Curves, K3s and Hyperkählers
Workshop

Speakers:

1. Ignacio Barros

2. Xi Chen

3. Yajnaseni Dutta

4. Phil Engel

5. Salvatore Floccari

6. Franco Giovenzana

7. Giovanni Mongardi

8. Federico Moretti

9. Gianluca Pacienza

10. Roberto Svaldi

11. Adrian Zahariuc

Organisers: Xi Chen and Frank Gounelas

Participation: Everyone is welcome, but to keep track of numbers, please send an email to: gounelas@math.uni-bonn.de

Location: The talks on Tuesday and Wednesday will take place in the Lipschitzsaal (the big room opposite the main entrance on the 1st floor), whereas on Thursday we will be in Seminar room 0.011.

Program: 

Titles and abstracts:

• Ignacio Barros (Antwerp):
  Title: Extremal divisors on moduli spaces of K3 surfaces
  Abstract: We establish numerical criteria for when a Noether–Lefschetz divisor on a moduli space of quasi-polarized K3 surfaces F_{2d}, or more generally on an orthogonal modular variety, generates an extremal ray in the cone of pseudoeffective divisors. In particular, for all d, we exhibit many extremal rays of the cone of pseudo-effective divisors of both F_{2d} and any normal projective Q-factorial compactification over its Baily–Borel model. This is based on joint work with L. Flapan and R. Zuffetti.

• Xi Chen (Alberta):
  Title: Real Regulators for Products of Elliptic Curves
  Abstract: Higher Chow groups were introduced by Spencer Bloch as the natural generalization of Chow groups. The corresponding cycle maps from higher Chow groups to cohomologies are called regulators. Unlike the images of cycle maps of ordinary Chow groups, which are conjecturally the Hodge groups, the images of regulators are quite mysterious, even for simple varieties such as products of elliptic curves. For a product of two general elliptic curves, by passing to the corresponding Kummer surface, we proved that the real regulator is surjective, i.e., the so-called Hodge-D conjecture. However, for a product of four or more general elliptic curves, despite some published construction of non-trivial higher Chow cycles on such variety, we proved that the Hodge-D conjecture fails, assuming the Kunneth decomposition of the Chow groups of products of very general Kummer surfaces. This is a joint work with James D. Lewis.

• Yajnaseni Dutta (Leiden):
  Title:  Tate—Shafarevich twists of intermediate Jacobians
  Abstract: Introduced and studied from number theoretic interests, the Tate—Shafarevich group, the group of torsors of a group scheme, has recently seen renewed interest from the perspective of commutative group spaces arising from Lagrangian fibrations. In this talk, I will discuss the twists of certain relative intermediate Jacobian and its deep analogy to the twists of the Beauville--Mukai system, after Markman. This talk will be based on an on-going joint work with Mattei and Shinder.

• Phil Engel (UIC):
  Title: Boundedness theorems for abelian fibrations
  Abstract: I will report on forthcoming work, joint with Filipazzi, Greer, Mauri, and Svaldi, on boundedness results for abelian fibrations. We will discuss a proof that irreducible Calabi-Yau varieties admitting an abelian fibration are birationally bounded in a fixed dimension; and that Lagrangian fibrations of symplectic varieties, in a fixed dimension, are analytically bounded. Conditional on the generalized semiampleness/hyperkahler SYZ conjecture, this bounds the number of deformation classes of hyperkahler varieties in a fixed dimension, with second Betti number at least 5.

• Salvatore Floccari (Bielefeld):
  Title: The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties
  Abstract: The Hodge conjecture for abelian varieties of dimension 4 has been fully established only very recently, thanks to the work of Markman. He had previously proven the algebraicity of Weil-Hodge classes on all abelian fourfolds of Weil type with discriminat 1, via the construction of hyperholomorphic bundles on varieties of generalized Kummer type. I will present an independent proof of the Hodge conjecture for all Weil fourfolds with discriminat 1 and their powers, obtained in joint work with Lie Fu. Our argument avoids hyperholomorphic sheaves and relies instead on a direct geometric relation between these fourfolds and certain families of hyperKähler varieties of OG6-type. As a consequence, we also establish the Hodge conjecture for many families of OG6-varieties which are of codimension 1 in their moduli spaces.

• Franco Giovenzana (Paris):
  Title: Coble projective duality of Kummer fourfolds
  Abstract: Despite being among the foundational examples of irreducible holomorphic symplectic manifolds, generalised Kummer varieties remain poorly understood from the point of view of explicit projective models. In this talk, we describe two birational projective models of Kummer fourfolds associated to Jacobian surfaces, and show that they are projectively dual. The construction relies on the classical Coble cubic and extends a duality result conjectured by Dolgachev and proven by Ortega and Nguyen in the context of moduli spaces of sheaves on genus 2 curves.This is joint work with D. Agostini, P. Beri, and Á. D. Ríos Ortiz.

• Giovanni Mongardi (Bologna):
  Title: Regenerations, degenerations and applications, II
  Abstract: Chen-Gounelas-Liedtke recently introduced a powerful regeneration technique, a process opposite to specialization, together with a controlled degeneration technique to prove existence results for rational curves on projective K3 surfaces. In two papers with G. Mongardi and with P. Beri and G. Mongardi we generalized both techniques to projective irreducible holomorphic symplectic manifolds, to prove existence of uniruled divisors, significantly improving known results. In this talk I will present the joint work with P. Beri and G. Pacienza, proving the existence of countably many ruled divisors on some IHS manifolds under mild hypothesis.

• Federico Moretti (Stony Brook):
  Title: Degree of irrationality of K3 surfaces and their covers
  Abstract: The degree or irrationality of a variety X is the minimal degree of a rational dominant map from X to a projective space of the same dimension. I will present some vector bundles techniques to study this problem. Among other things, I will prove that every K3 surface of genus at most 14 has degree of irrationality at most 4. In the second part of the talk I will show how a variation of the same techniques proves that a very general K3 surface of genus g is covered by a surface having degree of irrationality 2 (in progress). Most of the results are based on joint work with Andrés Rojas.

• Gianluca Pacienza (Nancy):
  Title: Regenerations, degenerations and applications, I
  Abstract: Chen-Gounelas-Liedtke recently introduced a powerful regeneration technique, a process opposite to specialization, together with a controlled degeneration technique to prove existence results for rational curves on projective K3 surfaces. In two papers with G. Mongardi and with P. Beri and G. Mongardi we generalized both techniques to projective irreducible holomorphic symplectic manifolds, to prove existence of uniruled divisors, significantly improving known results. In the talk I will present the joint work with G. Mongardi about the higher dimensional regeneration principle.

• Roberto Svaldi (Milan):
  Title: Boundedness theorems and birational geometry of fibered CY
  Abstract: I will explain ideas and techniques behind recent results showing that certain classes fibered CY varieties are bounded (see also P. Engel's talk for more on this), discussing several (birational) aspects of the problem (e.g., bases of such fibrations and their structure, the Kawamata-Morrison Cone Conjecture).

• Adrian Zahariuc (Windsor):
  Title: Interpolation of fat points on K3 and abelian surfaces
  Abstract: This talk will be concerned with questions in the style of the Segre-Harbourne-Gimigliano-Hirschowitz (SHGH) Conjecture. I will sketch a proof of the fact that any number of general fat points of any multiplicities impose the expected number of conditions on the primitive linear system of a very general K3 or abelian surfaces of any degree/polarization.