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Mathématiques Denis AUROUX Th`ese pr¶esent¶ee pour obtenir le titre de DOCTEUR DE L'ECOLE¶ POLYTECHNIQUE Sp¶ecialit¶e : Math¶ematiques par Denis AUROUX Titre : Th¶eor`emes de structure des vari¶et¶es symplectiques compactes via des techniques presque complexes. soutenue le 22 janvier 1999 devant le jury compos¶e de : M. Jean Pierre BOURGUIGNON M. Mikhael GROMOV M. Fran»cois LAUDENBACH M. Dietmar SALAMON M. Jean-Claude SIKORAV M. Claude VITERBO Rapporteurs : M. Simon DONALDSON et Mme Dusa McDUFF Remerciements Je tiens a` remercier Jean Pierre Bourguignon et Misha Gromov pour leur soutien constant tout au long de la r¶ealisation de ce travail ; Christophe Margerin et Pierre Pansu qui ont pris sur leur temps pour r¶epondre `a mes questions ; Fran»cois Lauden- bach pour l'accueil qui m'a ¶et¶e r¶eserv¶e au Centre de Math¶ematiques ; et en¯n tous ceux qui par des discussions enrichissantes ont contribu¶e a` ma compr¶ehension du sujet : tout particuli`erement Simon Donaldson, Cli® Taubes et Jean-Claude Sikorav, ainsi que Fabrice Lembrez, Emmanuel Ferrand et Stefano Vidussi. Adresse de l'auteur : Denis AUROUX Centre de Math¶ematiques, Ecole Polytechnique 91128 Palaiseau Cedex, France. e-mail : [email protected] Table des mati`eres Chapitre I. Introduction 3 1. Introduction 3 2. Sous-vari¶et¶es symplectiques : ¶enonc¶es et exemples 5 2 3. Rev^etements rami¯¶es de CP 8 Chapitre II. Asymptotically Holomorphic Families of Symplectic Submanifolds 11 1. Introduction 11 2. The local result 14 3. The globalization process 17 3.1. Statement of the result 17 3.2. Local coordinates and sections 17 3.3. General setup and strategy of proof 18 3.4. Obtaining transversality near a point of Sj 19 3.5. Constructing σt;k;j+1 from σt;k;j 22 = 3.6. Transversality to 0 over Uk 23 4. The main result 23 4.1. Proof of Theorem 2 23 4.2. Symplectic isotopies 24 5. Properties of the constructed submanifolds 25 5.1. Proof of Proposition 2 25 5.2. Homology and Chern numbers of the submanifolds 26 5.3. Geometry of the submanifolds 27 6. Conclusion 30 2 Chapitre III. Symplectic 4-manifolds as branched coverings of CP 31 1. Introduction 31 2. Nowhere vanishing sections 36 2.1. Non-vanishing of sk 36 2.2. Non-vanishing of @fk 40 2.3. Proof of Proposition 2 42 3. Transversality of derivatives 46 3.1. Transversality to 0 of Jac(fk) 46 3.2. Nondegeneracy of cusps 50 1 2 TABLE DES MATIERES` 4. Dealing with the antiholomorphic part 58 4.1. Holomorphicity in the neighborhood of cusp points 58 4.2. Holomorphicity at generic branch points 63 4.3. Proof of the main theorems 65 5. Generic tame maps and branched coverings 66 5.1. Structure near cusp points 66 5.2. Structure near generic branch points 70 5.3. Proof of Theorem 3 72 6. Further remarks 73 2 6.1. Branched coverings of CP 73 6.2. Symplectic Lefschetz pencils 75 6.3. Symplectic ampleness 77 Bibliographie 79 CHAPITRE I Introduction 1. Introduction Une approche de la topologie symplectique qui s'est r¶ev¶el¶ee extr^emement fruc- tueuse au ¯l des ann¶ees a pour point de d¶epart l'observation suivante : toute vari¶et¶e symplectique (X; !) peut ^etre munie d'une structure presque-complexe compatible avec la structure symplectique, c’est-`a-dire un endomorphisme J du ¯br¶e tangent T X, v¶eri¯ant J 2 = 1, et tel que la forme bilin¶eaire g(x; y) = !(x; Jy) d¶e¯nit une m¶etrique riemannienne¡ sur X (voir par exemple [McS1], p. 116). L'¶etude des vari¶et¶es symplectiques se pr¶esente alors comme une g¶en¶eralisation naturelle de la g¶eom¶etrie kÄahl¶erienne : en e®et la vari¶et¶e X est kÄahl¶erienne d`es lors que la structure presque-complexe J est int¶egrable, c’est-`a-dire permet de d¶e¯nir localement des coordonn¶ees complexes sur X. Les vari¶et¶es kÄahl¶eriennes fournissent un grand nombre d'exemples de vari¶et¶es symplectiques, puisqu'elles englobent entre autres toutes les vari¶et¶es alg¶ebriques projectives complexes. Toutefois, de nom- breuses vari¶et¶es symplectiques n'admettant pas de structure kÄahl¶erienne ont ¶et¶e construites, notamment par Thurston [Th] et plus r¶ecemment par Gompf [Go]. La donn¶ee d'une structure presque-complexe J sur X conduit naturellement `a ¶etudier les sous-vari¶et¶es de X dont l'espace tangent est en tout point J-invariant, c’est-`a-dire un sous-espace complexe de l'espace tangent `a X. Pour un choix g¶en¶erique de la structure presque-complexe compatible avec !, de telles sous-vari¶et¶es n'existent qu'en dimension complexe 1 : ce sont les c¶el`ebres courbes pseudo-holomorphes, intro- duites par Gromov [Gro1] et dont la th¶eorie a connu de constants d¶eveloppements (voir par exemple [McS2]). Un autre exemple frappant de l'analogie entre vari¶et¶es symplectiques compactes et vari¶et¶es kÄahl¶eriennes compactes est donn¶e par les invariants de Seiberg-Witten (voir par exemple [Mor]), dont l'interpr¶etation en termes de courbes pseudo-holo- morphes r¶ecemment obtenue par Taubes ([T1], [T2] et suivants) pour les vari¶et¶es symplectiques pr¶esente des similarit¶es remarquables avec l'interpr¶etation en termes de courbes complexes donn¶ee par Witten [W] pour les vari¶et¶es kÄahl¶eriennes. N¶eanmoins, certaines constructions de g¶eom¶etrie alg¶ebrique complexe semblaient jusqu’`a r¶ecemment ne pas pouvoir ^etre transpos¶ees dans un contexte symplectique : ainsi, le lieu d'annulation d'une section holomorphe g¶en¶erique d'un ¯br¶e tr`es ample sur une vari¶et¶e projective complexe d¶e¯nit une sous-vari¶et¶e complexe, tandis que la construction analogue ne fonctionne pas en g¶eom¶etrie presque complexe. L'id¶ee introduite par Donaldson [D1] consiste `a autoriser de petites variations de la structure presque-complexe : ainsi, une structure presque-complexe J compatible avec ! ¶etant ¯x¶ee, il s'agit de construire non pas des sous-vari¶et¶es J-holomorphes 3 4 I. INTRODUCTION (qui n'existent en g¶en¶eral pas au-del`a de la dimension 1), mais plut^ot des sous- vari¶et¶es approximativement J-holomorphes. De telles sous-vari¶et¶es sont obtenues comme lieux d'annulation de sections approximativement holomorphes de ¯br¶es convenablement choisis : l'observation fondamentale de Donaldson est que, de m^eme qu'un ¯br¶e holomorphe su±samment positif sur une vari¶et¶e projective admet un grand nombre de sections holomorphes, un ¯br¶e en droites su±samment positif sur une vari¶et¶e symplectique compacte admet de nombreuses sections approximative- ment holomorphes. Toutefois, contrairement au cas projectif ou` un argument facile de transversalit¶e permet d'obtenir imm¶ediatement des hypersurfaces complexes, un raisonnement assez long est n¶ecessaire pour prouver l'existence d'une section dont les propri¶et¶es de transversalit¶e `a la section nulle garantissent que le lieu d'annulation est une sous-vari¶et¶e lisse et approximativement J-holomorphe. Il est ais¶e de v¶eri¯er qu'une sous-vari¶et¶e approximativement J-holomorphe W X est symplectique, c’est-`a-dire que la restriction de ! munit W d'une structure½ symplectique. En outre, il existe une structure presque-complexe J 0 compatible avec !, proche de J (mais d¶ependant de W ), telle que W soit une sous-vari¶et¶e J 0-holomorphe de X. Un int¶er^et majeur du r¶esultat de Donaldson est donc de fournir le premier proc¶ed¶e g¶en¶eral de construction d'hypersurfaces symplectiques (codimension r¶eelle 2) dans une vari¶et¶e symplectique compacte arbitraire [D1]. Dans [A1] (voir aussi 2 et chapitre II), ce r¶esultat a ¶et¶e ¶etendu au cas de ¯br¶es de rang sup¶erieur : la constructionx de sections approximativement holomorphes v¶eri¯ant des propri¶et¶es convenables de transversalit¶e permet alors d'obtenir des sous-vari¶et¶es symplectiques (approximativement J-holomorphes) de codimension quelconque, dont on d¶etermine mieux le type topologique que par simple it¶eration du r¶esultat de [D1]. Il a de plus ¶et¶e ¶etabli que, en d¶epit de la grande flexibilit¶e de la construction de Donaldson, les sous-vari¶et¶es que l'on est susceptible d'obtenir `a partir de sections approximativement holomorphes d'un ¯br¶e su±samment positif donn¶e sont uniques a` isotopie symplectique pr`es [A1] (cf. 2 et chapitre II) ; en outre, ce r¶esultat reste vrai m^eme si l'on fait varier la structurex presque-complexe. Cela signi¯e que les sous-vari¶et¶es symplectiques construites `a partir de ¯br¶es suf- ¯samment positifs peuvent ^etre utilis¶ees pour d¶e¯nir des invariants symplectiques de la vari¶et¶e consid¶er¶ee : ainsi des invariants topologiques d¶e¯nis pour des vari¶et¶es de petite dimension (par exemple ceux de Seiberg-Witten en dimension 4) peuvent ^etre utilis¶es pour caract¶eriser des vari¶et¶es symplectiques de dimension sup¶erieure. D'autres r¶esultats classiques de g¶eom¶etrie projective complexe peuvent ^etre transpos¶es a` la g¶eom¶etrie symplectique de fa»con similaire. Ainsi, Donaldson a ¶et¶e le premier `a montrer que deux sections approximativement holomorphes convenable- ment choisies d'un ¯br¶e en droites su±samment positif d¶eterminent une structure de pinceau de Lefschetz symplectique sur une vari¶et¶e symplectique compacte [D2].
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