existene of uählerEiinstein metriF Theorem 6.30. If bar DH (2P + ) ∈ 4ρ + Ξ, then X admits a Kähler-Einstein metric.

Proof. essume (rst tht X dmits no uählerEiinstein metriF hen s t i → t ∞ we hve the equlity

t ∞ (bar DH (2P + ) -4ρ)(ξ ∞ )V = (t ∞ -1)(v -4ρ)(ξ ∞ )V orD dividing y V D t ∞ (bar DH (2P + ) -4ρ)(ξ ∞ ) = (t ∞ -1)(v -4ρ)(ξ ∞ ). sn prtiulrD sine v is the support funtion of 2P nd 2ρ ∈ Int(P )D nd t ∞ ≤ 1D we hve t ∞ (bar DH (2P + ) -4ρ)(ξ ∞ ) ≤ 0.
essume tht bar DH (2P + ) ∈ 4ρ + ΞF hen y the de(nition of ΞD the only possiility is tht ξ

∞ ∈ a t nd (bar DH (2P + ) -4ρ)(ξ ∞ ) = 0F
o prove tht this is impossile we hve to give slightly di'erent proofF st is simpler nd in ft the sme s in the tori seF e onsider the vnishing

a + ∂ν t ∂ξ ∞ e -νt dx = 0.
he di'erene with wht we hve done efore is tht we (x ξ ∞ insted of onsidering ξ t F ine ξ ∞ ∈ a t D we hve ∂j ∂ξ∞ = 0 nd so we dedue from the vnishing of the integrl the following equlityD vlid for t

< t ∞ F t a + ∂u t ∂ξ ∞ e -νt dx = (t -1) a + ∂u ref ∂ξ ∞ e -νt dx
he left hnd side term is zero euse we ssumed

0 = (bar DH (2P + ) -4ρ)(ξ ∞ ) = bar DH (2P + )(ξ ∞ ). IIT e thus hveD for ll t < t ∞ D a + ∂u ref ∂ξ ∞ e -νt dx = 0. his is ontrditionX let m := min{v(ξ); ξ ∈ a, |ξ| = 1} > 0F por ny δ > 0 (xedD there exists n > 0 suh tht if t ∞ -< t < t ∞ D ∂u ref ∂ξ∞ ≥ m/2 on B(x t , δ)F his is euse |x t | goes to ∞ nd u ref is symptoti to vF ghoose now δ = δ(m/4) given y vemm TFPUD then for t lose to t ∞ D we otin a + ∂u ref ∂ξ ∞ e -νt dx ≥ m/4 > 0.
gomined with the ostrution proved erlierD it gives our neessry nd su0ient ondition for the existene of uählerEiinstein metriF essume now t ∞ < 1D then we n write

t ∞ t ∞ -1 (bar DH (2P + ) -4ρ)(ξ ∞ ) = (v -4ρ)(ξ ∞ ), or 4ρ + t ∞ 1 -t ∞ (-bar DH (2P + ) + 4ρ) (ξ ∞ ) = v(ξ ∞ ). he funtion t → t 1-t is stritly inresing nd its imge is [0, ∞[D nd 4ρ ∈ Int(2P )F fesidesD sine v is the support funtion of 2P D the vlue v(ξ ∞ ) is ttined y m, ξ ∞ if
X is R(X) = sup t; t 1 -t (-bar DH (2P + ) + 4ρ) + 4ρ ∈ 2P + + -Ξ .
sf this se hppensD R(X) = 1 with no uählerEiinstein metris mens bar DH (P + ) ∈ ∂(4ρ + Ξ)F

Examples

(P + ) = 2 0 x 3 dx 2 0 x 2 dx -1 = 3/2 > 1
gonsider now the qudri whih is the wonderful ompti(tion of SL 2 (C)F hen with the sme identi(tionsD

P + = [0, 3 2 ]D so bar DH (P + ) = 3/2 0 x 3 dx 3/2 0 x 2 dx -1 = 9/8 > 1
6.6.2 Rank two examples e omputed numerilly @with silA the ryenter with respet to the huistermtErekmn mesure of P + for some ompti(tions of irreduile rnk two groupsF por ll ses we omputed the oordintes in the sis given y simple rootsD nd hose reliztion of the root systems in the euliden plne to determine the ryentersF he ryenter n lso e omputed extlyD either y hnd or with nother omputer progrmF e omputed the ext vlues for the wonderful ompti(tion of PSL 3 (C) nd for the non uählerE iinstein exmpleF vet us give some detils out the three rnk two root systemsD nd the results we otinedF Root system A 2 por the root system Φ = A 2 D denote y α 1 nd α 2 the simple rootsF here re three positive rootsX α 1 D α 2 nd α 1 + α 2 F ghoosing to relize A 2 in the unit irle in the euliden plneD i.e. tking α

1 = (1, 0) nd α 2 = (-1/2, √ 3/2)D we hveD for p = xα 1 + yα 2 D α∈Φ + (α(p)) 2 = (x -y/2) 2 (-x/2 + y) 2 (x/2 + y/2) 2 .
e omputed the ryenter y omputing (rst the integrls of 1D x nd y on P + with respet to the mesure with potentil α∈Φ + (α(p)) 2 ginst the IIV pigure TFIX fryenter for PSL 3 (C) wond veesgue mesureF e the funtion intPd in silD tking for tringultion the tringles with verties the origin nd other verties of P + F e give the oordintes of the ryenters here with preision estimted y the progrm to e lower thn or of the order of the lst digit we write hereF emrk tht the oordintes of 2ρ in the sis given y the simple roots re (2, 2) for A 2 F por the wonderful ompti(tion of PSL 3 (C) we otined pigure TFI gives representtion of P + + (-Ξ) in the plne with oordintes in α 1 nd α 2 s sisss nd ordintesF he one strting from the origin is the positive eyl hmer @remrk tht in this representtion the uilling form does not gree with the euliden produtAF he point 2ρ is represented s little squre nd the zone in whih the ryenter hs to e to ensure the existene of uählerEiinstein metri is the intersetion of the one strting from 2ρ nd P + F he ryenter is represented y the rossF gonsider now the low up of the wonderful ompti(tion of PSL 3 (C) t the losed oritF e otined the oordintes (2.2169041 . . . , 2.2169041 . . .) for the ryenter @see pigure TFPAD so this mnifold gin dmits uählerE iinstein metriF IIW pigure TFPX fryenter for Bl PSL 3 (C)

wond Root system B 2 o relize Φ = B 2 in the euliden plneD one n hoose α 1 = (1, 0) nd α 2 = (-1, 1) s simple rootsF he other positive roots re α 1 + α 2 nd 2α 1 + α 2 F e then omputeD for p = xα 1 + yα 2 D α∈Φ + (α(p)) 2 = x 2 y 2 (x -y) 2 (-x + 2y) 2 .
e n write here 2ρ = 4α 1 + 3α 2 F e hve desried in ghpter R three exmples of toroidl pno omptE i(tions of groups of type B 2 F he (rst is the wonderful ompti(tion of SO 5 (C)F he oordintes of bar DH (P + ) in the sis given y the simple roots re @see pigure Proof. he hnge of vriles for ones S σ in the proof of roposition eFIQ gives @y gvIID heorem QFQFRA n equivrint isomorphism etween X σ nd C n D whih we denote gin y S σ F eny linerized line undle on C n is trivilD so L dmits glol equivrint triviliztion t on X σ F emrk thtD t the (xed point z σ D we hve g • t(z σ ) = -v σ (t(z σ )) y de(nition of v σ F estriting to T nd rememering tht s is n invrint triviliztion of L on T D we dedue tht up to renormliztion y onstntD t(z) = v σ (z)s(z) on T F e n now look t the potentil ψ of h with respet to the triviliztion tD nd remrk thtD on T D nd if ϕ denotes the potentil of h with respet to s on T D we hve ψ

(z) = -v σ , ln |z| + ϕ(z)F vet y ∈ N R F sing gin the isomorphism T c ×N R T D we onsider T c ×(y - σ) s suset of T D nd denote y C y the losure of this set in X σ F ih set C y is neighorhood of z σ in X σ D nd
they form sis of neighorhoodsF yserve tht the olletion of the trnsltes of -σ over N R nd so the orresponding sets over X σ F wore preiselyD for ny point z in X σ D there is trnslte of -σ whih orresponds to neighorhood of zF e onsider (rst the omplex singulrity exponent of h t z σ F uppose c > 0 is suh tht e -2cψ is integrle in neighorhood of z σ F hen it is integrle in neighorhood C y F e hve (rst thtD e -2c(f (x)+ -vσ,x ) e 2 i xi dx.

ine i x i is equl to S * σ (1, . . . , 1), x D we onlude y using roposiE tion eFIQ tht the omplex singulrity exponent c zσ (h) is the supremum of the c > 0 suh tht 0

∈ Int(N -σ (2c(f + -v σ , • ) -2 S * σ (1, . . . , 1), • ))F o otin simpler onditionD remrk tht for ny funtion g nd positive slr λD N -σ (λg) = λN -σ (g)D nd tht if g 1 nd g 2 re two onvex funtions then N -σ (g 1 + g 2 ) is the winkowski sum of N -σ (g 1 ) nd N -σ (g 2 )F o we get c zσ (h) = sup{c > 0|cv σ ∈ Int(N -σ (cf )) -S * σ (1, . . . , 1
)}. purthermoreD for ny c < c zσ (h)D the roposition eFIQ shows tht e -2cψ is integrle on every C y for y ∈ N R F yserve now tht for ny point z ∈ X σ D there exists C y ontining zF o for ny point z ∈ X σ D c z (h) ≥ c zσ (h)F his onludes the proof of the propositionF A.2.5 lct on a compact smooth toric manifold Theorem A.15. Let X Σ be a smooth compact toric manifold, L a linearized line bundle on X Σ and h a T c -invariant non-negatively curved metric on L. Then lct(h) = sup{c > 0|cP L ⊂ Int(cP h + P -K X Σ )}.

