Hyperkähler Manifolds and Sheaves

Abstract

Moduli spaces of hyperkähler manifolds or of sheaves on them are often nonseparated. We will discuss results where this phenomenon reflects interesting geometric aspects, e.g. deformation equivalence of birational hyperkähler manifolds or cohomological properties of derived autoequivalences. In these considerations the Ricci-flat structure often plays a crucial role via the associated twistor space providing global deformations of manifolds and bundles.The aim of this note is to review a few scattered results for which non-separation phenomena and twistor spaces play a decisive role. We will touch upon questions concerning the birational geometry of hyperkähler manifolds, derived categories of coherent sheaves on K3 surfaces and their autoequivalences, Brauer classes, hyperholomorphic bundles, Chow groups, etc. There is no attempt at completeness and I apologize for not covering the material in a more concise form. I believe that some of the techniques can be pushed further to treat other interesting open problems in the area, some of which will be mentioned at the end.

The unreasonable effectiveness of wall-crossing in algebraic geometry

“…Derived categories of coherent sheaves on varieties have been hugely influential in recent years; we refer to [17,19,46,74] for an overview of the theory. Moduli spaces of sheaves on K3 surfaces have largely been influenced by [61]; we refer to [63] for an overview of applications of these techniques and to [39] for the higher-dimensional case of hyper-Kähler manifolds. For the original motivation from physics, we refer to [41,26].…”