Supersingular O'Grady varieties of dimension six

TL;DR

This paper extends O'Grady's 6-dimensional irreducible holomorphic symplectic varieties to positive characteristic fields, demonstrating supersingular OG6 varieties are unirational with algebraic cohomology and Tate-type Chow motives, supporting key conjectures.

Contribution

The authors construct supersingular OG6 varieties in positive characteristic and prove their unirationality, algebraic cohomology, and Tate-type Chow motives, confirming several conjectures.

Findings

Supersingular OG6 varieties are unirational.

Their rational cohomology is generated by algebraic classes.

Their Chow motives are of Tate type.

Abstract

O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields of positive characteristic p greater than 2, called OG6 varieties. We show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin--Shioda conjecture, the supersingular Tate conjecture and the supersingular Bloch conjecture proposed in our previous work, in analogy with the theory of supersingular K3 surfaces.