Proof. he ompt mnifold X Σ is overed y the 0ne tori mnifolds X σ D for σ ∈ Σ(n)F fy de(nition of the log nonil thresholdD lct(h) = min σ∈Σ(n) lct(h| Zσ ).

enother wy to sy this is tht lct(h) is the sup of c > 0 suh tht c ≤ lct(h| Xσ ) for ll σ ∈ Σ(n)F xow this ondition mensD y roposition eFIRD tht for ll σ

∈ Σ(n)D cv σ ∈ Int(N -σ (cf h + -S * σ (1, . . . , 1), • ).
fy roposition eFIID this is equivlent to the ondition tht for ll σ ∈ Σ(n)D

cv σ + σ ∨ ⊂ Int(N -σ (cf h + -S * σ (1, . . . , 1), • ).
his is further equivlent to the ondition tht for ll σ sndeedD the support funtion of the ntinonil undle isD from ixmple eFSD f -K X Σ (x) = -S * σ (1, . . . , 1), x .

∈ Σ(n)D σ∈Σ(n) (cv σ + σ ∨ ) ⊂ Int(N -σ (cf h + -S * σ (1, . . . , 1), • ). IQI ell from roposition eFIP tht σ∈Σ(n) (v σ + σ ∨ ) = N (f L ) = P L D
A.3 Alpha-invariant he liner systems in multiple of L give singulr metris on LD tht we will ll lgeri metrisD in the following wyF vet δ 1 , . . . , δ r ∈ H 0 (X, mL) e linerly independent setionsD nd denote y ∆ the liner system generted y theseF hen it de(nes n lgeri metri h ∆/m on L y settingD in ny triviliztionD

||ξ|| 2 h ∆/m =
|ξ| 2 ( |δ j (z)| 2 ) 1/m , for ny ξ ∈ L z F he lol potentil ϕ ∆/m (z) = 1 2m ln |δ j (z)| 2 is pshF sf ∆ is one dimensionlD generted y δD we denote y h δ/m the orresponding metriF ell the following result of hemillyD relting the αEinvrint with log nonil thresholds of lgeri metrisX Theorem A.17. [CS08, Appendix A] Let K be a compact group, let X be a compact complex K-variety and L a big and nef K-linearized line bundle on X. Then α K (L) = inf m∈N * inf ∆⊂H 0 (X,mL), ∆ K =∆ lct(h ∆/m ).

yne n slightly improve this resultD nd give the following sttementD whih is only given in the se of trivil group K y hemillyF Corollary A.18. Let K be a compact group, let X be a compact complex Kvariety and L a big and nef K-linearized line bundle on X. Then α K (L) = inf m∈N * inf ∆∈Irr(H 0 (X,mL)) lct(h ∆/m ),

where Irr(H 0 (X, mL)) denotes the set of all irreducible K-subrepresentations of H 0 (X, mL).

IQP

Proof. vet ∆ e KEinvrint suspe of H 0 (X, mL)D then ∆ = ∆ 1 ⊕ • • • ⊕ ∆ s with ∆ i irreduile suspesF por ll iD one n hoose sis δ i j of ∆ i F ogether they form sis of ∆ nd we n otin the metri h ∆ this wyF sn prtiulrD ϕ ∆/m (z) = 1 2m ln i j |δ i j (z)| 2 F ine the logrithm is inE resing we n write ϕ ∆/m (z) ≥ 1 2m ln |δ 1 j (z)| 2 = ϕ ∆1/m (z).

his impliesD y elementry properties of the omplex singulrity exponentD huHID IFR tht lct(h ∆/m ) ≥ lct(h ∆1/m )F e onlude tht the log nonil threshold of metri ssoited to KEinvrint liner system is greter thn the log nonil threshold of t lest one metri ssoited to n irreduile liner systemD so it is enough to onsider only theseF ).

e hve thus shown tht it is enough to ompute the log nonil thresholds of lgeri metris ssoited to one dimensionl GEinvrint suliner systems of multiples of LF e use heorem eFIS to onludeF sndeed if p ∈ mP L genertes one dimensionl GEinvrint suliner system in H 0 (X Σ , mL)D nd f p/m denotes the onvex funtion ssoited to the potentil of the orresponding lgeri metri h p/m D we hve N (f p/m ) = {p/m}F epplying heorem eFIS gives lct(h p/m ) = sup{c > 0|cP L ⊂ Int(cp/m + P -K X Σ )}. pinllyD oserve tht s p nd m vryD they desrie the set P G L (Q) of GE invrint points of P L with rtionl oordintesF Remark A.20. yne n lso proveD without the use of gorollry eFIVD tht we n onsider only metris orresponding to points of P L @not neessrily with rtionl oordintesAD y onsidering the expression of the log nonil threshold of ny metriF sndeedD if f is onvex funtion on N R D orresponding to metri h on LD nd p is point in N (f )D then the metri h p ssoited to the onvex funtion x → p, x is lso nonEnegtively urved metri on LD nd lct(h p ) ≤ lct(h)F gonsider the hlfEline strting from p nd ontining the originF st intersets ∂P t w p F henote y r its intersetion with ∂(p + P )F hen it is esy to see tht the log nonil threshold of h p is equl to the quotient of the distne etween p nd r y the distne etween p nd w p F he trnsltion sending 0 to p lso sends w p to rD so |r -p| = |w p |F he result followsF Remark A.23. sf P h = {0} then lct(h) = 1F Example A.24. gonsider the se P h = {b}D where b is the ryenter of the polytope P L F hen lct(h) is equl to the gretest lower ound for ii urvture R(X)D introdued y zékelyhidi zéIID nd omputed for tori mnifolds y vi viIIF prom this formul we reover the previous results of ong nd ghel9tsovE hrmovF Theorem A.25. [Son05] [CS08,Lemma 6.1] Let X be a smooth Fano toric manifold, and G be a subgroup of W . Then if S G is empty,

α K G (X) = 1; else, α K G (X) = 1 1+max p∈S G |p| |wp | ≤ 1 2 .
Proof. fy heorem eFIWD it is enough to onsider only the @rtionlA GEinvrint points of P F he (rst se follows immeditely using emrk eFPQF sn the seond seD we otin the formul using roposition eFPPF sndeedD it is enough to onsider points p in S G euse if q = 0 is not in ∂P D nd p is the intersetion of ∂P with the hlf line strting from the origin nd going through qD then lct(h q ) ≥ lct(h p )F purthermoreD max p∈S G |p| |wp| ≥ 1 euse otherwise if p ws suh point t whih this mximum ws ttined nd it ws < 1 then we would hve e ompute the αEinvrint of ny linerized line undle on the low up X of P 2 t one point whih we denote X in the followingF sdentify N with Z 2 F he fn of X hs four rysD with genertors u 1 = (1, 0)D u 2 = (1, 1)D u 3 = (0, 1) nd u 4 = (-1, -1)F he group W is isomorphi to Z/2Z nd ts on M R y exhnging the oordintes (x, y) → (y, x)F e de(ne the polytope P (k, l) to e the polytope whose verties re (0, k)D (0, l)D (k, 0) nd (l, 0)D for k, l ∈ N with l > kF st is esy to see tht the polytopes of nef nd ig divisors re the P (k, l)D up to trnsltion y hrterF por exmpleD the polytope of the ntinonil undle is Q := (-1, -1) + P (1, 3)F Proposition A.26. The α-invariant with respect to K W of the nef and big line bundle corresponding to P (k, l) is equal to inf( 1 l-k , 2 l ). Proof. fy heorem eFIWD it is enough to onsider points @with rtionl oordiE ntesA in the intersetion of P (k, l) with the (rst digonlF roweverD one esily remrks tht it is enough to onsider only the point (l/2, l/2)D similrly to the proof of heorem eFPSF e wnt to ompute sup{c > 0|cP (k, l) ⊂ Int(c(l/2, l/2) + Q)}.

his is of ourse equl to sup{c > 0|P (k, l) ⊂ Int((l/2, l/2) + 1 c Q)}.

yserve tht l/2 is the lest positive onstnt b suh tht {(0, l), (l, 0)} ⊂ (l/2, l/2) + bQ.

sf k ≥ l/2D then we hve lso {(0, k), (k, 0)} ⊂ (l/2, l/2) + l/2QD so P (k, l) ⊂ (l/2, l/2) + l/2Q. hus α K W (P (k, l)) = 2/l when k ≥ l/2F por the other seD oserve tht l -k is the lest positive onstnt b suh tht (k/2, k/2) ∈ (l/2, l/2) + bQF sf k ≤ l/2D then we hve lso P (k, l) ⊂ (l/2, l/2) + (l -k)Q.

  et d9utre prtD Ric(ω ref ) = λω ref + i∂∂f où ϕ et f sont deux fontions lisses sur XF v9éqution de uählerEiinstein peut lors s9érire X Ric(ω) = λω Ric(ω ref + i∂∂ϕ) = λ(ω ref + i∂∂ϕ) = Ric(ω ref ) -i∂∂f + λi∂∂ϕ en trvillnt lolementD ve ω ref = i∂∂ϕ ref D nous otenons -∂∂ ln det Hess C (ϕ ref + ϕ) = -∂∂(ln det Hess C (ϕ ref ) + f -λϕ). oujours lolementD el est équivlent à -∂∂ ln det Hess C (ϕ ref + ϕ) det Hess C (ϕ ref ) = -∂∂(f -λϕ). wis le memre de guhe peut s9érire -∂∂ ln ω n ω n ref où n est l dimension de XD e qui fit que les deux expressions sont dé(nies glolementF xous vons don ω n = e f +c-λϕ ω n ref pour une onstnte c @une fontion ∂∂Eexte sur X qui est ompteAF v onstnte c est déterminée pr le volume de ω n D et peut être prise en ompte S dns l fontion f qui n9est pour l9instnt dé(nie qu9à une onstnte dditive prèsF pinlementD nous otenons (ω ref + i∂∂ϕ) n = e f -λϕ ω n ref et ette éqution u dérivées prtielles en φ est en fit équivlente à l9éqution de uählerEiinsteinF

