Supersingular Irreducible Symplectic Varieties

Abstract

In complex geometry, irreducible holomorphic symplectic varieties, also known as compact hyper-Kähler varieties, are natural higher-dimensional generalizations of K3 surfaces. We propose to study such varieties defined over fields of positive characteristic, 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 characteristic p > 0, under natural numerical conditions:

(1) smooth moduli spaces of semistable (twisted) sheaves on K3 surfaces,

(2) smooth Albanese fibers of moduli spaces of semistable 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 expected in general.

On the geometric side, 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 rational chain connectedness is equivalent to supersingularity.

On the motivic side, we conjecture that algebraic cycles on supersingular symplectic varieties are 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 evidence for this, we prove all these conjectures on algebraic cycles for supersingular varieties in both cases (1) and (2).