Title and Abstract

A foliation F on X, complex projective smooth, is showed to have algebraic leaves if its dual is not pseudo-effective. In the particular case where F has positive minimal slope with respect to some movable class on X, the closures of the leaves are rationally connected. Combined with the existence of Viehweg-Zuo sheaves, this permits to show several versions of the Shafarevich-Viehweg ̀ hyperbolicity conjecture'. In the general case (dual of F not pseudo-effective), the algebraicity result leads to several questions (with presently few answers) about projective manifolds having a non pseudo-effective cotangent bundle. (Joint with M. P\u aun) Junyan Cao Title: Variation of singular Kähler-Einstein metrics Abstract: We study families of Kähler-Einstein metrics in various geometric situations and give some applications in complex geometry. First, we prove positivity of the variation of singular KE metrics in the general type case. Next, we analyze the log Calabi-Yau case and derive the local triviality of families of pairs for which the direct image of the log pluricanonical bundle is hermitian flat. It is a joint work with Henri Guenancia and Mihai Paun. Paolo Cascini Title: Minimal Model Program for foliations Abstract: I will discuss some recent results on the study of the birational geometry of foliations over complex projective threefolds. Joint work with C. Spicer. Fabrizio Catanese Title: Hypersurfaces in Abelian Varieties : Moduli spaces and canonical maps. Abstract: I shall mostly consider in my talk ample and smooth hypersurfaces X contained in an Abelian variety A, and investigate the open set of the moduli space of compact complex manifolds X' with ample canonical bundle which are topologically equivalent to X (this case is a special one of the more general investigation of Inoue type Varieties, initiated in joint work with Ingrid Bauer). Using recent joint work with Yongnam Lee on deformation to hypersurface embeddings, I shall show that not all such X’ are again Hypersurfaces in an AV, but we can calculate their limits. The classical theorem of Lefschetz gives sufficient conditions for the very ampleness of the linear system |X| on A, e.g. that the polarization is divisible by 3. Since the canonical divisor K_X of X is X restricted to X, we obtain very ampleness of the canonical system in the Lefschetz range. As a consequence of the embedding obstructions (given by Severi’s double point formulae), K_X cannot be very ample unless its dimension is at least 2 n + 1, where n = dim X. This can be translated into an inequality for the Pfaffian p of the Chern form of the divisor X, p >= n+2. At any rate, the canonical map of X is a finite map, and I will present several steps towards the proof of the Conjecture: Unless we have a principal polarization, then for general (A,X) the canonical map is birational onto its image. I shall then mention the results obtained by Cesarano in his Ph D Thesis, proving very ampleness for a general (A,X) in case of a polarization of type (1,2,2). His results motivates the following Question: if p >= n+2, then for general (A,X) the canonical map is an embedding. Ivan Cheltsov Title: Smooth Fano threefolds of degree 22 with infinite automorphism groups Abstract: This talk is about smooth Fano threefolds of Picard rank one and anticanonical degree 22 that have infinite automorphism groups. Donaldson conjectured that all but one of them must be Kahler-Einstein, see https://arxiv.org/abs/0803.0985. I will explain how to use equivariant birational geometry of these threefolds to prove Donaldson's conjecture (with 2 possible explicitly described exceptions). This is a joint work with Costya Shramov from Moscow, see https://arxiv.org/abs/1803.02774. Our proof is based on recent results by Sasha Kuznetsov and Yura Prokhorov obtained in https://arxiv.org/abs/1711.08504.