  (ω ref + i∂∂ϕ t ) n = e f -tϕtω n ref où f est dé(nie omme plus tôtD et où les métriques sont normlisées de mnière à voir λ = 1F gel orrespondD u niveu des PEformesD et en notnt ω ref +i∂∂ϕ t pr ω t D à l9éqution

  nd on the other hnd Ric(ω ref ) = λω ref + i∂∂f where ϕ nd f re two smooth funtions on XF he uählerEiinstein eqution n then e rewrittenX Ric(ω) = λω Ric(ω ref + i∂∂ϕ) = λ(ω ref + i∂∂ϕ) = Ric(ω ref ) -i∂∂f + λi∂∂ϕ working lolly with ω ref = i∂∂ϕ ref D we get -∂∂ ln det Hess C (ϕ ref + ϕ) = -∂∂(ln det Hess C (ϕ ref ) + f -λϕ).

  semisimple group is redutiveF sn prtiulrD ll simple omplex lgeri groups re redutive @nd semisimE pleAF sn ft we n otin ll redutive groups from torus nd semisimple onesD s the following proposition showsFProposition 3.4. [Spr98, 7.3.1 and 8.1.6] Let G be a reductive group, then R(G) is the identity component of the center Z(G) of G, and it is a torus, D(G) is a semisimple group, and G is the quotient of D(G) × R(G) by a nite central subgroup. enother hrteriztionD whih explins the nmeD is in terms of represenE ttionsF Proposition 3.5. A group G is reductive if and only if all rational representations of G are reducible, i.e. are direct sums of irreducible representations. he lst hrteriztion will e very useful for usF st gives strong link etween redutive group G nd mximl ompt sugroup K of GF Proposition 3.6. [see [Sch00], Chapter 5, and references therein] Let K be a maximal compact subgroup of G. Then G is reductive if and only if G is isomorphic to the omplexi(tion of K.Example 3.7. he generl liner group GL n (C) is redutive groupD it is the omplexi(tion of the unitry group U (n)F QQ 3.1.2 Lie algebras and Killing form yur referene ook on vie lgers is rumUPF prom now onD G will denote redutive groupD nd K mximl ompt sugroup of GF vet gD respetively kD denote the vie lgers of G nd KF hen g is lso the omplexi(tion of kX g = k ⊗ C. he indued omplex onjugtion on g is Cartan involution for GF hen K is (xed we will denote this involution y θF e vie lger (g, [•, •]) ts on itself through the adjoint vie lger tionX for x, y ∈ gD it is de(ned y ad(x)(y) := [x, y]F Denition 3.8. he Killing form B of g is the iliner symmetri form on g de(ned for x, y ∈ g y B(x, y) = Tr(ad(x)ad(y)).

  relgson proves tht we n hoose generting vetors e α of g α suh tht ! e α D e -α nd [e α , e -α ] = 2 α(hα) h α generte n sl 2 (C) sulgerD ! e -α = -θ(e α )F e just sle the vetors e α y 2 α(hα) to simplify the expressionsD nd thus we hve sis e α of g α suh tht e -α = -θ(e α ) nd [e α , e -α ] = h α F xow k α dmits e α + θ(e α )D ie α -iθ(e α ) s rel sisF his sis n lso e written e α -e -α D ie α + ie -α F e omplete this with ny sis of s nd we get rel sis of ikD thus lso omplex sis of gF RH e n pply this in the redutive se lso y onsidering the semisimple prt [g, g]D then ompleting the sis y hoosing ny rel sis of Z(g) ∩ kF

A

  decomposition in g vet a ∈ a + F vet Exp(ad(a)) e the liner pplition g → g de(ned y Exp(ad(a))(x) = ∞ n=0 ad(a) n (x) n! . ell tht G ts on g through the djoint tion AdD nd tht we hve the generl reltion Exp(ad(a))(x) = Ad(exp(a))(x) for x ∈ gF RS Lemma 3.26. Let l ∈ g and a ∈ a + . Then there exists A ∈ k, B ∈ a and C ∈ Ad(exp(a))( α∈Φ + k α ) such that

  and l denotes 1 α(a) ad(a)(il), then l ∈ k α and the decomposition above for il reads il = -cosh(α(a))l + 1 sinh(α(a)) (Ad(exp(a))(l );

  Proof. vet a ∈ a + F e egin y the two lst pointsF fy de(nition of l D we hveD if l = e α + θ(e α )D l = ad(a)(il)/α(a) = ad(a)(ie α )/α(a) + ad(a)(iθ(e α ))/α(a) = ie α -iθ(e α ) nd if l = ie α -iθ(e α )D l = ad(a)(il)/α(a) = ad(a)(-e α )/α(a) + ad(a)(θ(e α ))/α(a) = -e α -θ(e α ).sn prtiulrD in oth sesD l is in k α F fy linerity this is lso true of l for ny l ∈ k α F o prove the deomposition in the third pointD it su0es to ompute thtD using the de(nition of k α D Exp(ad(a))(l ) = cosh(α(a))l + sinh(α(a))il = Ad(exp(a))(x).

ST 4 . 1

 41 Linearized line bundles on reductive groups pirst we de(ne the generl notion of GElinerized undle on GEvrietyD for G ny vie groupF Denition 4.1. e GElinearization of line undle L on GEvriety X is GEtion on L suh thtX ! the GEtion on L lifts the GEtion on XD nd ! the mp etween the (ers L x nd L g•x de(ned y the tion of g ∈ G is linerF vet now G e onneted redutive group ginF he group G × G ts on G through the tions of G y left nd right trnsltions on itselfX (g 1 , g 2 ) • g = g 1 gg -1 2 F Proposition 4.2. The G × G-linearized line bundles on G are classied by characters of G. Furthermore, any G × G-linearized line bundle on G admits a G × {e} equivariant trivialization. sn prtiulrD if G is semisimpleD ny G × GElinerized line undle on G dmits G × GEequivrint triviliztionF Proof. vet L e G × GElinerized line undle on GF e (rst prove tht it dmits leftEGEequivrint triviliztionF ghoose nonzero element 1 e in the (er L e over the neutrl element e ∈ GF he setion s de(ned y s(g) = (g, e) • 1 e is well de(ned on ll GD euse the left tion is simply trnsitiveD nd non zero everywhere euse eh (g, e) indues liner isomorphism from L e to L g D so s trivilizes LF st is lso lerly G × {e} equivrint y onstrutionF essoite to L the hrter of G de(ned s the hrter of the liner tion of diag(G) ⊂ G × G on L e F gonverselyD given hrter χ of GD we get G × GElinerized undle on G y onsidering the trivil undle G × C together with the tion of

  varieties and lattice polytopes sn this setion we provide some generl results out tori vrietiesD tht either hve found n nlogue in the se of group ompti(tionsD or will e used in their studyF qenerl referenes for tori vrieties inlude gvIID pulWQD ydVVF vet T (C * ) r e torusD denote y M D respetively N D its group of hrE tersD respetively lgeri one prmeter sugroupsF Denition 4.3. e polarized toric variety (X, L) of dimension r is norml projetive T Evriety X with n open dense orit isomorphi to T D equipped with T Elinerized mple line undle LF vet us rell the theorem tht lssi(es suh ojets omintorillyF he preise orrespondene will e progressively explinedF sn the next setion we will see how elexeev nd utzrkov euHSD uilding on the work of elexeev nd frion efHRD efHRD extended this theorem to group ompti(tionsF Theorem 4.4. Polarized toric variety (X, L) are in bijective correspondence with convex, full-dimensional lattice polytopes P in M ⊗ R. fy lttie polytope we men tht the verties of P re in M F sn ftD the underlying tori vriety X is fully determined y the normal fan Σ of the polytope P D de(ned s followsF qiven vertex v ∈ P D onsider the @losedD fullEdimensionlA one C v ⊂ M ⊗ R with vertex 0D generted y the lttie points of -v + P D the trnslte of P y -vF vet σ v denote the dul one to C v D iFeF

  11. e norml irreduile projetive G × GEvriety X is lledG × GEequivariant compactication of G if X dmits n open nd dense orit under G × GD equivrintly isomorphi to G on whih G × G ts y left nd right multiplitionF sf X is G × GEequivrint ompti(tion of GD we will lwys identify G with the open nd dense orit in XF e will more suintly ll X group compactication of GF Remark 4.12. ell tht spherical variety under the group G is GEvriety on whih forel sugroup B of G ts with n open oritF rere we re onsidE ering G × GEvrietiesD so we onsider the B × BEoritsF st su0es to look t the B × BEorits in GF hese re lled the fruht ells nd the fruht deompoE sition shows tht there is n open fruht ellD nmely BB -where B -is the opposite forel sugroupF o group ompti(tions re spheril vrietiesF xow hoose T mximl torus in GF Proposition 4.13. [BK05, Corollary 6.2.14] Let X be an equivariant group embedding of G, then the closure Z of T in X is a normal toric variety.

  4.4.3 Rank one example: P 3 he only djoint semisimple group of rnk one is PGL 2 (C)F sts wonderE ful ompti(tion is the projetive spe P 3 F vet us desrie in detils this exmpleF gonsider P 3 s P(M 2 (C)) y identi(ying two times two mtrix a b c d with the point of C 4 with oordintes (a, b, c, d)F hen PGL 2 (C) is the open set {[a : b : c : d]; ad -bc = 0} ⊂ P 3 . purthermoreD PGL 2 (C) × PGL 2 (C) ts on P(M 2 (C)) with two orits deterE mined y the rnk of the representtivesF he (rst orit is preisely the open set PGL 2 (C) nd the seond the losed set formed y the lsses of rnk one mtriesX {[a : b : c : d]; ad -bc = 0}.

