Remarks on bundles on hyper-Kahler manifolds

Date: 2021-11-04

Time: 14:00 - 15:00

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Claudio Onorati


The geometry of moduli spaces of sheaves on K3 surfaces is very rich and led to very deep results in the last decades. Moreover, under certain hypotheses, these varieties are smooth projective and have a hyper-Kahler structure, providing non-trivial examples of compact hyper-Kahler manifolds. In higher dimensions the situation is much more complicated, nevertheless in the ’90s Verbitsky introduced a class of vector bundles on hyper-Kahler manifolds, called hyper-holomorphic, whose moduli spaces are singular hyper-Kahler (but not compact in general). Recently O’Grady proved that such bundles belong to a larger set of sheaves for which there exists a good wall-and-chamber decomposition of the ample cone. This suggests an analogy between the study of moduli spaces of hyper-holomorphic bundles on hyper-Kahler manifolds and the study of moduli spaces of bundles on K3 surfaces. After having recalled the needed definitions and results, in this talk I will face the formality problem for such set of bundles. More precisely, I will show that the dg Lie algebra governing the infinitesimal deformations of a stable and hyper-holomorphic vector bundle is formal, i.e. quasi-isomorphic to its cohomology. As a corollary one gets a different proof of a quadraticity result of Verbitsky. This is a joint work in progress with F. Meazzini (INdAM – Rome).