Remarks and questions on coisotropic subvarieties and 0-cycles of hyper-K\"ahler varieties

TL;DR

This paper explores the structure of the Chow ring of hyper-K"ahler varieties, proposing a Beauville decomposition and analyzing 0-cycle groups, with some conjectures proved in specific cases like Hilbert schemes of K3 surfaces.

Contribution

It introduces a conjectural framework for the Chow ring of hyper-K"ahler varieties and proves some conjectures for Hilbert schemes of K3 surfaces and Fano varieties.

Findings

Conjectural structure for the Chow ring of hyper-K"ahler varieties.

Construction of a Beauville decomposition for 0-cycles.

Proofs of some conjectures in specific cases like Hilbert schemes of K3 surfaces.

Abstract

This paper proposes a conjectural picture for the structure of the Chow ring of a (projective) hyper-K\"ahler variety, and the construction of a Beauville decomposition, with emphasis on the Chow group of $0$-cycles, which is endowed with a natural filtration of Brill-Noether type. Some of the conjectures are proved in the case of Hilbert schemes of K3 surfaces and Fano varieties of lines of cubic fourfolds.