  2 t the three torusE(xed pointsF pigure RFPX onderful ompti(tion of PGL 3 (C) por type B 2 D the djoint group is SO 5 (C) nd the orresponding polytope is represented in pigure RFQF pinlly the polytope of the wonderful ompti(tion of G 2 is represented in pigure RFRF 4.4.5 Wonderful compactications of non adjoint semisimple groups sn rnk oneD the simply onneted group SL 2 (C) dmits wonderful omE pti(tionF his is the qudri in P 4 = P(M 2 (C) ⊕ C) de(ned y the eqution det(A) = t 2 for (A : t) ∈ P(M 2 (C) ⊕ C)F sn higher rnksD qndini nd uzzi qIQ proved tht the only simple non djoint group whih dmits wonderful ompti(tion is the sympleE ti group Sp 2r (C)F por exmpleD the polytope orresponding to the wonderful ompti(tion of Sp 4 (C) is represented in pigure RFSF 4.4.6 Automorphism group wihel frion determined the utomorphism group of the wonderful omE pti(tion of semisimple djoint group in friHUD ixmple PFRFSF he most TT pigure RFQX onderful ompti(tion of SO 5 (C) pigure RFRX onderful ompti(tion of G 2 TU pigure RFSX onderful ompti(tion of Sp 4 (C)

  2n (C) × Sp 2n (C) @re studied in ft the utomorE phism groups of ll wonderful vrietiesAF he imge of Sp 2n (C) × Sp 2n (C) in Aut 0 (X) is furthermore the quotient of Sp 2n (C) × Sp 2n (C) y the enter of Sp 2n (C)D emedded ntidigonllyD nd is semisimpleF 4.5 Further examples of Fano group compactications vet us give the polytopes of some exmples of smooth pno group omptE i(tions tht re not wonderfulF e lssi(tion of suh mnifolds when the TV pigure RFTX xon wonderfulD pno toroidl ompti(tion of PSL 3 (C)

  6.2 Convex potential vet G e redutive omplex groupF vet X e G × GEequivrint smooth ompti(tion of GF vet L e G × GElinerized mple line undle on XF henote y P the ssoited polytopeF e identify G with its open dense orit in XD nd let s 0 e (xed leftE GEequivrint triviliztion of L| G given y roposition RFPF henote y ψ the potentil of h on G with respet to s 0 X ψ(z) := -ln(|s 0 (z)| 2 h ).

y

  his is in ft well de(ned on G euse dg nd φ reF vet u denote the onvex potentil of hD de(ned y u(a) = ϕ(exp(a)) for a KAKEintegrtionD this isD for onstnt C depending only on G nd the hoie of rr mesureD = C a + MA C (ϕ)(exp(a))J(a)da from the expression of the omplex wongeEempère we otin tht this is

  Denition 5.5. vet f e funtion a → RD nd σ onvex one in aF e ll Newton set of f the following set in a * N σ (f ) = {m ∈ a * ; ∃C, ∀x ∈ σ, f (x) -m(x) ≥ C}. por ny funtion f nd ny onvex one σD the xewton set of f is lerly onvexF ell the de(nition of the dul one σ ∨ of σX σ ∨ = {m ∈ a * ; m(x) > 0 ∀x ∈ σ}. he xewton set N σ (f ) is y de(nition stle under ddition of n element of the losure of the opposite of the dul one σ ∨ ⊂ a * F e write this lso N σ (f ) = N σ (f ) + (-σ ∨ ) where the plus sign mens the winkowki sumF Example 5.6. vet f e the 0ne funtion f (x) = m(x) + c where m ∈ a * nd c is onstntF hen N σ (f ) = m + (-σ ∨ )F vet us reord the following elementry properties of xewton setsF Proposition 5.7. Let f and g be two functions on a and c ∈ R.

  b 1 -s)α(ρ))ds = +∞.ogether with the ft tht g is greter thn m t t HD it mens tht for nyM D we n (nd b α > 0 suh tht g(s) ≥ M + m t for b 1 -s ≤ b α Femrk tht none of wht we hve done depends on the hoie of yF purE thermoreD sine there re only (nite numer of roots αD we n hoose b > 0 suh tht b < b α for ll αD nd it onludes the proofF WW 6.2.3 Reduction to estimates on m t and x t Lemma 6.14. Suppose we have uniform estimates |m t | < C m and |x t | < C x for t in some interval I ⊂ [0, 1]. Then there is an uniform upper bound for φ t on I.

  where µ is the exterior norml nd dσ is the surfe reF rite now e -ν = e -u JF his is ontinuous funtion on aD nd it vnishes on the eyl wllsF pixing M D we n thus let tend to HD nd we hve tht e -ν tends uniformly to H on S 1 ( , M )D so S1( ,M ) e -ν ξ, µ dσ tends to HF IHS ξ, µ dσ.xow s we hve seen eforeD we hve e -ν ≤ Ce -(v-4ρ)+C D so e -ν (x) deE reses exponentilly s |x| tends to in(nityF ine the re of S 2 (0, M ) grows polynomillyD this ensures tht S2(0,M ) e -ν ξ, µ dσ tends to zero s M tends to ∞F his ends the proofF vet us pply this to prove our ostrution to the existene of uählerE iinstein metriF Proposition 6.21. Assume there exists a Kähler-Einstein metric on X. Then bar DH (2P + ) ∈ 4ρ + Ξ.Proof. e hve seen tht the uählerEiinstein eqution restrited to the open orit redsX MA R (u)α∈Φ + α(∇u) 2 = e -u J.uppose tht there exists solution uF epplying roposition TFPH to u gives a + ∂ν ∂ξ e -ν dx = 0, so y de(nition of ν = u + jD dx.ine u is solution to the uählerEiinstein equtionD we hve e -ν = e -u J = MA R (u)

  6.6.1 Rank one examplespor the two rnk one exmplesD whih re the wonderful ompti(tions of SL 2 (C) nd of PGL 2 (C)D we know tht there exists uählerEiinstein metriD IIU euse they re homogeneous pno mnifoldsF e n hek tht our is stis(ed in this situtionF vet us (rst del with P 3 whih is the wonderful ompti(tion of PGL 2 (C)F e n identify a with R nd the unique positive root with the multiplition y oneF hen the polytope P is [-2, 2]D P + is [0, 2]D nd 2ρ = 1F he ryenter with respet to the huistermtErekmn mesure is then bar DH

( 2 .

 2 4920105 . . . , 2.4920105 . . .) s the oordintes of bar DH (P + )D so PSL 3 (C) dmits uählerEiinstein metriF e lso omputed the ext vlue in this seD whih is

  Cye -2cψ(z) dz ∧ dz = Tc×(y-σ) e -2cψ(z) dz ∧ dz.IQH ell tht ψ(z) = -v σ , ln |z| + ϕ(z)D nd tht f is the funtion on N R suh tht f (x) = ϕ(e x)F y we hve hosen sis of N or equivlently of M D nd we denote y (x i ) i=1...n the oordintes of x ∈ N R long this sisF his determines lol holomorphi oordintes z i = e xi+iθi on T N R × T c F sing the ft tht dzi zi ∧ dzi zi = dx i ∧ dθ i D nd T c EinvrineD we otin thtD up to onstntD Cy e -2cψ(z) dz ∧ dz = y-σ

  so tht the ondition n e writtenXN (cf L ) ⊂ σ∈Σ(n) Int(N -σ (cf h + -S * σ (1, . . . , 1), • ) = Int(N (cf h + f -K X Σ )).

  A.3.2 General formula vet X Σ e smooth ompt tori mnifoldF vet N (T ) e the normlizer of T in Aut(X Σ )D nd denote y W = N (T )/T the eyl group otined from T F he group N (T ) nturlly ts on M nd sine T ts trivilly on M D this indues n tion of W on M F fy dulity one lso gets n tion on N F prom the desription of morphisms etween tori vrieties gvIID heorem QFQFRD we n see tht W is isomorphi to the sugroup of GL(N ) omposed of the ρ suh tht ρ(Σ) = ΣF sn prtiulrD W is (niteF qiven sugroup G of W D we denote y T G the preimge in N (T ) of GD nd let K G := K ∩ T G F sf P is polytope in M R we let P G e the set of GEinvrint points of P F pinllyD if P is polytope in M R D we denote y P (Q) the set of rtionl points in P D iFeF points p suh tht there exists m ∈ N * with mp ∈ M F Theorem A.19. Let L be a T G -linearized line bundle on X Σ . Thenα K G (L) = inf p∈P G L (Q) sup{c > 0|cP L ⊂ Int(cp + P -K X Σ )}.Proof. he gorollry eFIV shows tht it is enough to onsider lgeri metris on L ssoited to K G Eirreduile liner system in multiple of LF he T c Eirreduile surepresenttions of H 0 (X Σ , mL) re the dimension one suspes orresponding to integrl points of the polytope P mL ssoited to mLF ell thtP mL = mP L F xow K G Eirreduile surepresenttion of H 0 (X Σ ,mL) is the union of the imges y G of T c Eirreduile representtionF vet p e n integrl point in mP L D nd denote y ∆ the K G Eirreduile liner system generted y the GEorit of pF he potentil of h ∆/m is ϕ ∆/m (zof this inequlity is the potentil of the lgeri metri h g∈G (g•p) m|G| orresponding to the liner system of H 0 (X Σ , m|G|L) generted y the setion g∈G (g • p)F sing gin the ft tht the omplex singulrity exponent is inresing huHID IFRD we get lct(h ∆/m ) ≥ lct(h g∈G (g•p) m|G|

A

  .3.3 Case of the anticanonical line bundle e ssume in this setion tht L = -K XΣ F his line undle dmits nturl Aut(X)ElineriztionD nd the polytope ssoited to this lineriztion ontins the origin in its interiorD euse-K X is igF por ny sugroup G of W D let S G := {p ∈ ∂P L |g • p = p ∀g ∈ G}F sf 0 = p ∈ P L D let w p e the point ∂P L ∩ {-tp|t ≥ 0}F Remark A.21. ! S G is empty if nd only if {0} is the only point (xed y G in P F ! sf S W is emptyD X Σ is lled symmetriF IQR Proposition A.22. Assume that P h = {p} with 0 = p ∈ P L . Then lct(h) = |w p | |w p | + |p| . p H w pProof. fy heorem eFIS we hve lct(h) = sup{c > 0|cP ⊂ Int(cp + P )}.

