# Master Lectures-- “Hyper-Kaehler Manifolds, Hodge Theory and Chow Groups”: Claire Voisin

Synopsis: The subject lies at the interface of algebraic geometry and analytic geometry. Projective K3 surfaces have been studied since the nineteenth century, and they still hold a privileged place in algebraic geometry. K3 surfaces are nothing else but compact hyperkaehler manifolds of (complex) dimension 2. Higher dimensional compact hyperkaehler manifolds have been seriously studied only after Yau's solution of Calabi's conjecture. Interesting classes of examples hyperkaehlers have been produced, and a series of beautiful general results have been proved, but it is clear that what we know is only the tip of the iceberg. It is a remarkable fact that all known examples have been constructed by algebraic geometry methods while the available general results concerning the topology of these varieties and their moduli spaces pass through uniformization and thus are not actually algebraic. In the projective setting, the study of projective hyperkaehlers has used in a complementary way methods from Hodge theory, derived categories, and algebraic cycles and from any of these viewpoints, these varieties have very special properties, not yet fully understood. The subject is closely related to moduli spaces of sheaves, as most known examples have been constructed as moduli spaces.

Voisin will lecture on Chow rings of hyperkaehler's, a subject which has started with Beauville-Voisin's conjectures and will be central in the workshop. Other important topics covered in the workshop are derived category methods in the study of hyperkaehler's, applications of Hodge theory to geometry, and construction methods for hyperkaehler's.

**Organizers**

Name | University |
---|---|

Claire Voisin | Centre National de la Recherche Scientifique, France |

Radu Laza | Stony Brook University, USA |

Kieran G. O'Grady | Università degli Studi di Roma La Sapienza, Italy |