Timetable, titles and abstracts
HK days in Sheffield
The University of Sheffield, 12-14 February 2025
Research talks
Ignacio Barros: Geometric theta correspondence and extremal divisors on moduli spaces of K3 surfaces
I will survey Kudla's classical geometric theta correspondence relating automorphic forms with algebraic/cohomological cycles on orthogonal Shimura varieties. In the codimension one setting we use this framework to show the extremality of many Noether-Lefschetz divisors on moduli spaces of K3 surfaces and HK varieties. This is based on joint work in progress with L. Flapan and R. Zuffetti.
Alessio Bottini: A modular construction of OG10
The construction of hyper-Kähler manifolds is notoriously difficult. Currently, all known examples, up to deformations, emerge from moduli spaces of sheaves on symplectic surfaces. This raises the question: can moduli spaces of sheaves on higher-dimensional hyper-Kähler manifolds also be hyper-Kähler? This question remains largely unexplored and is particularly challenging to address. In this talk I will discuss a strategy to realize the Laza—Saccà—Voisin compactification of the intermediate Jacobian fibration as a moduli space of stable bundles on a hyper-Kähler fourfold.
Francesco Denisi: MMP for Enriques pairs and singular Enriques varieties
An Enriques manifold is a connected complex manifold that is not simply connected and whose universal covering is an irreducible holomorphic symplectic (IHS) manifold. Given a projective IHS manifold X and an effective R-divisor D such that the pair (X,D) is log canonical, C. Lehn and G. Pacienza proved that any MMP starting from (X,D) terminates. The goal of this talk is twofold. First, we discuss an analogous result for Enriques pairs, which we define as log canonical pairs (Y, D), where Y is an Enriques manifold and D is an effective R-divisor. Second, we characterize the underlying variety of the resulting minimal model (Y’, D’) of (Y,D). This leads naturally to the definition of primitive Enriques varieties, for which we provide examples and explore some of their properties. The talk is based on joint work with Á. D. Ríos Ortiz, N. Tsakanikas and Z. Xie.
Yajnaseni Dutta: The relative intermediate Jacobian
Intermediate Jacobians for smooth projective varieties play a very similar role as Jacobians play for smooth projective curves. While relative Jacobian for families of curves is a well-studied concept, the relative intermediate Jacobians for families of higher odd dimensional varieties are rather uncharted territories. In this talk, I will construct a sheaf of intermediate Jacobian for a family of cubic threefolds (not necessarily smooth) and talk about possible applications of this gadget. This is a joint work in progress with Mattei and Shinder.
Franco Giovenzana: On the Projective Duality of Kummer Fourfolds and Their Equations
Kummer fourfolds, a generalization of K3 surfaces (and, in some sense, of abelian surfaces), belong to the class of Hyperkähler manifolds, which exhibit rich but intricate geometry. In this talk, we explore the projective duality of certain special Kummer fourfolds and explain how O'Grady's theory of theta groups can be used to derive their equations. This work, carried out in collaboration with Agostini, Beri, and Ríos-Ortiz, contributes to a broader framework of classical results involving moduli spaces of sheaves on curves and embeddings of abelian surfaces.
Yoon-Joo Kim: The Néron model of a Lagrangian fibration
Singular fibers in minimal elliptic fibrations were classified by Kodaira and Néron in the 1960s. In his proof, Néron constructed and systematically used a special group scheme acting on an elliptic fibration. This group scheme is now called the Néron model. Néron’s theory is restricted to 1-dimensional bases, so one cannot use Néron’s original approach to study higher-dimensional Lagrangian fibrations. The higher-dimensional analog of Néron’s definition is proposed by David Holmes. Quite unfortunately, Holmes also showed that such a generalized Néron model often fails to exist, even in simple cases.
