Titles and abstracts

  • Frédéric Campana: Critère d'algébricité pour les feuilletages sur les variétés projectives complexes

Résumé : Si $X$ est une variété complexe projective et $F$ un feuilletage holomorphe sur $X$, une condition simple de positivité assure que les feuilles de $F$ sont toutes algébriques, c'est-à-dire que l'on a une fibration rationnelle $f:X\to Z$ dont les fibres génériques sont des adhérences de feuilles. Ce résultat a des applications dans différentes directions: caractérisation des variétés rationnellement connexes, décomposition de Bogomolov-Beauville des variétés singulières à première classe de Chern nulle, conjecture de Shafarevich-Viehweg. Il s'agit d'un travail issu d'idées dues à Miyaoka, Mori, Bogomolov-Mc Quillan, puis developpées en collaboration avec Thomas Peternell, Mihai Paun, et pour l'application aux variétés avec $c_1=0$, d'une collaboration entre Thomas Peternell et nombre de ses collaborateurs, dont Andreas Hoering.



  • Andreas Höring: Stein complements in projective manifolds

Abstract: Let X be a complex projective manifold, and let Y be a prime divisor in X. If Y is ample, it is well-known that the complement X \setminus Y is an affine variety. Vice versa, assume that X \setminus Y is affine, or more generally a Stein manifold. Then X \setminus Y does not contain any curve, in particular Y has positive intersection with every curve. This leads to our main question: if X \setminus Y is Stein, what can we say about the normal bundle of Y ? After some general considerations I will focus on the case where Y is the projectivised tangent bundle of some manifold M, and X is a ``canonical extension''. We will see that the Stein property leads to many restrictions on the birational geometry of M. This is joint work with Thomas Peternell.


  • Vlad Lazić: The MMP for generalised pairs and applications to the geometry of varieties

Abstract: I will present recent developments in the birational geometry of generalised pairs, which were recently introduced to deal with outstanding conjectures in the MMP. In particular, I will discuss work with Thomas Peternell on the Generalised Abundance conjecture, which has surprising applications to the geometry of varieties, even in low dimensions.


  • Gianluca Pacienza: Deformations of rational curves on primitive symplectic varieties and applications.

Abstract: The study of « singular » holomorphic symplectic varieties is mainly motivated by the singular version of Beauville-Bogomolov decomposition theorem, obtained by Thomas Peternell, together with several other mathematicians.

In the talk I will present a joint work with Ch. Lehn and G. Mongardi in which we study the deformation theory of rational curves on a (possibly singular) primitive symplectic variety and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus. As applications of our technique, I will present the extension of Markman's deformation invariance of prime exceptional divisors to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties which are locally trivial deformation of any moduli space of sheaves on a projective K3 surface or fibers of the Albanese map of those on an abelian surface.

  • Mihai Păun: On a theorem of Green-Lazarsfeld.

Abstract: We will report on a joint work in progress with J. Cao and Y. Deng. One of our main results is a new proof of the structure of the so-called Green-Lazarsfeld sets, (namely that they are translate of subtori). The main technique involved in our arguments is an extension results for adjoint line bundles whose twisting is not semi-positive.