Moduli Spaces of Stably Irreducible Sheaves on Kodaira Surfaces

TL;DR

This paper investigates the structure of moduli spaces of stably irreducible sheaves on Kodaira surfaces, revealing they are non-Kähler and not simply connected, thus challenging existing classification schemes of holomorphic symplectic manifolds.

Contribution

It introduces a natural Lagrangian fibration on these moduli spaces and analyzes their properties to distinguish them from known deformation types.

Findings

Moduli spaces are not Kähler.

They are not simply connected.

They do not fit into existing classification types.

Abstract

Moduli spaces of stably irreducible sheaves on Kodaira surfaces belong to the short list of examples of smooth and compact holomorphic symplectic manifolds, and it is not yet known how they fit into the classification of holomorphic symplectic manifolds by deformation type. This paper studies a natural Lagrangian fibration on these moduli spaces to determine that they are not K\"ahler or simply connected, ruling out most of the known deformation types of holomorphic symplectic manifolds.