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I: On the geometry of hyperkähler manifolds

Compact hyperkähler (HK) manifolds (also known as Irreducible holomorphic symplectic manifolds) a rehigher-dimensional analogs of K3 surfaces and actually good ones with many properties in common. However, one of the most valuable theorems, the Torelli theorem for K3 surfaces, is slightly different in the case of HK manifolds. In this part, we introduce some of the main facts about HK manifolds, the known examples, and the notion of deformation type, which is crucial in order to construct new examples of such varieties. In particular, we focus on the moduli space of stable sheaves of K3 surfaces, an example introduced in the seminalworks by Mukai, and later continued by Göttsche-Huybrechts, O'Grady, and Yoshioka.

II: The birational geometry of hyperkähler manifolds of K3-ntype admitting symplectic actions

In this second part, we study a particular type of deformation of HK manifolds admitting a special property: they carry a non-trivial symplectic birational map. We will see that using some standard facts in lattice theory, such varieties are birational to moduli space of stable sheaves on K3 surfaces. Furthermore, by applying techniques in a setting of Bridgeland stability conditions, we can prove that they are isomorphic to some moduli spaces. This last part motivates us to study the wall-crossing of such moduli spaces in order to characterize their Mukaivector. This is a work in collaboration with D. Mattei and Y. Dutta.