Abstract: I will report on some recent results on the geometry of irreducible symplectic varieties that are deformation of moduli spaces of sheaves on K3 surfaces. In the first part of the talk I will introduce the main characters of this story and I will talk about their deformations: this will also include a joint work in progress with A. Perego and A. Rapagnetta on (undesingularisable) singular moduli spaces of sheaves. In the second part I will mostly focus on moduli spaces of O'Grady type and their symplectic desingularisation, presenting an older result on their ample cone (joint with G. Mongardi), and illustrating a recent application to symplectic automorphisms (joint with L. Giovenzana, A. Grossi and D.C. Veniani).
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