Program
Each lecturer will deliver 5 lectures on the following topics.
Olivier Debarre: "The geometry of Gushel-Mukai varieties"
Gushel–Mukai varieties are Fano varieties of dimensions 3, 4, 5 or 6 which are intimately related to double EPW sextics, which are hyper-Kähler varieties of dimension 4 (this relation is very similar to the relation between another important class of Fano varieties, cubic fourfolds, and the hyper-Kähler variety of lines contained in the cubic). This rich connection gives a lot of information on the geometry and periods of Gushel–Mukai varieties. I will review the many classical constructions that relate these families of varieties, explain what is known and what remains to be proved.
Christian Lehn: "Singular varieties with trivial canonical class"
In this series of lectures, we will discuss the recent advances in the field of singular varieties with trivial canonical class. This includes the decomposition theorem which says that such a variety is up to a finite cover isomorphic to a product of a torus, a Calabi-Yau and an irreducible symplectic variety. For irreducible symplectic varieties, we explain in more detail the recent advances in understanding their bimeromorphic geometry and the Kähler cone.
Emanuele Macrì: "Non commutative K3 surfaces, with application to hyper-Kähler and Fano manifolds"
The aim of this series of lectures is to explain the relation between polarized hyper-Kähler manifolds, non-commutative K3 surfaces, and certain Fano manifolds. To start with, I will introduce non-commutative K3 surfaces and moduli spaces of objects in them. Then I will present several examples, including non-commutative K3 surfaces associated to cubic fourfolds and Gushel-Mukai manifolds. Finally, we will discuss possible ways to associate a non-commutative K3 surface to a polarized hyper-Kähler fourfold and possible applications to Chow groups.
Kieran G. O'Grady: "The geometry of projective HyperKähler manifolds"
Projective K3 surfaces play a prominent rôle in Algebraic Geometry. They have a very rich geometry, like rational surfaces, and on the other hand they are linked to abelian varieties. K3 surfaces are the 2 dimensional HyperKähler manifolds. Higher dimensional projective HyperKähler manifolds have a geometry which is at least as rich as that of K3's, but of course it is not as well understood as in dimension 2. The aim of the course is to show examples of this intriguing geometry.
Link to the O'Grady's notes: https://www1.mat.uniroma1.it/people/ogrady/index-cetraro-2022.html
Short contributed talks:
Chenyou Bai: "Lagrangian families of hyper-Kähler manifolds"
Abstract: Motivated by the conjectural plan proposed by Beauville and Voisin on the algebraic cycles in projective hyper-Kähler manifolds, we propose to consider the following question: given two Lagrangian subvarieties that can be deformed from one to the other in a projective hyper-Kähler manifold, are they necessarily rationally equivalent to each other? I will try to explain why we consider this question, to what extent we may expect it gives a positive answer, and why it gives in general a negative answer, adding to the subtlety of Voisin's conjecture. Article reference: https://arxiv.org/abs/2203.06242
Franco Giovenzana: "Hodge structure of O'Grady's singular moduli spaces"
Abstract: O'Grady's Hyperkähler manifolds are constructed as resolutions of some moduli spaces of sheaves on a K3 surface or an abelian surface. Despite the cohomology of O'Grady's manifolds is well known, the same is not true for singular O'Grady's varieties. We investigate the Hodge structure of the singular moduli spaces and we draw some conclusion on their singularities. (from joint work with V. Bertini). Article reference: https://arxiv.org/abs/2203.07917
Lisa Marquand: "Classification of Symplectic Birational involutions of manifolds of OG10 type"
Abstract: In this talk, we will obtain a partial classification of birational symplectic involutions of manifolds of OG10 type. We do this from two vantage points: firstly following classical techniques relating birational transformations to automorphisms of the Leech lattice. Secondly, we look at involutions that are obtained from cubic fourfolds via the intermediate Jacobian construction. An involution of a cubic fourfold induces a birational transformation of the corresponding compactified intermediate Jacobian. We use the classification of involutions of a cubic fourfold previously studied to obtain a classification of symplectic birational involutions that fix a copy of the hyperbolic lattice U. If time permits, we will mention on going work to complete the classification. Article reference: https://arxiv.org/abs/2206.13814
Mihai Pavel: "Moduli spaces of semistable sheaves"
Abstract: In this talk we present the construction of some moduli spaces of semistable sheaves over a smooth projective variety (over the field of complex numbers). We will use a notion of stability for pure coherent sheaves, which lies between Gieseker- and slope-stability. This is defined with respect to the Hilbert polynomial of the sheaf, truncated up to a certain degree. We call it l-(semi)stability, where l marks the level of truncation. Before we proceed with the construction, we give a Mehta-Ramanathan type restriction theorem for l-(semi)stability, which applies in particular to Gieseker-semistable sheaves. With this ingredient in place, we construct moduli spaces of l-semistable sheaves in higher dimensions following ideas of Le Potier and Jun Li. We can think of these moduli spaces as intermediate compactifications between the Gieseker- and the Donaldson-Uhlenbeck compactification. Article reference: https://arxiv.org/abs/2105.09395 and https://arxiv.org/abs/2204.01762
Benedetta Piroddi: "K3 surfaces with a symplectic automorphism of order four"
Abstract: Given X a K3 surface admitting a symplectic automorphism t of order 4, we describe the isometry t* induced by t on the second integral cohomology lattice of X. Having called Z and Y respectively the minimal resolutions of the quotient surfaces X/t^2 and X/t, it is possible to describe the maps induced in cohomology by the rational quotient maps between X, Y, Z: with this knowledge, we are able to give a lattice-theoretic characterization of Z, and find the relation between the Néron-Severi lattices of X, Y and Z in the projective case. Article reference: https://arxiv.org/abs/2208.01962