HYPERKÄHLER EVENT

June 19-23, 2019
Laboratory of Algebraic Geometry


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Andrea Cattaneo
Title: Real structures on hyperkähler manifolds
Abstract: One of the problems lying on the boundary berween complex and real geometry is the classification of the so called real structures on a given manifold. Given a complex manifold, a real structure is nothing but an an anti-holomorphic diffeomorphism of order two of the underlying real manifold, which plays the role of complex conjugation. The fixed locus of such an involution is called a real form of the manifold, and in fact we can recover our original manifold by extension of scalars from any of its real forms. Of course, a manifold can have pairwise non-isomorphic real forms and the first problem is to decide wether there are only finitely many of them. The answer to this question is known in low dimension, i.e., for curves and surfaces, and in this talk we will provide an answer in the case where the manifold under consideration is hyperkähler, showing the finiteness (up to isomorphism) of the number of real forms in this case.

Grzegorz Kapustka
Title: Exceptional divisors of contractions of hyper-Kahler fourfolds.
Abstract: We study the geometry of conic bundles being exceptional divisors of birational contractions of hyper-Kahler fourfolds.

We relate them to constructions from classical geometry. This is a joint work in progress with B.van Geemen.

Grzegorz Kapustka
Title: Fano manifolds of K3 type and quadric fibrations.
Abstract: In order to construct Fano manifolds of K3 type, we study birational transformations of quadric fibrations associated to elementary transformations of Azumaya algebras. This is a joint work in progress with G.Bini, M.Kapustka, R.Laterveer

Enrica Mazzon
Title: Berkovich approach to hyper-Kähler degenerations and character varieties.
Abstract: In the late nineteen-nineties Berkovich developed a new approach to non-archimedean analytic geometry. This theory has quickly found many applications in algebraic and arithmetic geometry. In particular it turned out that there are strong and interesting connections between Berkovich spaces, birational geometry and mirror symmetry. In these talks, I will introduce the central objects of this theory: dual complexes, weight functions and essential skeletons. As an application, I will explain how the non-archimedean approach applies to the study of some degenerations of hyper-Kähler varieties, and to the study of character varieties, central objects in non-abelian Hodge theory.

Giovanni Mongardi
Title: All what you wanted to ask about OG6 but never dared to ask.
Abstract: In this talk, I will review most of the works on O'Grady six dimensional manifold, from its construction to recent results on its Hodge numbers, monodromy group and birational geometry.

Andrey Soldatenkov
Title: Andre motives of generalized Kummer type varieties.
Abstract: I will explain how to use the Kuga-Satake construction for hyperkähler manifolds to prove that, in some cases, their motives in the sense of Deligne or Andre are abelian. As an application I will show that on projective deformations of generalized Kummer varieties all Hodge classes are absolute.

Egor Yasinsky
Title: Infinite automorphism groups of hyperkahler manifolds.
Abstract: We survey results on infinite automorphism groups (both birational and biregular) of hyperkahler manifolds. Although there are many results about their finite subgroups (especially in the case of K3 surfaces), we are still far from understanding of the "global" structure of such groups.