Locally trivial monodromy of moduli spaces of sheaves on K3 surfaces

TL;DR

This paper investigates the monodromy groups of moduli spaces of sheaves on K3 surfaces, showing that the monodromy of non-primitive cases can be understood via the primitive case, extending previous results.

Contribution

It establishes that the monodromy group of non-primitive moduli spaces is isomorphic to that of the primitive case through a specific inclusion, generalizing Markman's work.

Findings

Monodromy groups of non-primitive moduli spaces are isomorphic to those of primitive cases.

The inclusion of the most singular locus induces the monodromy group isomorphism.

Extension of Markman's results to non-primitive Mukai vectors.

Abstract

In this paper we study monodromy operators on moduli spaces $M_v(S,H)$ of sheaves on K3 surfaces with non-primitive Mukai vectors $v$. If we write $v=mw$, with $m>1$ and $w$ primitive, then our main result is that the inclusion $M_w(S,H)\to M_v(S,H)$ as the most singular locus induces an isomorphism between the monodromy groups of these symplectic varieties, allowing us to extend to the non-primitive case a result of Markman.