Program

For the booklet of the conference, please click here.

Speakers:

Benjamin Bakker (University of Illinois at Chicago)
Title: The Matsushita alternative
Abstract: Matsushita conjectured that a Lagrangian fibration of an irreducible hyperkahler manifold is either isotrivial or of maximal variation. In this talk I will show how to prove this conjecture by adapting previous work of Voisin and van Geemen. I will also deduce some applications to the density of torsion points of sections of Lagrangian fibrations.
Olivier Debarre (IMJ-PRG)
Title: Numerical invariants of hyper-Kähler varieties
Abstract: We prove some elementary results on the various constants attached to a hyper-Kähler variety, in particular, in the presence of a class that is isotropic for the Beauville-Fujiki form.
Lie Fu (Université de Strasbourg)
Title: Unpolarized Shafarevich conjecture for hyperkähler varieties
Abstract: Extending the Shafarevich conjecture for hyperkähler varieties proved by André, we show the following unpolarized version: there are only finitely many hyperkähler varieties in a fixed deformation class defined over a number field and with good reductions outside a finite collection of places. This generalizes the result on K3 surfaces by Yiwei She. It is a joint work with Zhiyuan Li, Teppei Takamatsu, and Haitao Zou.
Grzegorz Kapustka (Jagellonian University)
Title: Nikulin orbifolds
Abstract: We describe a locally complete family of projective irreducible holomorphic symplectic orbifolds as double covers of special complete intersections (3, 4) in P^6. This is a joint work with C. Camere, A. Garbagnati and M. Kapustka.
Christian Lehn (TU Chemnitz)
Title: Non-hyperbolicity of symplectic varieties
Abstract: In a joint work with Ljudmila Kamenova, we investigate the Kobayashi pseudometric for irreducible symplectic manifolds. Assuming the SYZ conjecture, we prove that the Kobayashi pseudometric vanishes identically on every irreducible symplectic manifold of second Betti number at least seven. This in particular shows the claim for all known examples. Our results also hold for singular symplectic varieties.
Emanuele Macrì (Université Paris-Saclay)
Title: The geometry of linear systems of low degree on hyper-Kähler fourfolds, II
Abstract: In a recent joint work with O. Debarre, D. Huybrechts, and C.Voisin, we established a topological characterization of hyper-Kähler fourfolds of K3^{[2]} deformation type. The criterion is the existence of two lagrangian integral cohomology classes l, m on X with intersection number l^2m^2=2. In the course of the proof, we can assume that the two classes are Chern classes of line bundles L, M on X. In this talk, I will present the second part of the proof, where the two classes are Chern classes of line bundles L, M on X, with L and M not both nef.
Laurent Manivel (Université Paul Sabatier)
Title: A birational involution in genus 10
Abstract: Birational automorphisms of Hilbert schemes of points on algebraic K3 surfaces of Picard rank one were recently classified by Beri and Cattaneo. In particular they predicted the existence of a birational involution of the Hilbert cube of a general K3 surface of genus 10. I will describe this involution explicitely, in terms of the Mukai model of the surface.(Joint work with Pietro Beri.)
Eyal Markman (University of Massachussets)
Title: Rational Hodge isometries of hyper-Kähler varieties of K3^[n] and generalized Kummer type are algebraic
Abstract: Let X and Y be compact hyper-Kähler manifolds of K3^[n] deformation type. A cohomology class in their product XxY is an analytic correspondence, if it belongs to the subalgebra generated by Chern classes of coherent analytic sheaves. Let f be a Hodge isometry of the second rational cohomologies of X and Y with respect to the Beauville-Bogomolov-Fujiki pairings. We prove that f is induced by an analytic correspondence. We furthermore lift f to an analytic correspondence F between their total rational cohomologies, which is a Hodge isometry with respect to the Mukai pairings, and which preserves the gradings up to sign. When X and Y are projective the correspondences f and F are algebraic. We will also comment on the analogous result for the generalized Kummer deformation type (work in progress).
Mirko Mauri (Institute of Science and Technology Austria)
Title: Hodge-to-singular correspondence for reduced curves
Abstract: We show that the cohomology of some non-compact hyperkähler moduli spaces of Higgs bundles decomposes in elementary summands depending on the topology of the symplectic singularities on a (fixed!) master object. This is based on a joint work with Luca Migliorini.
Giovanni Mongardi (Università di Bologna)
Title: Kawamata-Morrison cone conjecture for singular HK, I
