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Th`Ese De Doctorat Universit´ePierre et Marie Curie Ecole´ doctorale de sciences math´ematiques de Paris centre These` de doctorat Discipline : Math´ematiques pr´esent´eepar Liana Heuberger Deux points de vue sur les vari´et´esde Fano : g´eom´etrie du diviseur anticanonique et classification des surfaces `asingularit´es 1/3(1,1) dirig´ee par Andreas Horing¨ Soutenue le 23 juin 2016 devant le jury compos´ede : M. Sebastien Boucksom Universit´eParis VI Examinateur M. Fr´ed´eric Campana Universit´eNancy I Examinateur M. Ivan Cheltsov University of Edinburgh Rapporteur M. Olivier Debarre Universit´eParis VII et ENS Examinateur M. Antoine Ducros Universit´eParis VI Examinateur M. Andreas Horing¨ Universit´ede Nice Sophia Antipolis Directeur de th`ese 2 Institut de math´ematiquesde Jussieu-Paris Universit´ePierre et Marie Curie. Rive gauche. UMR 7586. Ecole´ doctorale de sciences Bo^ıtecourrier 247 math´ematiquesde Paris centre. 4 place Jussieu Bo^ıtecourrier 290 75 252 Paris Cedex 05 4 place Jussieu 75 252 Paris Cedex 05 3 4 Remerciements [English version follows] Je tiens `aremercier mon directeur de th`ese,Andreas H¨oring,qui m'a initi´e`ala recherche math´ematique`apartir du Master 2. Pendant ces cinq ans d'´echanges hebdomadaires il m'a soutenu, relanc´eet encourag´esans cesse, il a partag´eavec moi ses id´ees,son intuition math´ematiqueet la richesse de ses comp´etences techniques. Il m'a appris a toujours demander plus de moi-m^eme,et je lui remercie pour son honn^etet´e,sa patience et sa g´en´erosit´e.C'´etaitune ´enormechance de l'avoir eu en tant que directeur de th`ese. Je voudrais aussi exprimer ma reconnaissance `aGianluca Pacienza, d'avoir accept´ede rapporter cette th`ese,pour le temps qu'il y a d´edi´eet sa lecture attentive. Merci ´egalement `aS´ebastien Boucksom, Fr´ed´ericCampana, Olivier Debarre et Antoine Ducros, qui me font le grand honneur d'^etremembres de mon jury. Je voudrais ensuite remercier ceux qui m'ont enseign´eles fondements de la g´eom´etriealg´ebrique `al'Universit´ede Bucarest, leurs excellents cours ont ´et´ela motivation principale pour que je choisisse ce chemin. Merci `aPaltin Ionescu, Cristian Voica et Victor Vuletescu, c'´etaitun privil`egede les avoir rencontr´es.Merci aussi `aLiviu Ornea, qui m'a encourag´ede participer aux cours de la SNSB et de m'inscrire en Master 2 `aParis, ce que j'estime ^etreles deux d´ecisionsfondamentales de ma carri`erejusqu’`apr´esent. Je suis tr`esreconnaissante `aAndrei Iordan de m'avoir aid´e`aobtenir la bourse qui a financ´ecette th`ese. Un grand merci `aMa¨yliset `aFran¸cois,qui ont pratiquement ´et´ema famille d'accueil `aParis. Ils m'ont g^at´eavec ´enorm´ement d'amiti´eet de g´en´erosit´e.La liste de choses que j'ai `aleur remercier doublerait probablement la longueur de cette th`ese.Je remercie aussi mes amis de Jussieu - J´er´emy pour les ´echanges math´ematiqueset son optimisme contagieux, Juliette pour les fous rires, Thibaud pour la th´erapiepost-mariage rouge et (par adoption) Flora, Oljivier pour les toits, Amine pour le marteau, Lie pour les chansons en chinois, L´eo pour les cartons, Adrien BG pour les capuccinos, Macarena et Valentin pour avoir apport´etant de couleur aux discutions sur l'enseignement, Joaquin pour le mate, Louis pour m'avoir bien aim´ed`esle d´epart et tant d'autres que j'ai oubli´e.Merci au Blond pour tous les g^ateauxqui ont certainement am´elior´ela vie des th´esardsdu couloir 15-16 et partout ailleurs. Merci `aNicolina et `aCristina, qui ont ramen´eun peu de l'Universit´ede Bucarest `al'IMJ. Je tiens aussi `aremercier mes amis de Nice, o`uj'ai ´et´etr`eschaleureusement accueillie d`esle d´epart: d'abord Jean-Baptiste et Agn`esde m'avoir h´eberg´e,et surtout `aJB pour avoir ´et´emon principal soutien math´ematiqueet moral pendant cette ann´ee.Merci aussi `aCharles, Julie, Arthur et mon petit fr`erede th`eseJie pour les pauses midi tr`esagr´eablesque nous avons r´eguli`erement partag´es. Un grand merci `ama famille, surtout `ames parents et ma grande-m`ere,qui m'ont constamment soutenu et encourag´eavec tant d'amour pendant mes hauts et mes bas. It was a great privilege to work with Alessio Corti on an article that became a large part of this 5 thesis. I thank him for having directed me towards such an interesting topic in algebraic geometry, for being extremely resourceful when suggesting different approaches and, at certain key moments, saying exactly the right things at the right time (the opposite is also true). His help has been invaluable to my work and I thank him wholeheartedly. I would like to thank Ivan