Schedule for: 23w5100 - Geometry of HyperKahler Varieties

A set dinner is served daily between 5:30pm and 8:30pm in the Xianghu Lake National Tourist Resort.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

A brief introduction to IASM with important logistical information, technology instruction, and opportunity for participants to ask questions.

The perverse filtration is an important topological invariant associated with a proper morphism; it generalizes the Leray filtration for a smooth morphism. In this talk, I will discuss two surprising conjectures concerning perverse filtrations associated with abelian fibrations. The first is the P=W conjecture where the perverse filtration for the Hitchin system plays a key role. The second is the Oblomkov-Rasmussen-Shende conjecture concerning the perverse filtration for a compactified Jacobian associated with a locally planar singularity. In both cases the perverse filtrations are expected to satisfy some common features, namely the multiplicativity and the “Perverse= Chern” phenomenon. I will discuss these features and a uniform method explaining them. Our main tool is an extension of the Beauville decomposition for abelian schemes. Joint work in progress with Davesh Maulik and Qizheng Yin.

This serves as a complement to Junliang Shen’s talk. We explain how to extend the Beauville decomposition from abelian varieties/schemes to certain abelian fibrations with singular fibers, and how this extension is related to the multiplicativity of the perverse filtration and the “Perverse = Chern” phenomenon. Joint work in progress with Davesh Maulik and Junliang Shen.

Lunch is served daily between 12:00am and 1:30pm in the Xianghu Lake National Tourist Resort

Given a six dimensional vector space V, two families of explicit hyperkähler manifolds are linked to lagrangian spaces in $\Lambda^3 V$: double EPW sextics and double EPW cubes. They are double covers of special degeneracy loci in $\mathbb{P}(V)$ and $G(3,V)$ respectively. In this talk, we give some interesting geometric relations between the two, in terms of Hodge structures and elliptic curves of minimal degree. As an application, we prove that there is a natural relation with Gushel-Mukai fourfolds corresponding to the same lagrangian data. This is joint work with G. and M. Kapustka

We exhibit examples of slope-stable and modular vector bundles on a hyperkähler manifold of K3^[2]-type which move in a 20-dimensional family. These are obtained by performing standard linear algebra constructions on the examples studied by O’Grady of (rigid) modular bundles on the Fano varieties of lines of a general cubic 4-fold and the Debarre-Voisin hyperkähler.

Among irreducible holomorphic symplectic (ihs) manifolds there are two sporadic examples discovered by O’Grady around 2000 in dimension 6 and in dimension 10, OG6 and OG10 respectively. The classification of automorphisms of ihs manifolds is interesting for many reasons, for instance to study the fixed loci of these actions and to construct possibly singular ihs varieties via symplectic quotients of ihs manifolds. This talk is based on a joint work with Onorati and Veniani and on a joint work with L. Giovenzana, Onorati and Veniani about a rigidity behavior of symplectic automorphisms of ihs manifolds of OG6 type and of OG10 type. More precisely I will show how to exploit a sphere packing argument to prove that the unique symplectic automorphism of an ihs manifold of OG10 type is the identity.

There is a natural subcategory of the derived category of a Fano threefold, now called the Kuznetsov component. I will talk about a Brill-Noether reconstruction theorem for a series of Fano threefolds, using Bridgeland moduli spaces in Kuznetsov components. As an application, first I will provide a uniform proof of categorical Torelli theorems for del Pezzo threefolds of degree >1. Then I will give a complete description of the auto-equivalences of Kunetsov components for several Fano threefolds. This talk is based on a joint work with Augustinas Jacovskis and Shizhuo Zhang, and a joint work with Soheyla Feyzbakhsh and Shizhuo Zhang.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

I will discuss some basic results on the structure of isotrivial Lagrangian fibrations for hyper-Kaehler manifolds. First, we establish a basic dichotomy: Type A (Kummer type) and Type B (Deligne-Mostow type). Then, in both cases, we prove that the smooth fibers of the fibration are isogeneous to powers of elliptic curves. Finally, we classify the Type A case and show that only the K3^n and Kum_n can occur.