In this talk, we show that Holmes’s generalized Néron model does exist for an arbitrary projective Lagrangian fibration of a smooth symplectic variety, under a single assumption that the Lagrangian fibration has no fully-nonreduced fibers. This generalizes Néron’s result to many higher-dimensional Lagrangian fibrations. Such a construction has several applications. First, it extends Ngô's results on Hitchin fibrations to many Lagrangian fibrations. Second, it allows Lagrangian fibrations to be considered as a minimal model-compactification of a smooth commutative group scheme-torsor. Third, it provides a tool to study birational behaviors of Lagrangian fibrations. Finally, the notion of a Tate-Shafarevich twist can be understood via the Néron model.
Christian Lehn: Tannakian groups of perverse sheaves and E6 geometry
In a joint work with Krämer and Maculan, we prove that the Fano surfaces of lines on smooth cubic threefolds are the only smooth (less than half-dimensional) subvarieties of abelian varieties whose Tannaka group for the convolution of perverse sheaves is an exceptional simple group. The proof uses an upgrade of the Krämer-Weissauer formalism from perverse sheaves to Hodge modules, the Hodge number estimates for irregular varieties by Lazarsfeld-Popa and Lombardi, an intensive computer search, and the geometry of lines on cubic surfaces.
Dominique Mattei: Twisting Lagrangian fibrations
Starting with a hyperkähler Lagrangian fibration X → B, one can cut it into pieces and re-glue it non-trivially into a new fibration called a twist of X/B. These twists are well understood for elliptic K3s by Ogg-Shafarevich theory, and were later investigated in higher dimensions by Verbitsky, Markman, Abasheva-Rogov (among others). This talk aims to report on recent joint projects with Huybrechts / Meinsma / Dutta-Shinder, in which we study twists of Jacobians of curves on K3s / intermediate Jacobians of cubic threefolds.
Reinder Meinsma: Derived equivalence for moduli spaces of sheaves on K3 surfaces
The Derived Torelli Theorem for K3 surfaces asserts that a two-dimensional moduli space of sheaves on a K3 surface is a Fourier--Mukai partner if and only if it is fine. Moreover, we learned from Caldararu's work that non-fine moduli spaces are twisted derived equivalent to the original K3 surface, with the twist being given by a certain Brauer class which can be understood as the obstruction to the existence of a universal sheaf. In this talk, I will report on joint work with Dominique Mattei in which we generalise Caldararu's results to higher-dimensional moduli spaces, and explore consequences for the derived categories. In particular, I will show that the 'obvious' generalisation of the Derived Torelli Theorem is false for higher-dimensional moduli spaces.
Claudio Onorati: Schur functors and modular bundles
Recently O'Grady has introduced the concept of modular sheaf on a hyperkähler manifold as an attempt to generalise the moduli theory of sheaves on K3 surfaces to higher dimensional hyperkähler manifolds. Even if the definition is rather straightforward, checking it on concrete examples may be troublesome. O'Grady himself produced several examples on Hilb²(S), most of them being rigid. Since the ultimate goal would be to consider moduli spaces of modular sheaves, it is important to have non-rigid stable examples. In this talk I will explain a recent work in collaboration with E. Fatighenti where we exhibit infinitely many slope stable modular bundles on the Fano variety of lines X of a cubic fourfold, describing their Ext¹-groups. Time permitting, I will also present a work in progress with A. D'Andrea and E. Fatighenti on a connection between discriminants of vector bundles on smooth and projective varieties and representation theory of GL(N).
Andrey Soldatenkov: Metric structures on degenerate twistor families
I will talk about special families of Lagrangian fibrations on compact hyperkähler manifolds. Starting from one such fibration, one can deform the symplectic form by adding the pull-back of a closed two-form from the base, thus obtaining a family of complex structures on the original hyperkähler manifold and preserving the Lagrangian fibration.
The family is parametrized by the affine line and will be called a degenerate twistor family, because it may be viewed as a degeneration of an ordinary twistor family. We will discuss how to prove that all fibres of such families admit Kähler metrics.
Based on joint work with Verbitsky.