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

Moduli Spaces of Stably Irreducible Sheaves on Kodaira Surfaces