Two days on hyper-Kähler varieties and related topics

The workshop will take place from Thursday morning, June 25, to Friday noon, June 26, 2026. It will take place at Bâtiment Sophie Germain of Université Paris Cité.

Schedule

Thursday 25 June

9:30 - 10:30, Salle 1016,  Matteo Verni

Coffee break

11:00 - 12:00, Salle 1016, Xiaolei Zhao

Lunch

14:00 - 15:00, Salle 1016, Calla Tschanz

Coffee break

15:30 - 16:30, Salle 1016, Howard Nuer

Friday 26 June

9:30 - 10:30, Salle 0013, Andrés Rojas

Coffee break

11:00 - 12:00, Salle 0013, Daniele Agostini

Lunch

Abstracts

• Daniele Agostini
  Title: Fixed divisors on hyperkähler manifolds
  Abstract: We show that the fixed divisor of a big and nef line bundle on a hyperkähler variety is reduced. In particular, the square of this line bundle does not have fixed divisors. In dimension four, we also show that if this divisor is not zero, then the mobile part induces (birationally) a Lagrangian fibration. This is joint work with Andreas Höring.

• Howard Nuer
  Title: Finite Order Symplectic Birational self-maps on Kummer-type manifolds
  Abstract: In recent years, the study of symmetries on hyperkähler manifolds has revealed deep connections between birational geometry and moduli spaces of sheaves. In this talk, based on joint work with Y. Dutta, D. Mattei, and S. Muller, we explore symplectic birational self-maps of finite order on projective Kummer-type manifolds. 
  A projective hyperkähler manifold of Kummer-type is said to be twisted modular if it is birational to the Albanese fiber of a moduli space of twisted sheaves on an abelian surface.  After reviewing some motivation and background, we will establish in the first part of the talk that any Kummer-type manifold admitting a finite-order symplectic birational self-map with a non-trivial action on H2(X,Z) must be twisted modular. This holds with the exception of certain precisely characterized cases in Picard rank 3, which we will detail via their Néron-Severi lattices. In the second part of the talk, we will transition to explicit geometric realizations of symplectic birational self-maps of finite order on modular Kummer-type manifolds. By investigating wall-crossing in the space of Bridgeland stability conditions, we will determine exactly which Mukai vectors allow the transformation induced by crossing the vertical wall to correspond to a finite-order symplectic birational self-map acting as a reflection on cohomology.  Time permitting, we will discuss some further directions and open questions arising from our work.
  
  

• Andrés Rojas
  Title: Coble type hypersurfaces and hyperkähler fourfolds
  Abstract: A classical result, already observed by Coble, asserts that a genus 2 Jacobian (resp. the Kummer of a genus 3 Jacobian) can be embedded in \mathbb{P}^8 (resp. in \mathbb{P}^7) as the singular locus of a unique cubic (resp. quartic) hypersurface. We present a precise analogue of this result in the context of hyperkähler fourfolds, and discuss applications to the geometry of two moduli spaces of polarized hyperkähler fourfolds of K3-type. This is a joint work with Benedetta Piroddi, Ángel Ríos, and Jieao Song.

• Calla Tschanz
  Title: From logarithmic Hilbert schemes to degenerations of hyperkähler varieties
  Abstract: In this talk, I will discuss my previous work on constructing explicit models of logarithmic Hilbert schemes. This relates to work or Li-Wu on expanded degenerations, Gulbrandsen-Halle-Hulek on degenerations of Hilbert schemes of points and Maulik-Ranganathan on logarithmic Hilbert schemes. The constructions I consider are local. I will then explain how we globalise these in joint work with Shafi and apply them to construct minimal type III degenerations of hyperkähler varieties, namely Hilbert schemes of points on K3 surfaces. 

• Matteo Verni
  Title: The Brauer group of Known Enriques manifolds
  Abstract: Enriques manifolds are the higher dimensional analogue of Enriques surfaces. Their study is similar, where the role played by K3 surfaces in dimension two is played by hyper-kähler manifolds in higher dimension. In this talk, we interest ourselves in the Brauer group of Enriques manifolds. This group is isomorphic to Z/2Z for Enriques surfaces. We explain how to compute it in the known higher dimensional examples, by purely topological means: this is closely related to the problem of computing H^3(X,Z)_tors of hyper-Kähler manifolds. We then describe  generalization in higher dimension of a result of Beauville, characterizing those Enriques surfaces for which the pullback of the nontrivial Brauer class vanishes, in terms of the existence of a special line bundle on its associated hyper-Kähler. For any Enriques manifold for which such a line bundle exists, we construct explicit trivial Brauer-Severi varieties on the hyper-Kähler which descend to nontrivial ones on the Enriques manifold. This is joint work with A. Frassineti, F. Rizzo and F. Tufo.

• Xiaolei Zhao
  Title: Non-commutative abelian surfaces and Kummer type hyperkähler manifolds (with further updates)
  Abstract: 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 hyperkähler 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 hyperkähler manifolds deformation equivalent to a generalized Kummer variety is not yet available.
  In this talk we will construct 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 realize all hyperkähler manifolds of Kummer type as moduli spaces of stable objects on non-commutative abelian surfaces. Applications to algebraic cycles and abelian fourfolds of Weil type will be discussed.
  This is joint work in preparation with Arend Bayer, Alex Perry and Laura Pertusi.

Venue

Bâtiment Sophie Germain, Université Paris Cité
8 Place Aurélie Nemours,
75013 Paris
https://maps.app.goo.gl/rVsRtAmmkdtVnf6A6