Let X be a smooth projective real variety. We will first explain that if X has infinitely many real forms, then its automorphism group is necessarily quite large. We then obtain the folklore result that if a surface S has infinitely many real forms, then S is either rational or a non-minimal surface birational to either a K3 surface or an Enriques surface. Finally, we construct an Enriques surface whose blowup at one point has infinitely many real forms. (Joint work with T.-C. Dinh, C. Gachet, K. Oguiso, L. Wang, X. Yu.)

I investigate the Beauville-Voisin conjecture which predicts the vanishing of the cycles in the Chow group of a projective hyper-Kähler manifold, which are both homologous to zero and tautological (polynomials in Chern classes of X and divisor classes). Cohomological polynomial relations are given by the Huybrechts-Verbitsky relations. Some of them are easy to establish in the Chow setting in the case of Lagrangian fibered HK manifolds. Other relations are more subtle and are obtained only for the Laza-Saccà-Voisin Lagrangian fibration. In the course of this study, we investigate the structure of the rank loci for a Lagrangian fibration f:X-B, where the rank of f is given.

Examples of non-commutative K3 surfaces arise from semiorthogonal decompositions of the bounded derived category of certain Fano varieties. The most interesting cases are those of cubic fourfolds and Gushel-Mukai varieties of even dimension. Using the deep theory of families of stability conditions, locally complete families of hyperkahler manifolds deformation equivalent to Hilbert schemes of points on a K3 surface have been constructed from moduli spaces of stable objects in these non-commutative K3 surfaces. On the other hand, an explicit description of a locally complete family of hyperkahler manifolds deformation equivalent to a generalized Kummer variety is not yet available. In this talk we will construct families of non-commutative abelian surfaces as equivariant categories of the derived category of K3 surfaces which specialize to Kummer K3 surfaces. Then we will explain how to induce stability conditions on them and produce examples of locally complete families of hyperkahler manifolds of generalized Kummer deformation type. This is a joint work in progress with Arend Bayer, Alex Perry and Xiaolei Zhao.

I will describe results from a very recent on-going collaboration with Izadi, Kamenova and Marquand. Given a family of HK manifolds over a surface, we show that the locus admitting a Lagrangian fibration is dense. Such HK manifolds admit an isotropic (wrt the BB quadratics form) algebraic (1,1)-class. Conversely and conjecturally (SYZ), existence of such (1,1)-class ensures Lagrangian fibration on some birational model. We show that any family of HK manifolds has a dense subset consisting of those whose Picard group has a square zero element.

My presentation will delve into the realm of measuring the degree of irrationality, fibering tonality, and fibering genus of projective K3 surfaces with Picard number 1. By exploring these measures, we gain insights into the anticipated asymptotic behaviors of these attributes for projective K3 surfaces in this category.

For a Lagrangian fibration f:X->B of an irreducible hyperkahler manifold X, the base B is conjectured to always be isomorphic to projective space. In 2008 Hwang proved that this is the case if B is assumed to be smooth by using the theory of varieties of minimal rational tangents on Fano manifolds. In this talk I will present a simpler proof of this result which leans more heavily on Hodge theory. Specifically, the main input is a basic functoriality result coming from Hodge modules. This is joint work with C. Schnell.

The coniveau filtration was first defined by Grothendieck to understand the generalized Hodge conjecture. He also raised the question of comparing two versions of such filtrations. For rational coefficients, Deligne's theory of weights proves the equality of the coniveau filtration and a stronger version of the filtration. In this talk I will report on some recent work on the integral version of Grothendieck's question, and indicate some of its arithmetic applications. The approach is based on a study of the topology of the space of cycles.

Let X be a n-dimensional (smooth) intersection of two quadrics, and let T*X be its cotangent bundle. I will show that the algebra of symmetric tensors on X is a polynomial algebra in n variables. The corresponding map T*X —> $C^n$ is a Lagrangian fibration, which admits an explicit geometric description; its general fiber is an open subset of an abelian variety, again with a precise geometric description. This is joint work with A. Etesse, A. Höring, J. Liu, C. Voisin.