  |wp| |p| > 1 with w p ∈ S G D whih is ontrditionF IQS A.3.4 Example

  hus α K W (P (k, l)) = 1 l-k when k ≥ l/2F IQT Bibliography efHR F elexeev nd wF frionF tle redutive vrietiesF sF e0ne vriE etiesF Invent. Math.D ISU@PAXPPU!PURD PHHRF efHR F elexeev nd wF frionF tle redutive vrietiesF ssF rojetive seF Adv. Math.D IVR@PAXQVH!RHVD PHHRF euHS F elexeev nd vF utzrkovF yn KEstility of redutive vrietiesF Geom. Funct. Anal.D IS@PAXPWU!QIHD PHHSF evWP rF ezd nd tFEtF voeF lurisuhrmoni funtions nd uählerE in metris on omplexi(tion of symmetri spesF Indag. Math. (N.S.)D Q@RAXQTS!QUSD IWWPF em pF emroF rition of log nonil thresholds in liner systemsF erxiv X IRIIFPUUHF euUT F euinF Équtions du type wongeEempère sur les vriétés kähE leriennes omptesF C. R. Acad. Sci. Paris Sér. A-BD PVQ@QAXeiiiD eIIW!eIPID IWUTF fegHP eF fen edesselem nd F gherrierF iinsteinEuähler metris on lss of undles involving integrl weightsF J. Math. Pures Appl. (9)D VI@QAXPSW!PVID PHHPF ffIQ F tF fermn nd fF ferndtssonF el wongeEempère equtions nd uählerEii solitons on tori log pno vrietiesF Ann. Fac. Sci. Toulouse Math. (6)D PP@RAXTRW!UIID PHIQF fieHR F fielwskiF resriing ii urvture on omplexi(ed symmetri spesF Math. Res. Lett.D II@RAXRQS!RRID PHHRF fifWH iF fifetF yn omplete symmetri vrietiesF Adv. Math.D VH@PAXPPS! PRWD IWWHF fuHS wF frion nd F uumrF Frobenius splitting methods in geometry and representation theoryD volume PQI of Progress in MathematicsF firkhäuser fostonD snFD fostonD weD PHHSF flST eF flnhrdF ur les vriétés nlytiques omplexesF Ann. Sci. Ecole Norm. Sup. (3)D UQXISU!PHPD IWSTF forWI eF forelF Linear algebraic groupsD volume IPT of Graduate Texts in MathematicsF pringerEerlgD xew orkD seond editionD IWWIF IQU friVU wF frionF ur l9imge de l9pplition momentF sn Séminaire d'algèbre Paul Dubreil et Marie-Paule Malliavin (Paris, 1986)D volE ume IPWT of Lecture Notes in Math.D pges IUU!IWPF pringerD ferlinD IWVUF friVW wF frionF qroupe de ird et nomres rtéristiques des vriétés sphériquesF Duke Math. J.D SV@PAXQWU!RPRD IWVWF friHU wF frionF he totl oordinte ring of wonderful vrietyF

  Il existe une métrique de Kähler-Einstein sur X si et seulement

	si le barycentre
				-1
	bar DH (P + ) =	p	(α(p)) 2 dp	(α(p)) 2 dp
	P +	α∈Φ +	P +	α∈Φ +
	de P + par rapport à la mesure de Duistermaat-Heckman est dans 2ρ + Ξ.

strit m t en un unique point x t de a + F xotre ut est d9otenir des estimées sur m t et |x t |D quiD ominées ux informtions sur le omportement symptotique des potentiels onvexes @issues du hpitre RAD permettront d9oE tenir les estimées sur u t -u ref

  , où u ref est le potentiel onvexe de l métrique de référene hoisieD et J est l fontion pprissnt dns l formule d9intégrtion KAK et dns l9expression du wongeEempère omplexeF our prouver l9existeneD il su0t d9otenir une orne supérieure indépenE dnte de t sur u t -u ref F xous otiendrons ette orne en étudint l fontion ν t = tu t +(1-t)u ref -ln JD dé(nie sur a + F gette fontion est stritement onvexeD et propre u sens où ν t (x) tends vers +∞ lorsque |x t | tends vers l9in(ni où que x t s9pprohe d9un des murs de l hmre de eyl a + F ille dmet don un minimum

	tends vers +∞D dns une diretion donnée ξF our en déduire que l ondition
	de notre théorème n9est ps stisfiteD nous utilisons l9nnultion X
	a +	∂ν t ∂ξ	e -νt dx = 0,
	dont l preuveD élémentireD repose sur le omportement de ν t ux ords de a + F
	v prtie l plus tehnique de e trvil est lors de trduire ette nnultionD
	lorsque |x t | tends vers +∞ en une informtion sur le polytopeD en utilisnt d9une
	prt l9informtion sur l roissne de ν t qui permet de voir que e -νt dx est une
	mesure dont le poids est essentiellement onentrée près de x t D et d9utre prt le
	hngement de vrile p = ∇u t pour se rmener u polytope 2P + F pinlementD
	nous trduisons l9nnultion pr
	t ∞ (bar DH
	F
	our les estimées sur l vleur m t du minimumD nous utilisons extement
	le même proédé que ng et hu dns HRD et peu de hoses sont à moE
	di(erF ge proédé repose sur l9utilistion du prinipe de omprison pour les
	équtions de wongeEempère réelleD ompgné d9un résultt de tohn sur les
	orps onvexesF n petit rgument supplémentire permet d9otenir une orne
	inférieure indépendnte de t sur l rpidité à lquelle roît ν t @u moins linéiE
	rementA à prtir de e minimum X
	ν t (x) ≥ κ|x -x t | -C
	pour κ et C deux onstntes indépendntes de tF
	xous nous onentrons ensuite sur le point où le minimum est tteint x t F
	xous supposons qu9il n9existe ps de métrique uählerEiinsteinD et don que |x t |
	n9est ps ornéF uitte à prendre une sousEsuiteD on peut supposer que |x t |
			IV

  [BK05, Proposition 6.1.11] The Picard group of X is isomorphic to the group M sc . If λ ∈ M sc , denote by L(λ) the associated line bundle. Then we have also:

L(λ) is globally generated if and only if λ is a dominant weight ; L(λ) is ample if and only if λ is dominant and regular ; L(λ) can be G × G linearized if and only if λ ∈ M .

  Vol({ν t ≤ m t + s})ds xow remrk tht {ν t ≤ m t + s} ⊂ s • A t y onvexity of ν t D where s • A t is the sEdiltion of A t with enter x t F e dedue from tht Vol({w t ≤ ν t + s}) ≤ s r Vol(A t ) ≤ Cs r e mt/2 . Vol({w t ≤ ν t + s})ds ≥ e -mt Vol(A t ) t is ounded ove y onstnt independent of tF hus we hve showed estimtes on |m t |F xow for liner growthD the estimte on |m t | implies tht we know oth n upper ound C 1 nd lower ound C 2 independent of t for the volume of A t F ine this set is onvexD nd ontins ll B(x t , δ 0 ) of (xed rdius δ 0 independent of t y the proof of vemm TFIUD this implies tht A t is inluded in ll A t ⊂ B(x t , δ) where δ only depends on C 1 nd δ 0 F fy onvexity of ν t D this implies tht ν t (x) ≥ 1 δ |x -x t | + m t outside of the llD nd we n extend this inequlity to the whole of a + simply y sutrting Let u be the convex potential of a smooth positive metric on -K X . Dene ν = u + j. Let ξ be any vector in a + . Then

	epplying this to the formul ove we otin a + e -νt(x) dx ≤ e -mt Ce mt/2 ≤ C e -mt/2 . Proposition 6.20. a +	0	+∞	e -s s n ds
	he left hnd side eing onstntD this inequlity gives n upper ound on
	m t F
	por the lower oundD remrk tht
			+∞
	V =	e -νt(x) dx = e -mt		e -s Vol({w t ≤ ν t + s})ds
	a +		0
			+∞
		≥ e -mt	1	e -s +∞
					e -s ds.
				1
	fy vemm TFIUD Vol(A t ) dmits lower ound c independent of tD so
			+∞
		V ≥ e -mt c		e -s ds,
			1
	nd we dedue tht
				+∞
		-m t ≤ ln(V ) -ln c		e -s ds
				1
	a + everywhereF sing gin the ft tht m t is uniformly ounded we get the e -νt(x) dx = a + +∞ νt(x) e -s dsdx = +∞ -∞ e -s a + 1 {νt(x)≤s} dxds = +∞ mt e -s Vol({ν t ≤ s})ds = e -mt +∞ 0 so -m IX ν t (x) ≥ 1 δ |x -x t | + m t -1 resultF 6.4 Obstruction, and upper bound on R(X) iverything relies on the following vnishing sttementF e -s IHQ IHR

  DH (2P + ) -4ρ ∈ (a + ) ∨ .e hve lso tht (a + ) ∨ ⊂ a ss D nd the inequlity ∂j ∂ξ ≤ -4ρ(ξ) is in ft strit when ξ ∈ a + ss D so indeedbar DH (2P + ) -4ρ ∈ Ξ.Proposition 6.22. Assume that X admits no Kähler-Einstein metrics, then the greatest Ricci lower bound of X is lower than or equal to the supremum of all t < 1 such that ) 2 = e -νt .epply the proposition to w t F his gives for ny ξ ∈ a + D is the support funtion of 2P D we hve for ny ξ nd t ny x ∈ a + D

	sing the vegendre trnsformD we n rewrite the leftEhnd side s sf v ∂u ref
	∂ξ
			≤	p, ξ	α, p	2 dp
				2P +	α∈Φ +
			≤ ξ, bar DH (2P + ) V.
		hividing y V we otin tht for ny ξ ∈ a + D
				ξ, bar DH (2P + ) ≥ 4ρ(ξ)
	nd tht preisely mens
		bar e now turn to n upper ound on R(X)F
		4ρ +	t t -1	(bar DH (2P + ) -4ρ) ∈ 2P + + (-Ξ).
	Proof. gonsider the eqution t time tX
					α(∇u) 2 .
	α∈Φ + e -ν dx = V ∂ξ ≤ -4ρ(ξ) everywhere for ξ ∈ a + D we hve a + ine we hve the inequlity ∂j sn prtiulrD is onstntF MA R (u t ) α∈Φ + α(∇u t a + ∂ν t ∂ξ e -νt dx = 0.
		-e -νt + (1 -t) a + ∂j his is equivlent to ∂ξ o we hve t ∂u t ∂ξ	e -ν dx ≥ 4ρ(ξ)V. ∂u ref ∂ξ e -νt +	∂j ∂ξ	e -νt = 0.
	get	4ρ(ξ)V ≤ ithout loss of generlity we n ssume t < 1 nd divide y (t -1)V to a + ∂u e -ν dx ∂ξ ≤ a + ∂u ∂ξ MA R (u) α∈Φ + α(∇u) 2 dx. t t -1 ∂u t ∂ξ e -νt V + 1 t -1 ∂j ∂ξ e -νt V = ∂u ref ∂ξ e -νt V .
					IHT IHU

  will ompute the limit of eh of these terms s t i → t ∞ F vet us (rst onsider a + ∂ut ∂ξt e -νt dxF vet bar DH (2P + ) denote the ryenter of 2P + with respet to the mesure α∈Φ + (α(p)) 2 dpF ell lso tht V = a + e -νt is onstntF Lemma 6.24. We have

	o the vnishing integrl of vemm TFPQ gives
			t	a +	∂u t + j ∂ξ t	e -νt dx + (1 -t)	a +	∂u ref + j ∂ξ t	e -νt dx = 0,
	whih n lso e written
	t	a +	∂u t ∂ξ t	e -νt dx +	a +	∂j ∂ξ t	e -νt dx = (t -1)	a +	∂u ref + j ∂ξ t	e -νt dx.
						∂u t
						a +	∂ξ
									a +	∂ν t ∂ξ	e -νt dx = 0
	ell tht w t = tu t + (1 -t)u ref y de(nitionD so
		ν IHV

t = tu t + (1 -t)u ref + j = t(u t + j) + (1 -t)(u ref + j).