Abstract: In this talk, we will outline a fundamental step for the proof of the Kawamata-Morrison cone conjecture, which is of independent interest. We generalize results due to Markman in the smooth setting, proving that the reflection by a prime exceptional divisor is a monodromy operator for mildly singular HK varieties. This is joint work with Ch. Lehn and G. Pacienza.
Gianluca Pacienza (Université de Lorraine)
Title: Kawamata-Morrison cone conjecture for singular HK, II
Abstract: In this talk, we overview the strategy of the proof of the cone conjecture in the singular setting, and explain in detail the steps of the proof. This is Joint work with Ch. Lehn and G. Mongardi
Laura Pertusi (Università di Milano)
Title: Moduli spaces of stable objects in Enriques categories.
Abstract: Enriques categories are characterized by the property that their Serre functor is the composition of an involutive autoequivalence and the shift by 2. The bounded derived category of an Enriques surface is an example of Enriques category. Other interesting examples are provided by the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids. In this talk, we study moduli spaces of semistable objects in the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids with respect to Serre-invariant stability conditions. We provide a result of non-emptyness for these moduli spaces, by using the relation with certain moduli spaces on the associated K3 category. This is a joint work in preparation with Alex Perry and Xiaolei Zhao.
Giulia Saccà (Columbia University)
Title: Moduli spaces as Irreducible Symplectic Varieties
Abstract: Recent developments by Druel, Greb-Guenancia-Kebekus, Horing-Peternell have led to the formulation of a decomposition theorem for singular (klt) projective varieties with numerical trivial canonical class. Irreducible symplectic varieties are one the building blocks provided by this theorem, and the singular analogue of irreducible hyper-Kahler manifolds. In this talk I will show that moduli spaces of Bridgeland stable objects on the Kuznetsov component of a cubic fourfold with respect to a generic stability condition are always projective irreducible symplectic varieties. This builds on the recent work of Bayer-Lahoz-Macri-Neuer-Perry-Stellari, which, ending a long series of results by several authors, proved the analogue statement in the smooth case.
Alessandra Sarti (Université de Poitiers)
Title: Degenerations of IHS manifolds with automorphism
Abstract: In a previous paper in collaboration with Boissière and Camere we showed an isomorphism between the moduli space of smooth cubic threefolds and some IHS manifolds with non-symplectic automorphism of order three, by using the fact that these are isomorphic to a certain arithmetic quotient of a complement of an arrangement of hyperplanes in a 10-dimensional complex ball. In the talk I plan to recall briefly the isomorphism and in particular to explain what happen if one considers some degenerations of IHS manifolds which correspond to points in the hyperplanes arrangement corresponding to nodal cubic threefolds. I will show how the original automorphism can not be extended nor it can become birational and how this automorphism ''changes type''. As a motivation one should keep in mind what happen for the covering involution of a K3 surface which is a double plane ramified on a smooth sextic, respectively on a sextic with a node: the invariant lattice changes. I will then use some birational map between these IHS manifolds that are in fact Fano varieties of lines of singular cubic fourfolds and the Hilbert scheme of two points on a K3 surface which is strictly related to the previous manifolds, to understand in a concrete way the degeneration of the automorphism. The results come from a joint paper with Boissière and Camere and from a work in progress with Boissière.
Lambertus Van Geemen (Università di Milano)
Title: Contractions of hyper-Kähler fourfolds and the Brauer group
Abstract: The exceptional locus of a birational contraction on a hyper-Kähler fourfold of $K3^{[2]}$-type is a conic bundle over a K3 surface. These conic bundles are projectivized (twisted) rank two vector bundles. We discuss the associated Mukai vectors, Brauer classes (B-fields) and Heegner divisors. We also give various examples of such conic bundles.
Misha Verbitsky (IMPA)
Title: TBA
Abstract: TBA
Claire Voisin (IMJ-PRG)
Title: The geometry of linear systems of low degree on hyper-Kähler fourfolds I
Abstract: In a recent joint work with O. Debarre, D. Huybrechts, and E. Macrì, we established a topological characterization of hyper-Kähler fourfolds of K3^{[2]} deformation type. The criterion is the existence of two lagrangian integral cohomology classes l, m on X with intersection number l^2m^2=2. In the course of the proof, we can assume that the two classes are Chern classes of line bundles L, M on X. In this talk, I will discuss one part of the proof, in which we exclude the case where both L and M are nef.

