Regularity of plurisubharmonic upper envelopes in big cohomology classes

Robert Berman

Description

Niveau: Supérieur, Doctorat, Bac+8
Regularity of plurisubharmonic upper envelopes in big cohomology classes Robert Berman Department of Mathematics, Chalmers University of Technology, Eklandag. 86, SE-412 96 Goteborg, Sweden e-mail : Jean-Pierre Demailly Universite de Grenoble I, Departement de Mathematiques, Institut Fourier, BP 74, 38402 Saint-Martin d'Heres, France dedicated to Professor Oleg Viro for his deep contributions to mathematics Abstract. The goal of this work is to prove the regularity of certain quasi- plurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of hermitian metrics with minimal singularities on a big line bundle over a compact complex manifold. We prove that the complex Hessian forms of these envelopes are locally bounded outside an analytic set of singularities. It is furthermore shown that a parametrized version of this result yields a priori inequalities for the solution of the Dirichlet problem for a degenerate Monge- Ampere operator ; applications to geodesics in the space of Kahler metrics are discussed. A similar technique provides a logarithmic modulus of continuity for Tsuji's “supercanonical” metrics, which generalize a well-known construction of Narasimhan-Simha. Resume. Le but de ce travail est de demontrer la regularite de certaines enveloppes superieures de fonctions quasi-plurisousharmoniques. De telles en- veloppes apparaissent naturellement dans la construction des metriques hermi- tiennes a singularites minimales sur un fibre en droites gros au dessus d'une variete complexe compacte.

• construction des metriques hermi- tiennes


• big cohomology


• module de continuite loga- rithmique pour les metriques


• has analytic singularities


• kiselman- legendre transform


• points z ?


• monge-ampere measure

Sujets

• Berman
• Chalmers University of Technology
• Demailly

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 20
• Langue: English

Extrait

Regularity of plurisubharmonic upper envelopes in big cohomology classes Robert Berman Jean-Pierre Demailly DepartmentofMathematics,Universit´edeGrenobleI, ChalmersUniversityofTechnology,De´partementdeMath´ematiques, Eklandag. 86, Institut Fourier, BP 74, SE-41296G¨oteborg,Sweden38402Saint-Martind’He`res,France e-mail:[e-mail] [e-mail] dedicated to Professor Oleg Viro for his deep contributions to mathematics

Abstract.The goal of this work is to prove the regularity of certain quasi- plurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of hermitian metrics with minimal singularities on a big line bundle over a compact complex manifold. We prove that the complex Hessian forms of these envelopes are locally bounded outside an analytic set of singularities. It is furthermore shown that a parametrized version of this result yields a priori inequalities for the solution of the Dirichlet problem for a degenerate Monge- Ampe`reoperator;applicationstogeodesicsinthespaceofK¨ahlermetricsare discussed. A similar technique provides a logarithmic modulus of continuity for Tsuji’s “supercanonical” metrics, which generalize a well-known construction of Narasimhan-Simha. Re´sume´.betuLdestleaiavtrcedege´ralrertnome´dluratie´edectrianes enveloppessupe´rieuresdefonctionsquasi-plurisousharmoniques.Detellesen- veloppesapparaissentnaturellementdanslaconstructiondesme´triqueshermi- tiennesa`singularit´esminimalessurunfibre´endroitesgrosaudessusd’unevarie´t´e complexecompacte.Nousmontronsquecesenveloppesposse`dentunHessiencom- plexelocalementborne´endehorsd’unensembleanalytiquedesingularite´s;par ailleurs,uneversionavecparame`tresdecer´esultatpermetd’obtenirdesine´galit´es aprioripourlasolutionduproble`medeDirichletrelatifa`unop´erateurdeMonge- Ampe`rede´g´en´er´e.Unetechniquesimilairefournitunmoduledecontinuite´loga- rithmiquepourlesme´triques“super-canoniques”deTsuji,lesquellesge´ne´ralisent une construction bien connue de Narasimhan-Simha. Key words.Plurisubharmonic function, upper envelope, hermitian line bundle, singular metric, logarithmic poles, Legendre-Kiselman transform, pseudo-effective cone, l me, Monge-Amp` rcanonical metric, Ohsawa-Takegoshi vo u ere measure, supe theorem. Mots-cl´Fnoitrulppoepvnlereeius´pharmisubue,eoniqser´fibe,uritrondee es.onc hermitien,me´triquesingulie`re,pˆoleslogarithmiques,transforme´edeLegendre- Kiselman,coˆnepseudo-effectif,volume,mesuredeMonge-Amp`ere,m´etrique super-canonique,the´ore`medeOhsawa-Takegoshi. AMS Classification.32F07, 32J25, 14B05, 14C30

2 Regularity of plurisubharmonic upper envelopes in big cohomology classes 1. Main regularity theorem LetXbe a compact complex manifold andωa hermitian metric onX, viewed as a smooth positive (1 usual we put1)-form. Asd c = 4i 1 π (∂−∂) so thatdd c = 2i 1 π ∂∂. Consider thedd c -cohomology class{α}of a smooth real d-closed formαof type (11) onXone has to consider the Bott- general, [ in Chern cohomology group for which boundaries aredd c -exact (11)-formsdd c ϕ, but in the caseXipisosomhtsirguoohleoDgoylbearuplhticcothoomtahK¨r,leis groupH 11 (X that a function Recall) ].ψis said to be quasi-plurisubharmonic (or quasi-psh) if and onlyidd c ψis locally bounded from below, or equivalently, if it can be written locally as a sumψ=ϕ+uof a psh functionϕand a smooth functionu. More precisely, it is said to beα-plurisubharmonic (orα-psh) ifα+dd c ψ>0. We denote by PSH(X α) the set ofα-psh functions onX. (1.1) Definition.The class{α} ∈H 11 (XR)is said to bepseudo-effectiveif it contains a closed(semi-)positive currentT=α+dd c ψ>0, andbigif it contains aclosed“K¨ahlercurrent”T=α+dd c ψsuch thatT>εω >0for someε >0. From now on in this section, we assume that{α}isbig. We know by [Dem92] that we can then findT 0 ∈ {α}of the form (12)T 0 =α+dd c ψ 0 >ε 0 ω with a possibly slightly smallerε 0 >0 than theεin the definition, andψ 0 a quasi-psh function with analytic singularities, i.e. locally (13)ψ 0 =clog X |g j | 2 +uwherec >0,u∈C ∞ ,g j holomorphic By [DP04],Xcarries such a class{α}if and only ifXis in the Fujiki classC ofsmoothvarietieswhicharebimeromorphictocompactK¨ahlermanifolds.Our main result is (1.4) Theorem.LetXbe a compact complex manifold in the Fujiki classC, and letαbe a smooth closed form of type(11)onXsuch that the cohomology class{α} Pickis big.T 0 =α+dd c ψ 0 ∈ {α}satisfying(12)and(13)for some hermitian metricωonX, and letZ 0 be the analytic setZ 0 =ψ 0 −1 (−∞). Then the upper envelope ϕ:= sup  ψ60 ψ α-psh  is a quasi-plurisubharmonic function which has locally bounded second order derivatives∂ 2 ϕ∂z j ∂z k onXrZ 0 , and moreover, for suitable constantsC B >0, there is a global bound |dd c ϕ| ω 6C(|ψ 0 |+ 1) 2 e B|ψ 0 | according to whichdd c ϕblows up at worst “meromorphically” onZ 0 . In particular ϕisC 11−δ onXrZ 0 for everyδ >0, and the second derivativesD 2 ϕare in L l p oc (XrZ 0 )for everyp >0.