e t e -νt dx = ξ t , bar DH (2P + ) V.

  .

	e now del with the seond termX
	D	∂j ∂ξ t	e -νt + 4ρ(ξ t )V ≤	D	∂j ∂ξ t	+ 4ρ(ξ t ) e -νt +	a + \D	4ρ(ξ t )e -νt
			≤	D	∂j ∂ξ t	+ 4ρ(ξ t ) e -νt + 4ρ(ξ t )	a + \D	e -νt
			≤	D	∂j ∂ξ t	+ 4ρ(ξ t ) e -νt + |4ρ|	a + \D	e -νt
	nd y onstrution of D we dedueX
			≤	D	∂j ∂ξ t	+ 4ρ(ξ t ) e -νt + |4ρ| • 2θ
	e onsider now the quntity
				∂j ∂ξ t	(x) + 4ρ(ξ t )
	for x ∈ DF
	he (rst thing to remrk is tht it is negtiveF sndeedD rell tht
			∂j
			∂ξ t
					IIP

  nd the derivtives of j re ounded on b(θ)ρ + a + D so there is @negtiveA onstnt C B(x t , δ) D we lwys hve ∂u ref ∂ξt ≤ v(ξ t )F xow onsider the ry strting from x -x t nd going to xF fy onvexityD we hve

	qthering everything givesD for i > max(i 0 , i 1 )D
	a +	∂j ∂ξ t	e -νt + 4ρ(ξ ∞ )V ≤ (2d + 1)θ + 8|ρ|θ +	3	+	3	= .
	Proof of Proposition 6.26. vet > 0F et θ := 6d nd let δ = δ(θ)F pirstD y vemm TFPUD we hve
				a + \B(xt,δ)	∂u ref ∂ξ t	e -νt < dθ.
			∂u ref
				∂ξ t
	suh tht			0 ≥	∂j ∂ξ t	(x) + 4ρ	x t |x t |	≥	C |x t |	.
	epplying this to the sequene t i D we (nd tht there exists i 1 suh tht for
	i > i 1 D nd uniformly for x ∈ DD
					∂j ∂ξ t	(x) + 4ρ (ξ t ) ≤	3V	.
	hen for i > i 1 D
		D	∂j ∂ξ t	+ 4ρ(ξ t ) e -νt ≤	D	∂j ∂ξ t	+ 4ρ(ξ t ) e -νt
							≤	3V D	e -νt
							≤	3
							IIR

yn

  nd only if m is in the supporting hyperplne of 2P de(ned y ξ ∞ F e dedue tht neessrily t ∞ is the unique vlue of t for whih

	4ρ +	t 1 -t	(-bar DH (2P

+ ) + 4ρ) ∈ ∂(2P + + -Ξ), if it existsF sf it does not existD then t ∞ = 1F

gomining this with the upper ound on R(X)D we hve proved Theorem 6.31. The greatest Ricci lower bound of a smooth and Fano group compactication

  A.3.1 Log canonical threshold and α-invariantvet X e ompt uähler mnifoldD L ig nd nef line undle on XF Denition A.16. essume tht ompt group K ts on XD nd tht L is KElinerizedF he lph invrint α K (L) of L with respet to the group K is de(ned s the in(mum of the log nonil thresholds of ll KEinvrintD non negtively urved singulr hermitin metris on LF