Support talks:

Chenyu Bai (IMJ-PRG)
Title: The geometry of G2 and Mukai model of genus 10
Abstract: The general K3 surfaces of genus 10 are complete intersections in a rational homogeneous variety defined as a quotient of the exceptional Lie group G2. In this talk, we will discuss the geometry of G2 and the Mukai model of genus 10. Specially, we will discuss how the two fundamental representations of G2 are related in the geometric context. This is a support talk for Laurent Manivel's talk "A birational involution in genus 10".
Pietro Beri (IMJ-PRG)
Title: Double EPW sextics and O'Grady's conjecture
Abstract: Double EPW sextics are an important family of Hyper-Kähler manifolds, described by O'Grady in 2006. We review their construction and their basic properties.We present their relation with O'Grady's conjecture about numerical $K3^{[2]}$, which states: "Let $X$ be a Hyper-Kähler fourfold. Suppose that the integral second cohomology group of $X$ is isometric, as a lattice with the BBF form, to the second cohomology group of an Hilbert square on a K3 surface. Suppose moreover that the Fujiki constant of $X$ is the same as an Hilbert square on a K3 surface. Then $X$ is deformation equivalent to an Hilbert square on a K3 surface".
Valeria Bertini (Porto)
Title: An overview on singular symplectic varieties
Abstract: The theory of singular symplectic varieties has attracted increasing attention in recent years, after the singular Beauville-Bogomolov decomposition theorem and as they behave similarly to their smooth analogue in many fundamental aspects. In this talk I will introduce and compare some different notions of singular symplectic varieties, giving an overview of their theory and some examples of them.
Alessio Bottini (Tor Vergata)
Title: Derived categories of Hyper-Kähler varieties
Abstract: In this talk we will review some recent results due to Taelman, Beckmann and Markman on the structure of derived equivalences of Hyper-Kähler varieties. We will recall the notion of the Looijenga-Lunts-Verbitsky Lie algebra attached to a Hyper-Kähler manifold, and explain that it is invariant under derived equivalences. Time permitting, we will see how this fact allows one to define a Mukai vector with values in a low dimensional vector space for a restricted class of objects, called "atomic".
Francesco Denisi (Université de Lorraine)
Title: Some aspects of the (birational) geometry of hyperkähler manifolds
Abstract: The purpose of this talk is to review some topics in hyperkähler geometry, such as the Kawamata-Morrison cone conjecture. We will highlight the link between the birational geometry of hyperkähler manifolds and convex geometry and, time permitting, we will discuss when a hyperkähler manifold is a Mori dream space.
Salvatore Floccari (IAG Hannover)
Title: Moduli of polarized hyperkähler varieties
Abstract: My talk will be a basic exposition of hyperkähler varieties in an arithmetic setting. The main point discussed will be the construction of the moduli space, or rather stack, of polarized hyperkähler varieties over Q and of the associated period map.
Lucas Li Bassi (Université de Poitiers)
Title: Cubic threefolds and IHS manifolds with a non-symplectic automorphism

Abstract: In a paper of 2019 Boissi`ere-Camere-Sarti prove that there exists an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of IHS fourfolds of K3 [2]-type with a non-symplectic automorphism of order three, whose invariant lattice is generated by a class of square 6. They study then the degeneration of the auto-morphism along the locus of the generic nodal degeneration of a cubic threefold, showing the birationality of this locus with some moduli space of IHS fourfolds of K3[2]-type with a nonsymplectic automorphism of order three belonging to a different family. I will present their results together with an implementation which consents to go further in the analysis of the degeneration and give a similar result also in the non-generic nodal case.