Séminaires

Théorèmes de transfert pour la cohomologie galoisienne et conjecture de Serre II

La première partie de l’exposé sera consacrée à une présentation générale et accessible de la conjecture de Serre II, prédisant l’existence de points rationnels sur des torseurs sous certains groupes linéaires quand on travaille sur des corps de petite dimension cohomologique.

Dans la deuxième partie, je parlerai d’un travail récent avec Giancarlo Lucchini Arteche dans lequel on démontre notamment que la conjecture pour les corps de caractéristique nulle implique la conjecture pour les corps de caractéristique quelconque. Ce résultat repose notamment sur quelques théorèmes de transfert pour la dimension cohomologique des corps que j’énoncerai et expliquerai.

An elementary Mori contraction from a smooth variety $X$ is a morphism with connected fibres onto a normal variety which contracts a single extremal ray of $K_X$-negative curves. Thanks to a result by P. Ionescu and J. Wisniewsi, we know that the length of such a contraction (i.e. the minimal degree $-K_X$ can have on contracted rational curves) is bounded from above. In a paper which dates back to 2013, A. Höring and C. Novelli studied elementary Mori contractions of maximal length, that is, elementary Mori contractions for which the upper bound is met. Their main result exhibits the structure of a projective bundle for the locus of positive-dimensional fibres up to a birational modification. In my talk, I will move to the submaximal case, in other words the case where the length equals its upper bound minus one, and focus on the divisorial case.

Les variétés de Fano sont des variétés projectives complexes avec premier caractère de Chern positif. Cette condition de positivité a des implications profondes en géométrie et en arithmétique. Par exemple, les variétés de Fano sont recouvertes par des courbes rationnelles , et les familles de variétés de Fano sur des bases unidimensionnelles admettent toujours des sections holomorphes. Ces dernières années, il y a eu un effort important pour définir des analogues supérieurs à la condition de Fano, qui devraient présenter des versions renforcées de propriétés des variétés de Fano. Dans cet exposé, je parlerai donc des « variétés de Fano supérieures » définies en termes de positivité des autres caractères de Chern. Ce travail est en collaboration avec Carolina Araujo, Roya Beheshti, Ana-Maria Castravet, Kelly Jabbusch, Svetlana Makarova et Nivedita Viswanathan.

Extension of Differential Forms, Uniformization, Miyaoka-Yau inequalities and the topological characterization of certain klt varieties (with Daniel Greb and Thomas Peternell)

The first part of this overview talk begins with a non-technical overview of minimal model theory, explaining why any classification theory of complex-projective manifolds always needs to consider singular varieties. The talk describes the relevant singularities in brief, mentions methods that have been developed to study them and will ideally convey an idea what classification results one might hope to expect.

The second part describes some of the theory that has been developed over the last years and mentions some of the more concrete applications.

To an algebraic variety X we associate its group of birational transformations Bir(X). In this talk, we will see the following theorem: If X is an algebraic variety such that Bir(X) is isomorphic to Bir(P^n), where P^n is the n-dimensional projective space, then X is birational to P^n. In other words, the group structure of Bir(X) determines whether X is rational or not. In another direction, I will explain that Borel subgroups of Bir(X), i.e. maximal connected solvable subgroups, are of derived length <= 2 dim(X) with equality if and only if X is rational and the Borel subgroup is standard. This is joint work with Regeta and Van Santen.

In this talk I will present a joint work with C. Casagrande, in which we study smooth complex Fano 4-folds with a rational fibration onto a 3-fold. After an introduction on the setting and motivation, I will discuss our main result: if X is Fano 4-fold with a rational fibration onto a 3-fold and it is not a product of surfaces, then the Picard number of X is at most 9, and the bound is sharp. Moreover, I will present a classification result in a special case within the setting above, and show new examples of Fano 4-folds with large Picard number.

The Shafarevich conjecture asks whether the universal covering of a compact Kähler manifold is holomorphically convex. In this work, we explore a similar question regarding the holomorphic convexity of intermediate coverings. We prove that if the intermediate covering of a compact Kähler surface admits a faithful reductive representation for its covering group and does not have two ends, then it is holomorphically convex. The main techniques employed include the analysis of the degeneracy loci of the Levi form and the properties of subanalytic functions.