Schedule & Programme

Schedule

Thursday

13:30-14:30: Schuett

14:45-15:45: Billi

16:15-17:15: Halle

Friday

9-10: Onorati

10:15-11:15: Tschanz

11:45-12:45: Buelli

Lunch break

14:15-15:15: Verni

15:30-16:30: Thompson

Abstract

Simone Billi: On maximality of involutions for Hilbert schemes of points

The Smith inequality estimates the total Betti numbers with coefficients F_2 of the fixed locus of an involution. This gives for example topological constraints for the real locus of a complex manifold with a real structure, which is given by an anti-holomorphic involution. Of particular interest are the involutions for which the Smith inequality is in fact an equality, these are called maximal.

We give a criterion for when a (anti-)holomorphic involution on a surface S with h^1(S,F_2)=0 induces a maximal involution on the Hilbert scheme of n points S^[n]. We moreover prove that if S is a K3 surface, then there are no maximal (anti-)holomorphic involutions on any deformation of S^[n]. In other words, there are no maximal brane involutions on hyper-Kähler manifolds of K3^[n] type, n=>2. (joint with L. Fu, A. Grossi and V. Kharlamov)

Ludovica Buelli: Monodromy of moduli spaces of sheaves on Abelian surfaces

The locally trivial monodromy group is an important deformation invariant for irreducible symplectic varieties and plays a fundamental role in the study of their birational geometry. In this talk, I will give a description of the locally trivial monodromy group of moduli spaces of semistable sheaves on Abelian surfaces, with non-primitive Mukai vector. The outcome is that the locally trivial monodromy group of a singular moduli space of this type is isomorphic to the monodromy group of a smooth moduli space, extending Markman's and Mongardi's description to the non-primitive case. As a geometric application, I will sketch a proof of the SYZ conjecture for any singular moduli space of this type.

Lars Halle: Degenerations of generalized Kummer varieties

In this talk, I will present a method for constructing explicit degenerations of generalized Kummer varieties, for any n, when the underlying abelian surface admits a type 2 Kulikov degeneration. I will moreover discuss some features of these degenerations. This is joint work with K. Hulek and Z. Zhang.

Claudio Onorati: Birational automorphism groups in families of hyper-Kähler manifolds

Given a non-trivial polarised family of hyper-Kähler manifolds, we study the behavior of the groups of birational transformations of the members of the family. We prove that, at least when the members belong to one of the known deformation classes, there always exists a dense subset on the base, where this group is infinite for any member in this subset. This is a joint work with F. Denisi, F. Rizzo and S. Viktorova.

Matthias Schuett: Numerically and cohomologically trivial automorphisms of elliptic surfaces

Numerically and cohomologically trivial automorphisms form a classical topic in algebraic geometry, especially for algebraic surfaces. I will focus on the case of complex elliptic surfaces of Kodaira dimension one, which features some surprising results, also with respect to past claims. Much of this can be motivated by the instructive case of Enriques surfaces.

Alan Thompson: Mirror symmetry for fibrations and degenerations of K3 surfaces

I will describe recent progress on the problem of understanding the mirror symmetric correspondence between Type II degenerations on a K3 surface and elliptic fibrations on its mirror. Specifically, I will describe Type II degenerations of a K3 surface polarised by a certain rank 18 lattice, where the central fibre is a pair of rational surfaces glued along an anticanonical elliptic curve. The mirror to such a K3 surface is a generic elliptically fibred K3 (with section), and I will show how this fibration may be split into two pieces which look like the mirrors to the two components of the degenerate fibre. Finally, I will describe how this special case gives a potential approach to understanding the general mirror correspondence. This is joint work with Luca Giovenzana.

Calla Tschanz: From logarithmic Hilbert schemes to degenerations of hyperkähler varieties 

In this talk, I will discuss my previous work on constructing explicit models of logarithmic Hilbert schemes. This relates to work or Li-Wu on expanded degenerations, Gulbrandsen-Halle-Hulek on degenerations of Hilbert schemes of points and Maulik-Ranganathan on logarithmic Hilbert schemes. The constructions I consider are local. I will then explain how we globalise these in joint work with Shafi and apply them to construct minimal type III degenerations of hyperkähler varieties, namely Hilbert schemes of points on K3 surfaces. 

Matteo Verni: Topological Brauer group of Kum_2n-type varieties and relations to Enriques manifolds

The topological Brauer group of a smooth projective complex variety is by definition the torsion part of its third degree Betti cohomology. It is an important stale birational invariant, for example it has been used by Artin and Mumford to prove the existence of non-stably rational, unirational varieties. One can see it as the "purely topological" part of the Brauer group: for this reason, understanding the former is a natural problem arising when one works with the latter, for example in hyper-Kähler geometry. The topological Brauer group of hyper-Kähler varieties has been computed only in the cases of K3^n-type and Kum_2-type manifolds, and it turns out to be zero. One is then led to speculate whether it always vanishes for hyper-Kähler manifolds. In this talk, we confirm this for all Kum_2n-type manifolds, extending the previously known case of n=1, and give a low bound on its torsion for all other Kum-type manifolds: this is joint work with Moritz Hartlieb. If time permits, we will mention progress on the closely related problem of computing Brauer groups of Enriques manifolds: this is joint work with Alessandro Frassineti, Francesca Rizzo and Federico Tufo.