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These de Doctorat de l'Universite Joseph Fourier Grenoble I

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Niveau: Supérieur, Doctorat, Bac+8
These de Doctorat de l'Universite Joseph Fourier (Grenoble I) Inegalites de Morse et problemes en geometrie analytique Thierry Bouche preparee a l'Institut Fourier laboratoire de mathematiques associe au C.N.R.S. (LA 188) soutenue le vendredi 9 fevrier 1990 Jury : Gerard Besson, examinateur Jean-Pierre Demailly, directeur Alain Dufresnoy, examinateur Paul Gauduchon, examinateur Henri Skoda, president

  • estimations du tenseur de courbure

  • fibre lineaire

  • courbure degeneree

  • inegalites de morse holomorphes

  • variete symplectique

  • application du theoreme de repartition spectrale

  • estimations pour l'operateur de monge-ampere


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Published by
Published 01 February 1990
Reads 66
Language English
Th`esedeDoctoratdelUniversite´JosephFourier(GrenobleI)
´ Inegalit´esdeMorseetproble`mesen ´ ´trie an lytique geome a
Thierry Bouche
pre´pare´e`alInstitutFourier laboratoiredemathe´matiquesassocie´auC.N.R.S.(LA188)
soutenuelevendredi9fe´vrier1990
Jury : Ge´rardBesson,examinateur Jean-Pierre Demailly, directeur Alain Dufresnoy, examinateur Paul Gauduchon, examinateur HenriSkoda,pr´esident
ABSTRACT We study two generalizations of the Holomorphic Morse Inequalities proved by Jean-Pierre Demailly in 1985: the case where the curvature of the line bundle is degenerate (maximum rank different from the dimension of the variety) and the case where the manifold is not compact. For these two cases, we prove precise theorems similar to the original one of [De 1]. The second part of the thesis investigates on one hand the study of an equivalent for the distortion function of a positive line bundle over a projective manifold; on the other hand we study the coeffective cohomology of a symplectic manifold. The main tools employed are those of differential geometry: Weyl asymptotic formula-type theorems, an equivalent for the heat kernel associated to Δ′′, and estimates on the curvature tensor.
´ ´ RESUME Nouse´tudionsdeuxg´ene´ralisationsdesine´galit´esdeMorseholomorphesdeJean-PierreDemailly:lecaso`ulacourburedubr´eestde´g´en´er´eeetlecaso`ulavari´et´e nestpascompacte.Danscesdeuxcas,desthe´ore`mespr´ecisanaloguesa`celuide [De1]sontde´montr´es.Lasecondepartiedecetteth`eseestconsacre´edunepart a`le´tudedun´equivalentpourlafonctiondedistorsiondunbr´epositifsurune varie´t´ekahl´erienneetdautrepart`al´etudedelacohomologiecoeectivedune vari´ete´symplectique.Lesprincipauxoutilsemploy´essontceuxdelage´ome´trie die´rentielle:th´eor`emesder´epartitionspectrale,´equivalentdunoyaudelachaleur pour Δ′′, ou estimations du tenseur de courbure.
´ MOTS-CLES Ine´galite´sdeMorseholomorphes,d′′reurbun,coitiemreheriae´nile´rbe,gilomohoco de´ge´n´ere´e,vari´et´eqituq.eesymplecvari´et´lahc,rueuayoaledveon,nxe-c
´ CLASSIFICATION MATHEMATIQUE 32 L 10 (primaire), 32 F 10, 32 L 05, 53 C 15 (secondaire)
C’est Jean-Pierre Demailly qui a conduit mes premiers pas dans la recherche mathe´matiIl´age´sontempspourmeconseillertoutaulongdela que. n a pas men pre´parationdecetravail.Jelenremerciesince`rement. HenriSkodamefaitlhonneurdepre´siderlejury,PaulGauduchondy participer, je les en remercie vivement. Jaitrouve´unaccueilsympathique`alInstitutFourier,etenparticulierau sein du Groupe de Travail d’Analyse Complexe dont Alain Dufresnoy fait partie. Ge´rardBessonasumedonnerdesindicationsquimontfaitprogresser.Jeles remercietousdeuxdavoiraccepte´departiciperaujury. Jemedoisenndecitertoutparticuli`erementMoniqueMarchandetArlette Guttin-Lombardquiontsufairedunbrouillonparfoissauvagecetextesoign´eet, jelesp`ere,agr´eablea`lire.
Sommaire
Chapitre0.Introductionge´n´erale                                       0.Plandelathe`se                                                     1.Notationsetprincipalesd´enitions                                 2.Pre´sentationdesre´sultats                                         
ChapitreI.In´egalit´demorseholomorphespourunbre´lin´eairea` es courbed´´ene´r´ee                          ur eg               0. Introduction                                                        1.ComplexedeWitten(dapre`sDemailly)                             2. Spectre de la forme quadratiqueQX(cas constant)                 3. Spectre de la forme quadratiqueQXc(nee´sa´gral)                   4.D´emonstrationsduthe´or`eme0.1etducorollaire0.2               
ChapitreII.In´egalit´esdeMorseholomorphessurunevari´ete´non compacte                                                    0. Introduction                                                        1.Estimationdespremi`eresfonctionspropresdeΔ′′hors d’un compact 2.Applicationduthe´or`emeder´epartitionspectraleetconclusion       3.Estimationspourlope´rateurdeMonge-Amp`eresurari´ete´forte-une v mentm-convexe                                                  
ChapitreIII.Aplatissementdelam´etriquedeFubini-Studydunbr´e positifsurunevarie´teprojective                           ´ 0. Introduction                                                        1. Comportement asymptotique du noyau de la chaleur associe a un ´ ` ope´rateurdeSchro¨dingeravecchampmagnetique                   ´ 2.Fonctiondedistorsiondunbre´ample                             
ChapitreIV.Lacohomologiecoeectivedunevari´ete´symplectique   0. Introduction                                                        1. Les groupesHq(A)                                                2.Ellipticit´educomplexed:A                                      3. Rapports avec la cohomologie de de Rham                          
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