  J. Alge-braD QIQ@IAXTI!WWD PHHUF fWW F F ftyrev nd iF xF elivnovF iinsteinEuähler metris on symmetri tori pno mnifoldsF J. Reine Angew. Math.D SIPXPPS! PQTD IWWWF fIQ wF frionD F muelD nd F mF Lectures on the structure of algebraic groups and geometric applicationsD volume I of CMI Lecture Series in MathematicsF rindustn fook egenyD xew helhiY ghenni wthemtil snstitute @gwsAD ghenniD PHIQF ghIS F ghenD F honldsonD nd F unF uählerEiinstein metris on pno mnifoldsF sX epproximtion of metris with one singulritiesF J. Amer. Math. Soc.D PV@IAXIVQ!IWUD PHISF ghIS F ghenD F honldsonD nd F unF uählerEiinstein metris on pno mnifoldsF ssX vimits with one ngle less thn 2πF J. Amer. Math. Soc.D PV@IAXIWW!PQRD PHISF ghIS F ghenD F honldsonD nd F unF uählerEiinstein metris on pno mnifoldsF sssX vimits s one ngle pprohes 2π nd ompletion of the min proofF J. Amer. Math. Soc.D PV@IAXPQS!PUVD PHISF gvII hF eF goxD tF fF vittleD nd rF uF henkF Toric varietiesD volume IPR of Graduate Studies in MathematicsF emerin wthemtil oietyD rovideneD sD PHIIF gHV sF eF ghel tsov nd uF eF hrmovF vogEnonil thresholds for nonsingulr pno threefoldsF Uspekhi Mat. NaukD TQ@S@QVQAAXUQ!IVHD PHHVF hgVQ gF he gonini nd gF roesiF gomplete symmetri vrietiesF sn Invariant theory (Montecatini, 1982)D volume WWT of Lecture Notes in Math.D pges I!RRF pringerD ferlinD IWVQF helIS F helroixF elphEinvrint of tori line undlesF Ann. Pol. Math.D IIRXIQ!PUD PHISF her F hervnF elph invrints nd kEstility for generl polristions of fno vrietiesF rivXIQHUFTSPUF huHI tFEF hemilly nd tF uollárF emiEontinuity of omplex singulrity exponents nd uählerEiinstein metris on pno orifoldsF Ann. Sci. École Norm. Sup. (4)D QR@RAXSPS!SSTD PHHIF honHV F uF honldsonF uähler geometry on tori mnifoldsD nd some other mnifolds with lrge symmetryF sn Handbook of geometric analysis. 1D volume U of Adv. Lect. Math. (ALM)D pges PW!USF sntF ressD omervilleD weD PHHVF i F ikl nd eF ukhlikovF yn the glol log nonil threshold of fno omplete intersetionsF rxivXIRIPFRWSPF prWI F pulton nd tF rrrisF Representation theoryD volume IPW of Graduate Texts in MathematicsF pringerEerlgD xew orkD IWWIF e (rst ourseD edings in wthemtisF ptUV wF plenstedEtensenF pheril funtions of rel semisimple vie groupF e method of redution to the omplex seF J. Funct. Anal.D QH@IAXIHT!IRTD IWUVF pulWQ F pultonF Introduction to toric varietiesD volume IQI of Annals of Mathematics StudiesF rineton niversity ressD rinetonD xtD IWWQF he illim rF oever vetures in qeometryF putVQ eF putkiF en ostrution to the existene of iinstein uähler metrisF Invent. Math.D UQ@QAXRQU!RRQD IWVQF qIQ tF qndini nd eF uzziF xormlity nd smoothness of simple liner group ompti(tionsF Math. Z.D PUS@IEPAXQHU!QPWD PHIQF quII hF qunF ype s lmostEhomogeneous mnifolds of ohomogeneity one"sF Pac. J. Appl. Math.D Q@IEPAXRQ!UID PHIIF quII hF qunF ype s lmostEhomogeneous mnifolds of ohomogeneity one"ssF Pac. J. Appl. Math.D Q@QAXIUW!PHID PHIIF quII hF qunF ype ss lmostEhomogeneous mnifolds of ohomogeneity oneF Pacic J. Math.D PSQ@PAXQVQ!RPPD PHIIF quIQ hF qunF ype s lmost homogeneous mnifolds of ohomogeneity oneD sssF Pacic J. Math.D PTI@PAXQTW!QVVD PHIQF queIP rF quenniF ori plurisuhrmoni funtions nd nlyti djoint idel shevesF Math. Z.D PUI@QERAXIHII!IHQSD PHIPF quiWR F quilleminF uehler strutures on tori vrietiesF J. Dierential Geom.D RH@PAXPVS!QHWD IWWRF relUV F relgsonF Dierential geometry, Lie groups, and symmetric spacesD volume VH of Pure and Applied MathematicsF edemi ressD snF rrourt fre tovnovihD ulishersD xew orkEvondonD IWUVF relVR F relgsonF Groups and geometric analysisD volume IIQ of Pure and Applied MathematicsF edemi ressD snFD yrlndoD pvD IWVRF snE tegrl geometryD invrint di'erentil opertorsD nd spheril funE tionsF rowHI tF eF rowldF wultiplier idels of monomil idelsF Trans. Amer. Math. Soc.D QSQ@UAXPTTS!PTUI @eletroniAD PHHIF rumUP tF iF rumphreysF Introduction to Lie algebras and representation theoryF pringerEerlgD xew orkEferlinD IWUPF qrdute exts in wthemtisD olF WF IQW tohRV pF tohnF ixtremum prolems with inequlities s susidiry ondiE tionsF sn Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948D pges IVU!PHRF sntersiene ulishersD snFD xew orkD xF FD IWRVF uzVU fF F uzrnovski%F xewton polyhedr nd fezout9s formul for mtrix funtions of (niteEdimensionl representtionsF Funktsional. Anal. i Prilozhen.D PI@RAXUQ!URD IWVUF uuvVW pF unopD rF urftD hF vunD nd F ustF vol properties of lgeE ri group tionsF sn Algebraische Transformationsgruppen und In-variantentheorieD volume IQ of DMV Sem.D pges TQ!USF firkhäuserD fselD IWVWF uuVW pF unopD rF urftD nd F ustF he ird group of GEvrietyF sn Algebraische Transformationsgruppen und InvariantentheorieD volE ume IQ of DMV Sem.D pges UU!VUF firkhäuserD fselD IWVWF unHP eF F unppF Lie groups beyond an introductionD volume IRH of Progress in MathematicsF firkhäuser fostonD snFD fostonD weD seond editionD PHHPF uVT xF uoiso nd F kneF xonhomogeneous uählerEiinstein metris on ompt omplex mnifoldsF sn Curvature and topology of Riemannian manifolds (Katata, 1985)D volume IPHI of Lecture Notes in Math.D pges ITS!IUWF pringerD ferlinD IWVTF uVV xF uoiso nd F kneF xonhomogeneous uählerEiinstein metris on ompt omplex mnifoldsF ssF Osaka J. Math.D PS@RAXWQQ!WSWD IWVVF viII gF viF qretest lower ounds on ii urvture for tori pno mnE ifoldsF Adv. Math.D PPT@TAXRWPI!RWQPD PHIIF vIS rF viD F hiD nd F oF e riterion for the properness of the KEenergy in generl uähler lssF Math. Ann.D QTI@IEPAXIQS!ISTD PHISF wVU F wuhiF iinsteinEuähler formsD putki invrints nd onvex geometry on tori pno vrietiesF Osaka J. Math.D PR@RAXUHS!UQUD IWVUF wiHH F willotF qéométrie d9erkelov des vriétés toriques et (rés en droites intégrlesF Mém. Soc. Math. Fr. (N.S.)D @VHAXviCIPWD PHHHF wtSU F wtsushimF ur l struture du groupe d9homéomorphismes nlytiques d9une ertine vriété kählérienneF Nagoya Math. J.D IIXIRS!ISHD IWSUF xII fF xill nd eF 'enholzF ixmples of uählerEiinstein tori pno mnifolds ssoited to nonEsymmetri re)exive polytopesF Beitr. Algebra Geom.D SP@PAXPWU!QHRD PHIIF ydVV F ydF Convex bodies and algebraic geometryD volume IS of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]F pringerEerlgD ferlinD IWVVF en IRH introdution to the theory of tori vrietiesD rnslted from the tpneseF yIP F ydk nd F noF elph invrint nd uEstility of QEpno vrietiesF Adv. Math.D PPW@SAXPVIV!PVQRD PHIPF ezHW qF ezziniF eutomorphisms of wonderful vrietiesF Transform. GroupsD IR@QAXTUU!TWRD PHHWF IH pF odestà nd eF piroF uählerEii solitons on homogeneous tori undlesF J. Reine Angew. Math.D TRPXIHW!IPUD PHIHF oWU F F okfellrF Convex analysisF rineton vndmrks in wtheE mtisF rineton niversity ressD rinetonD xtD IWWUF eprint of the IWUH originlD rineton perksF uz eF uzziF rojetively norml omplete symmetri vrieties nd fno omplete symmetri vrietiesF hh hesisF uzIP eF uzziF pno symmetri vrieties with low rnkF Publ. Res. Inst. Math. Sci.D RV@PAXPQS!PUVD PHIPF hHH qF F hwrtzF uotients of ompt nd omplex redutive groupsF sn Théorie des invariants et Géométrie des variétés quo-tientsD volume TI of Travaux en coursD pges R!VQF rermnnD PHHHF iuVV F F iuF he existene of uählerEiinstein metris on mnifolds with positive ntinonil line undle nd suitle (nite symmetry groupF Ann. of Math. (2)D IPU@QAXSVS!TPUD IWVVF onHS tF ongF he αEinvrint on tori pno mnifoldsF Amer. J. Math.D IPU@TAXIPRU!IPSWD PHHSF prWV F eF pringerF Linear algebraic groupsD volume W of Progress in MathematicsF firkhäuser fostonD snFD fostonD weD seond editionD IWWVF üÿIQ rF üÿF uählerEiinstein metris on symmetri pno T EvrietiesF Adv. Math.D PRTXIHH!IIQD PHIQF zéII qF zékelyhidiF qretest lower ounds on the ii urvture of pno mnifoldsF Compos. Math.D IRU@IAXQIW!QQID PHIIF i qF inF uEstility nd kählerEeinstein metrisF rxivXIPIIFRTTWF iVU qF inF yn uählerEiinstein metris on ertin uähler mnifolds with C 1 (M ) > 0F Invent. Math.D VW@PAXPPS!PRTD IWVUF iWH qF inF yn gli9s onjeture for omplex surfes with positive (rst ghern lssF Invent. Math.D IHI@IAXIHI!IUPD IWWHF iWU qF inF uählerEiinstein metris with positive slr urvtureF Invent. Math.D IQH@IAXI!QUD IWWUF imII hF eF imshevF Homogeneous spaces and equivariant embeddingsD volume IQV of Encyclopaedia of Mathematical SciencesF pringerD reidelergD PHIIF snvrint heory nd elgeri rnsformtion qroupsD VF IRI HR FEtF ng nd F huF uählerEii solitons on tori mnifolds with positive (rst ghern lssF Adv. Math.D IVV@IAXVU!IHQD PHHRF uUV F F uF yn the ii urvture of ompt uähler mnifold nd the omplex wongeEempère equtionF sF Comm. Pure Appl. Math.D QI@QAXQQW!RIID IWUVF IR F hng nd F hngF qenerlized uählerEiinstein metris nd energy funtionlsF Canad. J. Math.D TT@TAXIRIQ!IRQSD PHIRF IRP

IQV

No.

lct(h) = sup{c > 0|cv σ ∈ Int(N -σ (cf h )) -S *σ (1, . . . , 1)}.

Root system G 2 por the root system G 2 D we hd only one toroidl exmpleD whih is the wonderful ompti(tion of the group G 2 F sn this se we n relize the simple roots in the euliden plne s α 1 = ( √ 3/3, 0) nd α 2 = (-√ 3/2, 1/2)F he other positive roots re α 1 + α 2 D 2α 1 + α 2 D 3α 1 + α 2 nd 3α 1 + 2α 2 F e n thus omputeD for p

pigure TFT shows the ryenterD whose oordintes with respet to the simE ple roots re (10.260455 . . . , 6.0448053 . . .).

his mnifold dmits uählerEiinstein metriF sndeedD the oordintes of 2ρ re (10, 6)F IPP pigure TFTX fryenter for G 2 wond IPQ Appendix A

Alpha invariants of toric line bundles

his is the text from helISD pulished in Annales Polonici MathematiciF Introduction he αEinvrint of line undle L on omplex mnifold X is n invriE nt mesuring the singulrities of the nonEnegtively urved singulr hermitin metris on LF st ws introdued y in in the se of the ntinonil undle on pno mnifoldF in showed in iVU tht if the αEinvrint of the ntiE nonil undle is stritly greter thn n n+1 D then the pno mnifold dmits uählerEiinstein metriF he uEinEhonldson onjeture sserts in generl tht X dmits n exE treml metri in c 1 (L) if nd only if the line undle L is uEstleF st ws proved in ghISD ghISD ghISD i tht it holds when L is the ntinonil undleF sn prtiulr @s it ws shown lso in yIPAD if the αEinvrint of the ntinonil undle is greter thn n n+1 D then the ntinonil undle is uEstleF hervn her gve similr ondition of uEstility for generl line undleD involving gin its αEinvrintF his is one motivtion to ompute expliitly the αEinvrints of line undles when possileF sn gHVD ghel9tsov nd hrmov omputed for exmple the αEinvrint of the ntinonil undle for mny pno mnifolds of dimension threeF sn higher dimensionsD ong onHS proved formul giving the αEinvrint of the ntinonil undle on tori pno mnifold in terms of its polytopeF he only tori mnifolds stisfying in9s riterion re the symmetri tori mnifoldsF ftyrev nd elivnov fWW proved (rst tht their αEinvrint ws oneD so tht they dmit uählerEiinstein metriF ng nd hu HR fully settled the question of the existene of uählerEiinstein metris on tori pno mnifoldsD nd n illustrtion tht in9s riterion is only su0ient ondition n e found in the tori world xIIF IPR he αEinvrint of line undle L is strongly relted to the log nonil thresholds @ltA of metris on LF he log nonil threshold ws initilly n lgeri invrint de(ned for idel shevesD ut it ws shown to oinide with the omplex singulrity exponent nd hemilly de(nes the log nonil threshE old of ny nonEnegtively urved singulr hermitin metri on line undle in gHV for exmpleF yne of the min exmples of omputtion of log nonil threshold is in the se of monomil idelsF rowld rried out the omputtion of the lt of suh n idel in terms of its xewton polygon rowHIF yne n (nd in quenni queIP n nlyti proof of this resultD generlized to ompute the lt of n idel generted y 4tori4 psh funtion on neighorhood of 0 ∈ C n D iFeF funtion invrint under rottion in eh oordinteF ine the only smooth 0ne tori mnifolds without torus ftor re isoE morphi to C n D the omputtion of quenni in ft gives the log nonil threshold of ny invrint metri on n 0ne smooth tori mnifoldD s we explin in etion eFPF sn this noteD we give formul for the αEinvrint of ny line undle L on ompt smooth tori mnifold in terms of its polytopeF e lso ompute the log nonil threshold of ny invrint nonEnegtively urved singulr metri on LF Remark A.1. efter this rtile ws eptedD the uthor ws informed tht other uthors omputed similr invrints using other methods @rF viD F hiD F o vISD nd pF emro emAF A.1 Line bundles on smooth toric manifolds A.1.1 Toric manifolds vet us rell some si fts out tori vrieties @see pulWQD ydVVD gvIIAF vet T = (C * ) n e n lgeri torusF henote its group of hrters y M D whih is isomorphi to Z n through the hoie of sisD nd let

