Remarks on derived equivalences of Ricci-flat manifolds

After a finite étale cover, any Ricci-flat Kähler manifold decomposes into a product of complex tori, irreducible holomorphic symplectic manifolds, and Calabi–Yau manifolds. We present results indicating that this decomposition is an invariant of the derived category. The main idea to distinguish the derived category of an irreducible holomorphic symplectic manifold from that of a Calabi–Yau manifold is that point sheaves do not deform in certain (non-commutative) deformations of the former, whereas they do for the latter. On the way, we prove a conjecture of Caldararu on the module structure of the Hochschild–Kostant–Rosenberg isomorphism for manifolds with trivial canonical bundle as a direct consequence of recent work by Calaque, van den Bergh, and Ramadoss.