We will present recent advances in the field of singular varieties with trivial canonical class obtained in joint work with Bakker and Bakker-Guenancia building on work by many others. This includes the decomposition theorem which says that such a variety is up to a finite cover isomorphic to a product of a torus, irreducible Calabi-Yau (ICY) and irreducible symplectic varieties (ISV). For ISVs, we explain in more detail a general framework to approach the global moduli theory of these varieties, building among other things on recent progress in local deformation theory of ISVs or more generally of K-trivial varieties, on finiteness results of algebraic ISVs, and on Ratner theory. This approach culminates in a Global Torelli theorem for the varieties in question as soon as the second Betti number is at least 5.

We aim at constructing ihs varieties with trivial algebraic regular fundamental group starting from an ihs manifold X and a finite group G of symplectic actions on X. I will present some work in progress in the case where X is a generalized Kummer fourfold or an O’Grady’s sixfold. The case of generalized Kummer fourfolds is the content of a joint work with Bertini, Capasso, Mauri and Mazzon. Moreover I will present some classification results about symplectic birational transformations of manifolds of OG6 type (joint work with Onorati and Veniani) and how these results could be exploit for our purpose.

The example of irreducible symplectic orbifold the most studied in the literature can be obtained via a resolution in codimension 2 of a Hilbert scheme of two points on a K3 surface quotiented by a symplectic involution. In this talk, I will explain how can be described the wall divisors of this orbifold. This is a work in collaboration with U. Riess.

We compare two prominent classes of singular symplectic varieties with properties similar to IHSM. To show that they coincide in the smooth case, we prove that in the definition of IHSM being simply connected can be replaced by vanishing irregularity. The proof uses the Beauville-Bogomolov decomposition theorem as well as representation theory of finite groups to analyze quotients of complex tori. We provide new singular examples and information on their Lagrangian fibrations.

(Part II) We will present recent advances in the field of singular varieties with trivial canonical class obtained in joint work with Bakker and Bakker-Guenancia building on work by many others. This includes the decomposition theorem which says that such a variety is up to a finite cover isomorphic to a product of a torus, irreducible Calabi-Yau (ICY) and irreducible symplectic varieties (ISV). For ISVs, we explain in more detail a general framework to approach the global moduli theory of these varieties, building among other things on recent progress in local deformation theory of ISVs or more generally of K-trivial varieties, on finiteness results of algebraic ISVs, and on Ratner theory. This approach culminates in a Global Torelli theorem for the varieties in question as soon as the second Betti number is at least 5.

In this talk, I will first describe families of projective fourfolds of K3^[2]-type carrying a symplectic involution. To each of these families one can associate a family of projective irreducible symplectic orbifolds, obtained as partial resolution of their quotients, which we call Nikulin orbifolds. I will explain how to study projective models of these orbifolds and finally, if time allows, I will describe a locally complete projective family containing Nikulin orbifolds of degree two. This is joint work with A. Garbagnati, G. Kapustka and M. Kapustka.

In a joint work with A. Rapagnetta we recently proved that moduli spaces of semistable sheaves on K3 surfaces are irreducible symplectic varieties (with the only exception of symmetric product of K3 surfaces). The aim of this talk is to describe the Hodge structure and the Beauville form on the second integral cohomology of these moduli space, which is the content of a recent joint work with Rapagnetta: in particular, we will prove that if S is a projective K3 surface, v is a Mukai vector on S of positive Mukai square and H is a generic polarization, then there is a morphism from the orthogonal of v to the second integral cohomology of the moduli space M_v(S,H) of H-semistable sheaves on S with Mukai vector v, and that this morphism is a Hodge morphism, a group isomorphism and an isometry. This extends previous results obtained by O'Grady and Yoshioka for smooth moduli spaces, and by the author and Rapagnetta for moduli spaces having a symplectic resolution.