e denote y T c (S 1 ) n the ompt torus in T F gonsidering only ones for the tori settingD we will ll

e fan Σ onsists of (nite olletion of ones σ ⊂ N R suh tht every one is strongly convex @i.e. {0} is fe of σAD the fes of ones in Σ re in Σ nd the intersetion of two ones in Σ is union of fes of othF he support of Σ is |Σ| := σ∈Σ σ ⊂ N R F IPS ell tht fn Σ in N R determines tori vriety X Σ D tht isD norml T Evriety with n open nd dense orit isomorphi to T D nd every tori vriety is otined this wyF fy the oritEone orrespondene gvIID heorem QFPFTD mximl one σ of Σ orresponds to (xed point z σ in X Σ F elsoD oneEdimensionl one ρ in Σ orresponds to prime invrint divisor D ρ of X Σ D nd these divisors generte the group of eil divisors of X Σ F vet ρ e suh oneD then we denote y u ρ the primitive vetor in N generting this ryF e will denote y Σ(r) the set of rEdimensionl ones in ΣF wny properties of X Σ n e red o' from the fnF por exmpleD X Σ is smooth if nd only if every one in the fn Σ is generted y prt of sis of N F e will ll one smooth if it stis(es this onditionF he vriety X Σ is omplete if nd only if |Σ| = N R F e will ssume in generl in the following tht either |Σ| = N R or tht Σ is given y strongly onvexD full dimensionl one σ nd its fesD in whih se we will denote X σ the orresponding @0neA tori vrietyF A.1.2 Line bundles ell tht line undle L on GEvriety X is lled linearized if there is n tion of G on L suh tht for ny g ∈ G nd x ∈ XD g sends the (er L x to the (er L g•x nd the mp de(ned this wy etween L x nd L g•x is linerF o T Elinerized line undle L on X Σ is ssoited set of hrters v σ D for σ ∈ Σ(n)F e de(ne v σ s the opposite of the hrter of the tion of T on the (er over the (xed point z σ F his de(nes the support funtion g L of LD whih is funtion on the support |Σ| of ΣD liner on eh oneD whih tkes integrl vlues t points of N D y x → v σ , x for x ∈ σF enother equivlent dt is the eil divisor D L ssoited to LD whih is relted to g L y the followingX

sf L is e'etiveD then to L is ssoited polytope P L in M R F his polytope n e de(ned s

he properties of the line undle n e red o' from the polytope or the support funtionF sn prtiulrD we n ssoite to eh point of P L ∩ M glol setion of LD nd the olletion of these setions form sis of the spe of lgeri setions of LF ell lso the followingD where we ssume tht |Σ| = N R F Proposition A.2. [CLS11, Theorem 6.1.7] The following are equivalent:

L is nef L is generated by global sections {v σ } is the set of vertices of P L g L is concave. IPT Proposition A.3. [CLS11, Lemma 9.3.9] L is big i P L has nonempty interior. Proposition A.4. [CLS11, Lemma 6.1.13] The line bundle L is ample i g L is concave and v σ = v σ whenever σ = σ ∈ Σ(n).

Example A.5. he ntinonil divisor -K XΣ on tori mnifold is given y -K XΣ = ρ D ρ F st is lwys ig on tori mnifoldF

A.1.3 Non-negatively curved singular metrics on line bundles

Potential on the torus vet L e T Elinerized line undle on X Σ F ell tht ny linerized line undle on T (C * ) n is trivilF pix n invriE nt triviliztion s of L on T F qiven hermitin metri h on the line undle LD we denote y ϕ h the lol potentil of h on T D whih is the funtion on T de(ned yX

he lol potentils of smooth hermitin metri re smoothF e will work here with singulr metrisD whose lol potentil re a priori only in L 1 loc F e singulr hermitin metri h is sid to hve non negtive urvture @in the sense of urrentsA if nd only if every lol potentil of h is psh funtionF e T c Einvrint funtion ϕ on T is determined y funtion f on N R D idenE ti(ed with the vie lger of T c D through the equivrint isomorphismX

Behavior at innity of the potentials Denition A.6. vet L e nef line undle on X Σ F he funtion f L : x → -g L (-x) is onvex funtion on N R D nd it is the potentil of ontinuousD T c EinvrintD non negtively urved metri on L lled the ftyrevEshinkel metri @see wiHHAD whih we denote y h L F Proposition A.7. The map h → f h denes a bijection between the singular hermitian T c -invariant metrics on L with non-negative curvature, and the convex functions on N R , such that there exists a constant C with

Proof. ee lso ffIQD roposition QFQF vet h e singulr hermitin T c E invrint metris on L with nonEnegtive urvtureF rite h = e -v h L D nd let ω L e the urvture urrent of h L F hen v is ω L Epsh funtion on XF sn IPU prtiulrD v is ounded from ove on XF henote y u the onvex funtion on R n ssoited to the T c Einvrint funtion v| T F hen we see tht

gonverselyD the stndrd ft tht psh funtionD whih is ounded from oveD extends uniquely over n nlyti setD llows one to extend u := f -f L to n ω L Epsh funtion on the whole of X if f stis(es the ondition of the propositionF A.2 Log canonical thresholds A.2.1 Denition vet X e ompt omplex mnifoldD nd L line undle on XF vet h e singulr hermitin metri on LF e rell the de(nition of the log nonil threshold of h @see the ppendix of gHVAF Denition A.8. vet z ∈ XF he omplex singulrity exponent c z (h) of h t z is the supremum of the rel c > 0 suh tht e -2cϕ is integrle in neighorhood of zD where ϕ is lol potentil of h ner zF Denition A.9. he log nonil threshold lct(h) of h is de(ned s

A.2.2 Newton body of a function

Denition A.10. vet σ e oneF vet f e funtion de(ned on N R F he(ne the xewton ody of f on σ s

sf σ = N R we will write N (f )F he following properties of the xewton ody will e usefulF Proposition A.11. For any function f , N σ (f ) is convex, and

Proof. he (rst two properties re trivilF vet us rie)y prove the lst stteE mentF vet m e in the rightEhnd setD i.e.

he right hnd side is the sum of lowerEounded funtion of t ∈ y + σ nd onstntD so the left hnd side is lowerEounded funtion of x ∈ σF his shows one inlusion nd the other is proved y similr rgumentF qiven non negtively urved T c Einvrint metri h on LD we de(ne the ssoited onvex suset P h of M R D s the xewton ody of f h F Proposition A.12. For the Batyrev-Tschinkel metric h L , we recover the polytope P L . For any T c -invariant, non-negatively curved metric h on L, P h ⊂ P L . If h is smooth, we also have P h = P L Proof. por the (rst sttementD oserve tht m ∈ P L if nd only if for ny one σ ∈ ΣD for ll x ∈ σD g L (x) = v σ , x ≤ m, x F his inequlity is equivlent to -v σ , x + m, x ≥ 0 nd sine the funtions involved re linerD it is stis(ed for ll x ∈ σ if nd only if -v σ , x + m, x is ounded elow on σF ine f L (-x) = -g L (x) = -v σ , x for x ∈ σD we get tht m ∈ P L if nd only if for every one σ ∈ ΣD the funtion f L (-x)-m, -x is ounded elow on σF pinllyD this n e trnslted sX for every one σ ∈ ΣD the funtion f L (y) -m, y is ounded elow on -σF o onludeD we note tht

he seond sttement is n esy onsequene of the (rst nd roposition eFU sine whenever two funtions f nd g stisfy f ≤ g + C for onstnt CD we hve trivilly N σ (f ) ⊂ N σ (g)F por the lst sttementD remrk tht in this seD f h -f L extends to onE tinuous funtion on X Σ D so we hve f L -C ≤ f h ≤ f L + C for some onstnt CF he sme property of xewton odies llows one to onludeF A.2.3 Integrability condition he (rst result on log nonil thresholds on tori vrieties ws the omE puttion y rowld rowHI in the se of monomil idelsF quenni gve n nlyti proof of this resultD extending the omputtion to the se of non lgeE ri psh funtionsF he key ingredient in this nlyti version is the following integrility onditionF Proposition A.13. (see [Gue12]) Let σ be a smooth cone of maximum dimension. Let f be a convex function on N R . Then e -f is integrable on all translates of σ if and only if 0 ∈ Int(N σ (f )).

his is essentilly the result in quénni queIP euse ny smooth 0ne tori mnifold with no torus ftor is isomorphi to C n F rowever we desrie the hnge of vriles used preiselyD to use it lter in the ompt seF IPW Proof. ghoose sis of N formed y the genertors of the extreml rys of σD then de(ne S σ to e the isomorphism from N to Z n sending the hosen sis to the nonil sis of Z n F vet f e funtion on N R D nd g the funtion on R n suh tht f = g • S σ F hen from the de(nition of xewton ody we hve N σ (f ) = S * σ (N D (g))D where S * σ is the dul isomorphism from Z n to M nd D is the one generted y the nonil sis of Z n F sing the hnge of vrilesD e -f is integrle on ll trnsltes of σ if nd only if e -f •S -1 σ is integrle on ll trnsltes of DF epply queIPD roposition IFW to the onve funtion -f • S -1 σ F his proves tht we hve integrility if nd only if 0 ∈ Int(N D (f • S -1 σ ))F sing S * σ D whih is linerD this indeed trnsltes to 0 ∈ Int(N σ (f ))F emrk tht the sttement in queIPD roposition IFW only mentions inteE grility on DD ut the equivlene with integrility on ll trnsltes is esily derived from roposition eFIIF A.2.4 lct on an ane smooth toric manifold Proposition A.14. Let σ be a smooth cone of maximum dimension, X σ the corresponding smooth ane toric manifold. Let L be a linearized line bundle on X σ , and h a T c -invariant metric with non-negative curvature. Then