Moduli spaces of stably irreducible sheaves on Kodaira surfaces

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 would fit into a possible classification of holomorphic symplectic manifolds by deformation type. This article studies a natural Lagrangian fibration on these moduli spaces to determine that they are not Kahler or simply connected, ruling out most of the known deformation types of holomorphic symplectic manifolds.