# Workshop on geometry and arithmetic of hyperkähler manifolds

February 9-11, 2015, 10:00 - 17:00 at Leibniz Universität Hannover

Room: B305 (Mon), F442 (Tue, Wed), Welfenschloss (university main building) - overall directions

**Program**

The goal of the workshop is to bring together young researchers working on the

subject and to give them a possibility to present their work and exchange ideas. There will be an advanced mini-course

**K3 categories of cubic fourfolds and Hodge theory**

by Daniel Huybrechts and talks by the following participants:

Benjamin Bakker

Chiara Camere

Francois Charles

Lie Fu

Giovanni Mongardi

Ulrike Rieß

Pawel Sosna

Malte Wandel

Benjamin Wieneck

**Schedule**

Mo | Di | Mi | |

10.00 | Huybrechts | Huybrechts | Huybrechts |

11.30 | Wandel | Rieß | Bakker |

12.30 | Lunch | Lunch | Lunch |

14.30 | Charles | Sosna | Fu |

16.00 | Camere | Wieneck | Mongardi |

**Abstracts**

**Ben Bakker - Classifying ruled subvarieties of K3^n type manifolds**

A conjectural picture of Hassett and Tschinkel describes the geometry of the locus swept out by a rational curve in terms of the intersection theory of the curve class. Using the results of Bayer and Macri on the ample cones of moduli spaces, we give a partial such classification for K3^n type manifolds and discuss some results in the general case. We also use the classification to understand the monodromy action on these loci.

**François Charles - Zarhin's trick for K3 surfaces**

If A is an abelian variety, then the abelian variety (AxA*)^4 is

principally polarized. This fact is known as Zarhin's trick. We will

explain the analogue of this result for K3 surfaces, with applications to

algebraic cycles.

** Chiara Camere - Lattice-polarized irreducible holomorphic symplectic manifolds.**In this talk I will discuss the notion of lattice-polarized irreducible holomorphic symplectic manifolds and the construction of their moduli spaces. Then I will explain a generalization of a lattice-theoretical mirror symmetry, as known for K3 surfaces, to higher dimensions. Finally we will see some examples in the case of fourfolds of K3^[2]-type.

**Lie Fu - On the motivic hyperkähler resolution conjecture**

Ruan's conjecture on hyperkähler resolution says that the orbifold cohomology ring of an smooth projective orbifold is isomorphic to the cohomology ring of its hyperkähler resolution of singularities. I would like to formulate the motivic analogue of this conjecture on the Chow ring (or more generally the Chow motive as an algebra in the category of Chow motives) of hyperkähler varieties. The particular interesting cases that I want to discuss are the Hilbert-Chow morphisms for the symmetric product of K3 surfaces and the similar generalized Kummer varieties. This is a joint work with Zhiyu Tian.

**Giovanni Mongardi - Monodromy of irreducible symplectic manifolds**

Exploiting recent results on the ample cone of irreducible symplectic

manifolds, we provide a different point of view for the computation of

their monodromy groups. In particular, we give the final step in the

computation of the monodromy group for generalised Kummer manifolds

and we prove that the monodromy of O'Grady's ten dimensional manifold

is smaller than what was expected.

**Ulrike Rieß - On Beauville's conjectural weak splitting property**

We present a recent result on the Chow ring of irreducible symplectic varieties.

The main object of interest is Beauville's conjectural weak splitting property, which

predicts the injectivity of the cycle class map restricted to a certain subalgebra of the

rational Chow ring (the subalgebra generated by divisor classes). For special irreducible

symplectic varieties we relate it to a conjecture on the existence of rational Lagrangian

fibrations. After deducing that this implies the weak splitting property in many new cases, we

present parts of the proof.

**Pawel Sosna - On the dynamical degrees of reflections on cubic fourfolds**

Given a birational self-map of a smooth complex projective variety,

one can associate certain numbers with it, the so-called dynamical degrees.

Considering all birational maps at once, gives the dynamical spectrum

of a variety, which is a birational invariant. It is an interesting

question whether this spectrum could be useful in determining the rationality of, for example, cubic hypersurfaces. In the talk I will explain the computation of the dynamical degrees of certain compositions of reflections in points on a smooth cubic fourfold.

This is joint work in progress with Chr. Böhning and H.-Chr. Graf von Bothmer.**Malte Wandel - Automorphisms of O'Grady's manifolds acting trivially on cohomology**

I will report on recent results with G. Mongardi about automorphisms on O'Grady's manifolds: For any hyperkaehler manifold there is a natural representation of the automorphism group on the second integral cohomology. This representation yields one of the main tools to translate geometric questions about the manifolds into lattice theory. The kernel of the representation is known to be a deformation invariant (Hassett-Tschinkel) and has been computed for Hilbert schemes of points on K3s (Beauville, Boissiere-Sarti: Here the kernel is trivial.) and for generalised kummer manifolds (Boissiere-Nieper-Wisskirchen-Sarti: Here the kernel is non-trivial.). In this talk I want to present a sketch of the proof that the afforementioned representation is injective for O'Grady's tendimensional examples and has a kernel of order 2^8 in the six-dimensional case.**Benjamin Wieneck - On polarization types of Lagrangian fibrations**

The generic fiber of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is an abelian variety. This fact is used to construct a deformation invariant, called the polarization type, for every Lagrangian fibration which is essentially a polarization type of a polarized generic fiber. Conjecturally this invariant should only depend on the deformation class of the total space. Indeed for K3^[n]-type fibrations it is always a principal polarization which can be shown using methods developed by E. Markman.

**Registration**

There is no formal registration, but you are welcome to inform us that you would like to participate in order to ease the planning on our side. If you need any assistance, please contact Ute Szameitat or either of the organisers.

**Organisers**

Christian Lehn

Matthias Schütt

**Sponsors**

Leibniz Universität Hannover

Graduiertenkolleg 1463 Analysis, Geometrie und Stringtheorie

Riemann Center for Geometry and Physics

Foundation Compositio Mathematica

ERC StG "SURFARI"