Elenco seminari del ciclo di seminari
“WORKSHOP ON IHS MANIFOLDS AND STABILITY CONDITIONS”

Universal functors associated to hyperkahler moduli spaces are conjectured to be P-functors. I will present some ideas of why I expect these functors to become spherical on the twistor space associated to the moduli space. This is work in progress.

Derived and triangulated categories are a fundamental object of study for many mathematicians, both in geometry and in topology. Their structure is however in many ways insufficient, and usually an enhancement is needed to carry on many important constructions on them. In this talk we will discuss existence and uniqueness of such enhancements.

After recent works of Gianluca Pacienza in collaboration with François Charles and Giovanni Mongardi, it seems possible to prove the existence of rational curves on irreducible symplectic varieties (IVS) of a fixed deformation type, starting from special points of their moduli spaces and using the parallel transport operators group. I will introduce this approach, used by Charles, Mongardi and Pacienza in the case of IVS that are deformation equivalent to one of the examples introduced by Arnaud Beauville, and I will tackle the open case of IVS deformation equivalent to the dimension 10 example due to Kieran O'Grady. Furthermore, I will explain some important consequence on the Chow group of 0-cycles of the variety.

In a recent paper Hoering and Peternell completed the proof of the Bogomolov decomposition in the singular projective setting: every normal projective variety which is smooth in codimension 2 and has canonical singularities and numerically trivial canonical bundles, admits a quasi-étale cover which is a product of complex tori, Calabi-Yau varieties and irreducible symplectic varieties. These last are the singular analogue of hyperkaehler manifolds, and share many features with them. In a joint work with A. Rapagnetta we show that all moduli spaces of semistable sheaves over projective K3 surfaces (with respect to generic polarizations) are irreducible symplectic varieties, with the only exception of symmetric products of K3 surfaces. Moreover, we describe their second integral cohomology, their Beauville form and their Fujiki constant.