Algebraic Geometry in Angers

On extension of pluricanonical forms defined on the central fiber of a Kähler family

We will report on a recent joint work with M. Paun. We obtain an extension criterion for the canonical forms defined on an infinitesimal neighborhood of the central fiber of a family of Kähler manifolds. We will present some aspects of the proof of our main result, as well as its application to the extension of pluricanonical forms.

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On an effective YTD conjecture for multiplicity free manifolds

The YTD conjecture relates the existence of constant scalar curvature Kähler metrics with an algebro-geometric condition called K-stability. The correct notion of K-stability to be used in this correspondence is not known precisely. An effective version of the YTD conjecture would be involving a K-stability condition that can be checked effectively on large families of examples. The resolution of the YTD conjecture for Kähler-Einstein metrics by Chen, Donaldson and Sun provided striking examples of this: in this case, the K-stability condition can be checked effectively for manifolds with a large group action such as multiplicity free manifolds or T-manifolds of complexity one. In this talk, I will report on various recent advances and ongoing work on the YTD conjecture and more precisely on an effective YTD conjecture for cscK metrics on multiplicity free manifolds.

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A decomposition theorem for singular Calabi-Yau varieties

Let $X$ be a compact Kähler manifold with trivial first Chern class. The Beauville-Bogomolov decomposition theorem (1984) asserts that there exists a finite unramified cover $X'\to X$ such that $X'$ is a product of a torus by irreducible varieties of two types (Calabi-Yau or holomorphic symplectic). The generalization of that statement to singular varieties arising from the Minimal Model Program was obtained in 2017 by Höring-Peternell in the projective case, relying among other things on earlier works of Druel and Greb-Guenancia-Kebekus. In this talk, I will explain how to obtain the general Kähler case thanks to deformation theoretic arguments. This is joint work with Ben Bakker and Christian Lehn.

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Bogomolov-Guan manifolds

The only known example of a non-Kähler irreducible holomorphic symplectic manifold was described in works of Bogomolov and Guan. I am going to explain the construction and tell some results about the geometry of such manifolds. In particular, I will show that the algebraic reduction of a BG-manifold is isomorphic to a projective space and prove that the group of regular automorphisms of it is Jordan. It is a joint work with F. Bogomolov, N. Kurnosov and E. Yasinsky.

Recent progress on the existence of minimal models

In this talk I will present some recent progress on the existence of minimal models for varieties with mild singularities, as well as on the termination of flips in low dimensions. This progress is made possible by considering the category of generalised pairs, which appears naturally in several contexts. I will present joint work with Nikolaos Tsakanikas and touch upon recent works of Chen-Tsakanikas and Moraga.

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Rationality of Fano 3-folds over nonclosed fields

The rationality problem for smooth Fano threefolds over algebraically closed fields is basically solved. In this talk I will discuss rationality of forms of these Fanos over nonclosed fields of characteristic 0, as well as, rationality singular Fano 3-folds with mild singularities. I will concentrate on the case where the Picard number equals 1. Part of the results are based on joint works with Alexander Kuznetsov.

Generalised Kummer structures and moduli of generalised Kummer surfaces

A generalised Kummer surface X is the resolution of the quotient of a complex 2-torus A by an order 3 automorphism group G_A. If (B,G_B) is another complex 2-torus such that the associated generalised Kummer surface is isomorphic to X, we say that (B,G_B) is a generalised Kummer structure on X. The aim of the talk will be to understand the number of these generalised Kummer structures, and the moduli space of these surfaces, when suitably polarised (in a certain sense when the surface is non-algebraic). We will see that when X is algebraic or not the results are quite different.

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