Supersingular irreducible symplectic varieties

TL;DR

This paper investigates supersingular symplectic varieties over fields of positive characteristic, exploring their geometric and motivic properties, and establishing conjectures about their structure, rationality, and algebraic cycles.

Contribution

It introduces equivalent definitions of supersingularity for these varieties, proves conjectures relating supersingularity to unirationality and chain connectedness, and analyzes their algebraic cycle structure.

Findings

Supersingularity is equivalent to unirationality for certain symplectic varieties.

Supersingular symplectic varieties have simplified algebraic cycle structures.

All conjectures about algebraic cycles are proven for these supersingular varieties.

Abstract

We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of positive characteristic, under natural numerical conditions: (1) smooth moduli spaces of sheaves on K3 surfaces and (2) smooth Albanese fibers of moduli spaces of sheaves on abelian surfaces. Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to be equivalent in both cases (1) and (2). Their equivalence is conjectured in general. On the geometric aspect, we conjecture that unirationality characterizes supersingularity for symplectic varieties. Such an equivalence is established in case (1), assuming the same is true for K3 surfaces. In case (2), we show that the rational chain connectedness is equivalent to supersingularity. On the motivic aspect, we conjecture that the algebraic cycles on supersingular symplectic varieties should be much simpler than their complex counterparts: its rational Chow motive is of supersingular abelian type, the rational Chow ring is representable and satisfies the Bloch--Beilinson conjecture and Beauville's splitting property. As evidences, we prove all these conjectures on algebraic cycles for supersingular varieties in both cases (1) and (2).