Titles and Abstracts

Daniel Huybrechts (Bonn)

Opening Talk: The work of Manfred Lehn

Nick Addington (Oregon)

Hermitian-Einstein connections on universal bundles

For a smooth, compact moduli space of stable vector bundles on a K3 surface, we construct a Hermitian-Einstein connection on the universal bundle, which restricts to a H-E connection on its "wrong-way" slices. Thus not only does the moduli space parametrize stable bundles on the K3 surface, but the K3 surface parametrizes stable bundles on the moduli space. Ingredients include the Bismut-Freed-Quillen connection on a determinant line bundle, and a careful analysis of the curvature of a certain principal bundle coming from gauge theory. This is joint work with Andrew Wray. 

Samuel Boissière (Poitiers)

The Fano variety of lines of a cuspidal cyclic cubic fourfold

In the framework of the compactification of the moduli spaces of prime order non-symplectic automorphisms of irreducible holomorphic symplectic manifolds, a key question is to understand the geometry of limit automorphisms. Starting from a nodal degeneration of cubic threefold, the general member of the family of Fano varieties of lines of the triple covering branched over the cubic is an IHS manifold equiped with the automorphism induced by the covering. It degenerates to a variety whose singular locus is a K3 surface.

I will present recent results obtained in collaboration with Chiara Camere and Alessandra Sarti that explain how the geometry of this K3 surface permits to define a limit automorphism in a suitable moduli space parametrizing pairs of IHS manifolds with automorphism.

Lothar Göttsche (Trieste)

(Refined) Verlinde and Segre formulas for Hilbert schemes of points

This is joint work with Anton Mellit. Segre and Verlinde numbers of Hilbert schemes of points have been studied for a long time. The Segre numbers are evaluations of top Chern and Segre classes of so-called tautological bundles on Hilbert schemes of points. The Verlinde numbers are the holomorphic Euler characteristics of line bundles on these Hilbert schemes. We give the generating functions for the Segre and Verlinde numbers of Hilbert schemes of points. The formula is proven for surfaces with K_S^2=0, and conjectured in general. Without restriction on K_S^2 we prove the conjectured Verlinde-Segre correspondence relating Segre and Verlinde numbers of Hilbert schemes. Finally we find a generating function for finer invariants, which specialize to both the Segre and Verlinde numbers, giving some kind of explanation of the Verlinde-Segre correspondence. 

Emanuele Macrì (Orsay)

Antisymplectic involutions on projective hyper-Kähler manifolds

An involution of a projective hyper-Kähler manifold is called antisymplectic if it acts as (-1) on the space of global holomorphic 2-forms. I will present joint work with Laure Flapan, Kieran O’Grady, and Giulia Saccà on antisymplectic involutions associated to polarizations of degree 2. We study the number of connected components of the fixed loci and their geometry. In particular, if the divisibility of the ample class is 2, one connected component of the fixed locus is a Fano manifold of index 3, thus generalizing the case of cubic fourfolds. Still in the case of cubic fourfolds, the second component is of general type, thus answering a question by Manfred Lehn.

Sergey Mozgovoy (Dublin)

Stability conditions on surfaces

In this talk I will discuss Bridgeland stability conditions on the derived categories of surfaces. I will introduce the global dimension of a stability condition, explain its relevance in establishing wall-crossing formulas for the refined DT invariants, and describe the global dimension of a large family of stability conditions on weak Fano surfaces. 

Kieran O'Grady (Rome)

High dimensional moduli spaces of (semi)stable sheaves on polarized HK varieties of type K3^{[2]} with general moduli

We will exhibit irreducible components of moduli spaces of (semi)stable sheaves on polarized HK varieties of type K3^{[2]} with general moduli which are HK varieties of type K3^{[2]}, or birational to HK varieties of type K3^{[m]} with arbitrarily high m, or deformations of birational models of singular  symplectic varieties. Our work was motivated by examples given by Enrico Fatighenti.

Michael Rapoport (Bonn)

G-Bundles on curves with a twist

In a Bourbaki talk in the 50's, Grothendieck showed that a G-bundle on a complex curve can be described by an equivariant G-bundle on a Galois cover of the curve. He pointed out that not every equivariant G-bundle is obtained in this way if there are ramification points, and  posed the problem of describing equivariant G-bundles in terms of objects on the base curve. In the talk I will show that the theory of  Bruhat-Tits group schemes allows one to give a satisfactory answer to this problem. Joint work with G. Pappas. 

Alessandra Sarti (Poitiers)

On the cone conjecture for Enriques Manifolds

Enriques manifolds are non simply connected manifolds whose universal cover is irreducible holomorphic symplectic, and as such they are natural generalizations of Enriques surfaces. The goal of the talk is to prove the Morrison-Kawamata cone conjecture for such manifolds when the degree of the cover is prime using the analogous result (established by Amerik-Verbitsky) for their universal cover. If time permits I will also show the cone conjecture for the known examples having non-prime degree. This is a joint work with Gianluca Pacienza.

Paolo Stellari (Milano)

Deformations of stability conditions with applications to Hilbert schemes of points and very general abelian varieties

The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is notoriously a difficult problem, especially when the canonical bundle is trivial. In this talk, I will illustrate a new and very effective technique based on deformations. A key ingredient is a general result about deformations of bounded t-structures (and with some additional and mild assumptions). Two remarkable applications are the construction of stability conditions for very general abelian varieties in any dimension and for some irreducible holomorphic symplectic manifolds, again in all possible dimensions. This is joint work with C. Li, E. Macrì and X. Zhao. 

Duco van Straten (Mainz)

Paramodular forms and Calabi-Yau geometry

In the talk I describe how a specific Siegel paramodular form has a (conjectural) incarnation in a Calabi-Yau threefold that arose from mirror symmetry in a Grassmannian variety. This insight leads to an unexpected  congruence of this form with hilbert-modular form. This is joint work with Vasily Golyshev.

Claire Voisin (Paris)

Complete intersections of two quadrics and Lagrangian fibrations

We show that, on  the cotangent bundle of a  n-dimensional smooth complete intersection X  of two quadrics, there are n quadratic functions (which are sections of Sym^2T_X)  providing a Lagrangian fibration whose fibers can be described as quotients of Jacobians of hyperelliptic curves by a group of translations of order 2. For n=3, X is a moduli space of rank 2 bundles and the fibration is the Hitchin fibration.

This is joint work with Beauville, Etesse, Höring and Liu.

Schedule

Wednesday

10h00 Registration, welcome coffee

10h45 Opening (Huybrechts)

11h00 - 12h00 Voisin

14h00 - 15h00 Göttsche

15h00 - 15h30 Coffee

15h30 - 16h30 Sarti

16h45 - 17h45 Rapoport

 Thursday

09h30 - 10h30 Macrì

10h30 - 11h00 Coffee

11h00 - 12h00 Mozgovoy

14h00 - 15h00 O'Grady

15h00 - 15h30 Coffee

15h30 - 16h30 Addington

19h00 Dinner

Friday

09h30 - 10h30 Stellari

10h30 - 11h00 Coffee

11h00 - 12h00 Boissière

13h30 - 14h30 van Straten