Abstract:
Compact Hyperkähler manifolds (also known as IHS) are notable Kähler manifolds renowned for their rich and intriguing geometry. Using classical results from quadratic forms and advance methods from Bridgeland stability conditions, we will show that any projective Hyperkähler manifold (deforming to the Hilbert scheme of n points on a K3 surface) with a Picard number at least 4 is always isomorphic to a moduli space of (twisted) stable sheaves on a K3 surface.
We will also discuss exampleswith Picard numbers less than 4, including cases that are not birational to a moduli space of twisted sheaves on a K3 surface. These examples reveal connections to an open problem in algebraic geometry regarding therationality of cubic fourfolds.