### A workshop on the Chow group of holomorphically
symplectic manifolds:

talks and abstracts

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.š 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.
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