Abstract
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 set of sheaves
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 sheaves 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 sheaves on hyper-Kahler manifolds and the
study of moduli spaces of sheaves on K3 surfaces. After having
recalled the needed definitions and results, in this talk I will
face the formality problem for such set of sheaves. In
particular, I will extend the notion of hyper-holomorphic to
complexes of locally free sheaves, and show how the associated
dg Lie algebra of derived endomorphism is formal, namely
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).