Abstract
Let X be a projective K3 surfaces. In two examples where there exists a fine moduli space M of stable vector bundles on X, isomorphic to a Hilbert scheme of points, we prove that the universal family {mathcal {E}} on Xtimes M can be understood as a complete flat family of stable vector bundles on M parametrized by X, which identifies X with a smooth connected component of some moduli space of stable sheaves on M.
Highlights
Let X be a projective K3 surface, and M a moduli space of semistable sheaves on X
By Mukai’s seminal work [15], when M is smooth, it is an example of the so-called irreducible holomorphic symplectic manifolds, which are an important class of building blocks in the classification of compact Kähler manifolds with trivial first Chern class
It is an interesting question to understand whether the moduli spaces M of semistable sheaves on M inherit any good properties from M
Summary
Let X be a projective K3 surface, and M a moduli space of semistable sheaves on X. Our choices of the K3 surfaces and the Mukai vectors, as well as the explicit constructions of the moduli space M and the universal family E in the above two cases, are motivated by [10, Example 5.3.7] and [16, Theorem 1.2] respectively. In both cases, the stable sheaves on X are given by the spherical twist (or its inverse) of the ideal sheaves of k points on X around OX , their corresponding moduli spaces M are isomorphic to the Hilbert scheme X [k] of k points on X. All objects in this text are defined over the field of complex numbers C
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