Symplectic varieties: their interplay with Fano manifolds and derived categories

Funded by the European Union - NextGenerationEU under the National Recovery and Resilience Plan (PNRR) - Mission 4 Education and research - Component 2 From research to business - Investment 1.1 Notice Prin 2022 - DD N. 104 del 2/2/2022

Algebraic varieties are of great interest within pure mathematics and relate to many problems in number theory, geometry and mathematical physics. In this project, we focus on a special class of algebraic varieties, Hyperkähler (HK) manifolds, and look at their
interactions with many other fields of algebraic geometry: abelian varieties, derived categories and Fano varieties.

The project has many aims and proposes a comprehensive study of the subject. First, we want to study the Chow ring of HK manifolds, and relate them with special Fano varieties called of K3 type.

An instance of this relation is that between a cubic fourfold (Fano of K3 type) and its scheme of lines (HK), where algebraicity of a special class on the HK manifold is related with stable rationality of the Fano variety.

Secondly, we want to study HK manifolds also through their images via finite maps, and to construct further projective families of them.

To do so, we plan to study degenerations of HK manifolds, which are also useful for two other topics of the project: on one hand we want to understand the Motivic zeta function associated to HK varieties, and prove the monodromy conjecture for its poles.

On the other hand, we want to study moduli spaces of lattice polarized HK manifolds, where degenerations naturally occur as representatives of points in the boundary of the moduli space.

The last topic in this part of the project is the study of singular HK varieties, which are an emerging and active topic in the field, where we aim to complete a classification of surfaces and to construct their moduli spaces of sheaves.

Furthermore, we want to understand the behavior of derived equivalent fibered varieties, which could apply also to HK manifolds with a Lagrangian fibration, and to understand when the equivalence is a relative equivalence.

A similar investigation will also be done for Enriques varieties, which are free quotients of HK manifolds, where we aim to study Torelli type theorems, both classical, derived, and twisted.

Finally, in the topic of derived categories, we want to test Orlov’s conjecture about the invariance of Hodge and motivic structures under derived equivalencies in some special cases.

Our project will also focus on two important topics for the study of HK manifolds: the first is the study of positivity properties in abelian varieties, which will lead to a better understanding of their geometry.

This is in strong connection with HK manifolds with Lagrangian fibration, where a relative version of positivity properties might be useful to study the possible fibration structures.

The second is a classification attempt for Fano varieties: we will focus on some Fano varieties of given invariants, e.g. the Lefschetz defect, and on Fano varieties of K3 type, where we aim for an algorithmic classification of fourfolds inside homogeneous spaces.

All the Fano varieties obtained in this way will then be studied in relation to HK manifolds constructed from them.