Moduli spaces of polarized irreducible symplectic manifolds are not necessarily connected

TL;DR

This paper demonstrates that the moduli space of polarized irreducible symplectic manifolds of a fixed type can be disconnected, using Markman's characterization of polarized parallel-transport operators.

Contribution

It establishes the non-connectedness of moduli spaces for polarized irreducible symplectic manifolds of $K3^{[n]}$-type, highlighting a new topological property.

Findings

Moduli space of polarized $K3^{[n]}$-type manifolds can be disconnected.

Connection between polarization type and moduli space topology.

Application of Markman's characterization to derive non-connectedness.

Abstract

We show that the moduli space of polarized irreducible symplectic manifolds of $K3^{[n]}$-type, of fixed polarization type, is not always connected. This can be derived as a consequence of Eyal Markman's characterization of polarized parallel-transport operators of $K3^{[n]}$-type.