On Irreducible Symplectic Varieties of $\mathrm{K3}^{[n]}$-type in Positive Characteristic

Abstract

We show that there is a good notion of irreducible sympelectic (IS) varieties of K3 [n] -type over an algebraically closed field of characteristic p when p > 2n. Then we generalize Ogus' supersingular crystalline Torelli theorem and Saint-Donat's boundedness results for K3 surfaces to these varieties. As applications, we answer a slight variant of a question asked by F. Charles on moduli spaces of sheaves on K3 surfaces and give a crystalline Torelli theorem for supersingular cubic fourfolds.

On the Shafarevich conjecture for irreducible symplectic varieties

“…Beauville-Bogomolov decomposition [Bea83], [Bog74]), though there are very few examples of irreducible symplectic varieties (namely, K3 [n] -type, generalized Kummer type, OG 6type, and OG 10 -type). Recently, the arithmetic properties of irreducible symplectic varieties have been studided by many people (see [And96], [Yan19], and [Bin21]).…”

Twisted Derived Equivalences and Isogenies between K3 Surfaces in Positive Characteristic

“…We first show that every isometry P Q ∼ → P ′ Q whose induced isomorphism L 2,t ⊗ Q ∼ → L 2,t ′ ⊗ Q sends δ 2,t to δ 2,t ′ is induced by a CSpin-isogeny ψ : A t → A t ′ by conjugation. Indeed, by [51,Prop. 3.2.4], there exists some CSpin-isogeny ψ ′ : A t → A t ′ , which induces some isomorphism P Q ∼ → P ′ Q whose induced isomorphism L 2,t ⊗ Q ∼ → L 2,t ′ ⊗ Q sends δ 2,t to δ 2,t ′ .…”

“…Let k be any algebraically closed field of characteristic 0 or p ≥ 7, X be any K3 surface over κ and Y := X [2] be the Hilbert scheme of 2 points on X. The assumption that p ≥ 7 ensures that Y has the same Hodge numbers as a complex hyperkähler manifold of K3 [2] -type ([51,Prop. 5.1.3]), so that Y is a K3 [2] -type variety in the sense of [51,Def.…”

“…The assumption that p ≥ 7 ensures that Y has the same Hodge numbers as a complex hyperkähler manifold of K3 [2] -type ([51,Prop. 5.1.3]), so that Y is a K3 [2] -type variety in the sense of [51,Def. 1].…”

Finiteness of reductions of Hecke orbits

“…We remark that the theorem above makes no mention of polarizations, so does not simply follow from the Kuga-Satake map for integral models of Shimura varieties. It instead requires an analysis of a Kuga-Satake construction at the level of p-divisible groups due to Yang [Yan19].…”