We give a lattice-theoretic characterization for a manifold of OG10 type to be birational to some moduli space of (twisted) sheaves on a K3 surface. We apply it to the Li-Pertusi-Zhao variety of OG10 type associated to any smooth cubic fourfold. Moreover we determine when a birational transformation is induced by an automorphism of the K3 surface and we use this to classify all induced birational symplectic involutions.

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We extend classical results of Perego and Rapagnetta on moduli spaces of sheaves of type OG10 to moduli spaces of Bridgeland semistable objects on the Kuznetsov component of a cubic fourfold. In particular, we determine the period of this class of varieties and use it to understand when they become birational to moduli spaces of sheaves on a K3 surface.

For the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane, we show that a natural Abel-Jacobi map from H 4 prim (Y ) to H 2 prim (Z) is a Hodge isometry. We describe the full H 2 (Z) in terms of the Mukai lattice of the K3 category A of Y . We give numerical conditions for Z to be birational to a moduli space of sheaves on a K3 surface or to Hilb 4 (K3). We propose a conjecture on how to use Z to produce equivalences from A to the derived category of a K3 surface.