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Regularity of plurisubharmonic upper envelopes in big cohomology classes

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28 pages
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


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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-MartindHe`res,France e-mail:robertb@chalmers.se demailly@fourier.ujf-grenoble.fr 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´esminimalessurunbre´endroitesgrosaudessusdunevarie´t´e complexecompacte.Nousmontronsquecesenveloppesposse`dentunHessiencom-plexelocalementborne´endehorsdunensembleanalytiquedesingularite´s;par ailleurs,uneversionavecparame`tresdecer´esultatpermetdobtenirdesine´galit´es aprioripourlasolutionduproble`medeDirichletrelatifa`unop´erateurdeMonge-Ampe`rede´g´en´er´e.Unetechniquesimilairefournitunmoduledecontinuite´loga-rithmiquepourlesme´triquessuper-canoniquesdeTsuji,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´be,uritrondee es.onc hermitien,me´triquesingulie`re,pˆoleslogarithmiques,transforme´edeLegendre-Kiselman,coˆnepseudo-eectif,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. Asdc=4i1π() so thatddc=2i1π∂∂. Consider theddc-cohomology class{α}of a smooth real d-closed formαof type (11) onXone has to consider the Bott- general, [ in Chern cohomology group for which boundaries areddc-exact (11)-formsddcϕ, but in the caseXipisosomhtsirguoohleoDgoylbearuplhticcothoomtahK¨r,leis groupH11(X that a function Recall) ].ψis said to be quasi-plurisubharmonic (or quasi-psh) if and onlyiddcψis 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+ddcψ>0. We denote by PSH(X α) the set ofα-psh functions onX. (1.1) Definition.The class{α} ∈H11(XR)is said to bepseudo-effectiveif it contains a closed(semi-)positive currentT=α+ddcψ>0, andbigif it contains aclosedK¨ahlercurrentT=α+ddcψsuch thatT>εω >0for someε >0. From now on in this section, we assume that{α}isbig. We know by [Dem92] that we can then findT0∈ {α}of the form (12)T0=α+ddcψ0>ε0ω with a possibly slightly smallerε0>0 than theεin the definition, andψ0a quasi-psh function with analytic singularities, i.e. locally (13)ψ0=clogX|gj|2+uwherec >0,uC,gjholomorphicBy [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(11)onXsuch that the cohomology class{α} Pickis big.T0=α+ddcψ0∈ {α}satisfying(12)and(13)for some hermitian metricωonX, and letZ0be the analytic setZ0=ψ01(−∞). Then the upper envelope ϕ:= supψ60 ψ α-pshis a quasi-plurisubharmonic function which has locally bounded second order derivatives2ϕ∂zj∂zkonXrZ0, and moreover, for suitable constantsC B >0, there is a global bound |ddcϕ|ω6C(|ψ0|+ 1)2eB|ψ0| according to whichddcϕblows up at worst “meromorphically” onZ0. In particular ϕisC11δonXrZ0for everyδ >0, and the second derivativesD2ϕare in Llpoc(XrZ0)for everyp >0.