Cheltsov for having accepted to referee this thesis, for having read and verified its every detail. I am deeply grateful for his time and dedication, which has greatly improved the quality of this work. I was fortunate enough to have wonderful friends amidst my colleagues both in Paris and Nice. I would especially like to thank Eduard and Fernanda for having stuck with me through thick and thin with remarkable resilience. And last but certainly not least, a giant thank you to my friends from the PRAGMATIC autumn school, and by extension the algebraic geometry group at the Collegio Imperiale, who to me have always embodied the perfect equilibrium between work and fun : Diletta, Enrica, Fano, Andrea, Roberto, Giulio, Ketil, Alessandro, Andy and Mohammad. I am looking forward to seeing them at another conference soon. 6 Deux points de vue sur les vari´et´esde Fano : g´eom´etriedu diviseur anticanonique et classification des surfaces `asingularit´es1/3(1,1) Le sujet principal de cette th`eseest l'´etudedes vari´et´esde Fano, qui sont des objets centraux de la classification des vari´et´esalg´ebriques. La premi`erequestion abord´eeconcerne les vari´et´esde Fano lisses de dimension quatre. On cherche a ´etudierles potentielles singularit´esd'un diviseur anticanonique de sorte qu'on puisse les ´ecriresous une forme locale explicite. En tant qu'´etape interm´ediaire,on d´emontre aussi que ces points sont au plus des singularit´esterminales, c’est-`a-direles singularit´esles plus proches du cas lisse du point de vue de la g´eom´etriebirationnelle. On montre ensuite que ce dernier r´esultatse g´en´eraliseen dimension arbitraire en admettant une conjecture de non-annulation de Kawamata. De fa¸concompl´ementaire, on s'int´eresse`ades vari´et´esde Fano de dimension plus petite, mais ad- 1 mettant des singularit´es.Il s'agit des surfaces de del Pezzo ayant des singularit´esde type 3 (1; 1). Ceci est l'exemple le plus simple de singularit´erigide, c’est-`a-direqui reste inchang´ee`aune d´eformation Q-Gorenstein pr`es.On classifie enti`erement ces objets en trouvant 29 familles. On obtient ainsi un tableau contenant des mod`elesde ces surfaces, qui pour la plupart sont des intersections compl`etes dans des vari´et´estoriques. Ce travail s'inscrit dans un contexte plus large, o`ules auteurs de [OP] calculent leur cohomologie quantique pour ensuite v´erifier si les Conjectures A et B de [ACC+16] sont valides dans ces cas. Bien que ces probl´ematiquessoient tr`esproches, elles font appel `adeux points de vue oppos´ees. L'´etudedes surfaces, abord´ee au Chapitre 7, commence par l'utilisation d'une variante du pro- gramme des mod`elesminimaux afin de les construire, et se conclut gr^ace`ades mod`eles obtenus par des techniques informatiques. Le cas des vari´et´esde Fano de dimension quatre utilise des techniques modernes des singularit´esdes paires pour retrouver le r´esultatde terminalit´e.Ce qui donne lieu `a une ´etudede cas purement locale, alors que les m´ethodes sus-cit´eesdans le cas des surfaces sont globales. 7 Vari´et´esde Fano et syst`emesanticanoniques La probl´ematiqueprincipale provient du constat suivant : en dimension strictement inf´erieure`a quatre, soit le syst`eme anticanonique j − KX j est globalement engendr´e,soit, par un r´esultatnon trivial de Shokurov, un diviseur g´en´eral D 2 j − KX j est lisse. Par contre, en dimension quatre, il existe des exemples o`uun tel D est singulier en un point. Les travaux de H¨oringet Voisin [HV11] montrent qu'il s'agit du cas extr´emal,i.e. D ne peut ^etrequ’`asingularit´esisol´ees.Le but du Chapitre 3 est de trouver une forme explicite des ´equationslocales de D afin de d´ecrire pr´ecis´ement ses singularit´es.En utilisant des m´ethodes des singularit´esdes paires, on relie la g´eom´etried'une vari´et´ede Fano ambiante X avec celle des diviseurs anticanoniques et on obtient le premier r´esultat suivant : Theorem 0.1. Soit X une vari´et´ede Fano de dimension quatre et D 2 j−KX j un ´el´ement g´en´eral. Alors D a au plus des singularit´esterminales. On remarquera que la notion de singularit´eterminale, introduite dans le deuxi`emechapitre, est le bon ´equivalent du r´esultatde Shokurov en dimension trois, puisque si D est une surface, un point terminal est bien un point lisse. On peut ensuite am´eliorerce r´esultat.En effet les singularit´es terminales des vari´et´esde dimension trois, bien que classifi´ees,sont en nombre infini, ce qui n'est pas le cas pour les familles de d´eformations de vari´et´esde Fano ambiantes. Il s'agit du point de d´epartdes travaux pr´esent´esensuite.
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