A workshop on the Chow group of holomorphically symplectic manifolds:
talks and abstracts

19-24 May 2014,
Laboratory of Algebraic Geometry, Moscow

Home | Venue | Schedule | Program

Arnaud Beauville (Université de Nice)
Introduction to the Chow ring of K3 surfaces and holomorphic symplectic manifolds

I will first discuss my paper with Claire Voisin on the Chow ring of K3 surfaces, then explain how it leads to conjectures on the Chow ring of holomorphic symplectic manifolds and discuss the current status of these conjectures.

Olivier Debarre (ENS)
Birational transformations and periods for certain Fano manifolds

So called elementary transformations along curves contained in Fano threefolds proved very useful for Iskovskikh's classification of prime Fano threefolds. We attempt to define these transformations in higher dimensions in order to study the fibers of the period map of certain Fano manifolds.

François Charles (IRMAR and MIT)
Zarhin's trick for K3 surfaces and the Tate conjecture

If A is an abelian variety over an arbitrary field, Zarhin's trick shows that (AxA)^4 can be endowed with a principal polarization. Using moduli spaces of stable sheaves, we prove a general version of Zarhin's trick for K3 surfaces over arbitrary fields. As an application, we will give a simple proof of the Tate conjecture for K3 surfaces over arbitrary finite fields.

François Charles (IRMAR and MIT)
Bertini irreducibility theorems over finite fields

Given a geometrically irreducible subscheme X in P^n over F_q of dimension at least 2, we prove that the fraction of degree d hypersurfaces H such that the intersection of H and X is geometrically irreducible tends to 1 as d tends to infinity. This is joint work with Bjorn Poonen.

Sergey Galkin (HSE)
Lines on rational cubic fourfolds, and associated K3 surfaces

This is a joint work with Evgeny Shinder.

It is expected (after Iskovskikh, Zarkhin, Tregub, Beauville-Donagi, Hassett, Kulikov, Kuznetsov, Addington-Thomas, and others) that generic cubic fourfolds are irrational, and rational ones are related in some way to K3 surfaces. For example, Pfaffian cubics were shown to be rational by Morin in 1940, and in 1984 Beauville and Donagi shown that their variety of lines is a Hilbert scheme of 2 points on a K3 surface, related to the original cubic by projective duality. We generalize this result to all rational cubic fourfolds, under the assumption of Denef-Loeser's conjecture. Namely, if a class of an affine line is not a zero divisor in the Grothendieck ring of varieties, then Fano variety of lines on a rational cubic fourfold is birational to a Hilbert scheme of two points on some K3 surface. The proof uses a theorem of Larsen and Lunts and a new unconditional relation between the classes of the variety of lines and of the symmetric square of any cubic hypersurface in the Grothendieck ring of varieties. The latter relation also reproduces many known results, such as 27 lines on a cubic surface.

Alexander Kuznetsov (Steklov Institute)
Calabi--Yau subcategories and Fano manifolds of degree 10

A cubic fourfold is known to share some properties with K3 surfaces. The reason for that is a semiorthogonal component of its derived category which is a deformation of the derived category of a K3 surface. I will give more examples of varieties which have a K3 or Calabi--Yau subcategory as a semiorthogonal component of their derived category. Among these are Fano fourfolds of degree 10. I will show that in some cases its K3 subcategory is equivalent to the derived category of a K3 surface.

Eyal Markman (University of Massachussetts, Amherst)
The Kawamata-Morrison conjecture for the ample cone of a hyper-Kahler variety

This work is joint with Kota Yoshioka. We derive the Kawamata-Morrison conjecture for the ample cone of a projective irreducible holomorphic symplectic manifold from the proven movable cone version of the conjecture and a lower bound on the Beauville-Bogomolov-Fujiki degree of exceptional curve. The latter bound in known in the K3 and generalized Kummer deformation types by work of Bayer-Macri, Yoshioka, and Bayer-Hassett-Tschinkel. Similar results were independently obtained by Amerik and Verbitsky for the Kahler cone, dropping the projectivity assumption.

Claire Voisin (Ecole Polytechnique)
On the canonical 0-cycle of a K3 surface

In joint work with Beauville, we proved that a projective K3 surface has a canonical 0-cycle of degree 1, satisfying many remarkable properties. Huybrechts, O'Grady and myself found more recently that this 0-cycle appears as Chern class of rigid bundles. I will describe these results and what is known in the higher dimensional case.

Claire Voisin (Ecole Polytechnique)
Modified diagonals and Chow rings of Hyper-Kaehler manifolds

It has been conjecture by Beauville and myself that any cohomological relation involving divisor classes and Chern classes of the tangent bundle of a hyper-Kaehler manifold already hold in the Chow ring. I will explain the relation between this conjecture and another one concerning the relations between the big diagonals in the powers of a hyper-Kaehler manifold. In particular, I will discuss recent works of Yin, O'Grady and myself on modified diagonals. The case of Calabi-Yau manifolds will be also discussed.

Home | Venue | Schedule | Program

Laboratory of Algebraic Geometry and its Applications