Curves, Ricci flat Varieties and their Interactions

PRIN Project 2020 · PI and local coordinator: Giovanni Mongardi

Algebraic varieties are of great interest within pure mathematics and are connected with many problems in number theory, geometry and mathematical physics. Within this project we focus on the interaction of two particular classes of algebraic varieties, namely varieties with trivial first Chern class and algebraic curves. This is inspired by the well-known result of Lazarsfeld that connects the Brill-Noether theory of a curve C and the geometry of a K3 surface containing it.

The minimal model program reduces the birational classification of smooth projective varieties to the study of three classes: Fano varieties, varieties with trivial first Chern class (Ricci flat varieties) and canonically polarised varieties. In this project we will also include singular varieties and study the respective counterparts of these classes.

Our first aim is to produce new examples of Ricci flat varieties and to make progress towards their classification in small dimension. Moreover we propose to study them as fibered varieties. Fano varieties arise naturally in this context, e.g, if we take a cubic fourfold (which is Fano), its scheme of lines is Ricci flat. In this part of the project we aim to identify which Fano varieties are naturally related to Ricci flat varieties. We also aim at obtaining a criterion for a bimeromorphic recognition of complex tori, the simplest class of Ricci flat varieties.

The second part of the project concerns (moduli of) projective curves. We intend to use compactified Jacobians of curves to improve our understanding of the Hitchin map, to study the cohomology of moduli spaces of curves and of abelian varieties and to use Ricci flat varieties to improve our understanding of the Brill-Noether theory of curves on abelian surfaces.
Furthermore, we want to determine whether it is possible to reconstruct an abelian surface from a general curve contained in it. Here we plan to use the derived category of symplectic varieties. In this context we will also study pairs of singular plane curves which are naturally linked with surfaces isogeneous with a product of curves, having unexpected Betti numbers.
The project has potential applications in the direction of the geometric Langlands correspondence for the Hitchin map, conjectures on the dimension and existence of Brill-Noether loci in the moduli space of curves, cohomology of toroidal compactifications of moduli spaces, rationality questions of Fano varieties and the quantization of classical field theories.

Principal Investigator