The moduli space of quasi-polarized K3 surfaces of low genus has several natural compactifications such as the Satake-Baily-Borel, GIT and Kahler Einstein compactifications. It is natural to reconcile these different compactifications from a geometric and birational point of view. This was done in the eighties by Shah and Looijenga for genus 2 case, and more recently by Laza-O’Grady and Ascher-DeVleming-Liu for the case of genus 3. In this talk, I will discuss the case when the K3 surfaces have a Mukai model and make some predictions. Specifically, we give an interpolation between the Baily-Borel and GIT models for polarized K3 surfaces of genus 4 following the general outlines of the Hassett-Keel-Looijenga (HKL) program. We then discuss the connection to the K-stability point of view.

This talk explores involution maps, categorizing them into two types: symplectic and non-symplectic. The inspiration for this work arises from the analysis of widely recognized involutions which can be obtained as autoequivalences within the derived category of coherent sheaves on a K3 surface. We extended these constructions to obtain new non-symplectic birational maps in the specific case of Hyperkähler manifolds presented as moduli spaces of sheaves on a K3 surface with a fixed Mukai vector. This ongoing work is a collaboration with D. Faenzi and G. Menet.

In this talk, I will give semiorthogonal decompositions of derived categories of coherent sheaves on moduli stacks of semistable objects on K3 surfaces. An each summand is given by the categorical Hall product of subcategories called quasi-BPS categories, which approximate the categorification of BPS cohomologies for K3 surfaces. When the weight and the Mukai vector is coprime, the quasi-BPS category is shown to be smooth and proper, with trivial Serre functor etale locally on the good moduli space. So it gives a twisted analogue of categorical crepant resolution of the singular symplectic moduli space, and reminiscents categorical analogue of chi-independence phenomena. This is a joint work in progress with Tudor Padurariu.

Due to a conjecture of Bloch, the transcendental lattice of a surface should control its transcendental motive. In particular, the conjecture implies that any symplectic automorphism of a K3 surface acts trivially on CH0. Voisin and Huybrechts have confirmed the conjecture for automorphism of finite order, which forces the Picard number to be large. We showed that any symplectic automorphism of a K3 surface acts trivially on CH_0 as long as the Picard number is larger than 2. This is a joint work with Zhiyuan Li and Xun Yu.

Fano varieties and compact hyperkähler varieties are two important objects in the study of algebraic geometry. In this talk, we focus on two specific families of Fano fourfolds, cubic fourfolds and Gushel—Mukai(GM) fourfolds. We investigate their connection with hyperkähler varieties from the perspective of the K3 nature of the derived categories. More precisely, we identify the double dual EPW sextic with a Bridgeland moduli space of stable objects in the Kuznetsov component of a very general GM fourfold. Then we discuss a moduli-theoretic approach to constructing Lagrangians of hyperkähler varieties arising from cubic fourfolds or GM fourfolds. This is based on the joint work with Soheyla Feyzbakhsh, Zhiyu Liu and Shizhuo Zhang.

Given a type II degeneration family of projective surfaces, there are two existing constructions of degenerations of their Hilbert schemes in the literature, due to Nagai and Gulbrandsen-Halle-Hulek respectively. The latter was constructed by taking a GIT quotient of the relative Hilbert scheme of the expanded degeneration family. I will compare the two constructions, and explain how variation of GIT helps determine the semistable locus. Based on joint work with M.G.Gulbrandsen, L.H.Halle and K.Hulek.

A smooth projective rational surface over an algebraically closed field admits a full exceptional collection. Building on work of Hille--Perling, Perling, and Vial, we study mutations of (numerically) exceptional collections by analyzing the lattice theoretic behavior of the numerical Grothendieck group. On the one hand, we show that some results, known for del Pezzo surfaces, can be extended to the blow-up of the projective plane in 9 points in very general position. On the other hand, on the blow-up of 10 points in general position we construct an exceptional collection of maximal length which is not full. This disproves a conjecture of Kuznetsov and a conjecture of Orlov. The talk is based on arXiv:2211.07724 and arXiv:2304.01269.

The Shafarevich problem concerns the finiteness of varieties over a number field with good reduction outside a fixed finite collection of places. This is originally asked for curves with a given genus, which was confirmed by Faltings in his proof of Mordell conjecture. In higher dimensional cases, such arithmetic problem has been investigated for various types of varieties. In this talk, I will introduce several kinds of Shafarevich problems for hyper-Kähler varieties and a recent joint work with Lie Fu, Zhiyuan Li and Teppei Takamatsu.