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.

We generalize Mukai and Shafarevich’s definitions of isogenies between K3 surfaces over ${\mathbb{C}}$ to an arbitrary perfect field and describe how to construct isogenous K3 surfaces over $\bar{{\mathbb{F}}}_p$ by prescribing linear algebraic data when $p$ is large. The main step is to show that isogenies between Kuga–Satake abelian varieties induce isogenies between K3 surfaces, in the context of integral models of Shimura varieties. As a byproduct, we show that every K3 surface of finite height admits a CM lifting under a mild assumption on $p$.

We study isogenies between K3 surfaces in positive characteristic. Our main result is a characterization of K3 surfaces isogenous to a given K3 surface X in terms of certain integral sublattices of the second rational ℓ-adic and crystalline cohomology groups of X. This is a positive characteristic analog of a result of Huybrechts [15], and extends results of [52]. We give applications to the reduction types of K3 surfaces and to the surjectivity of the period morphism. To prove these results we describe a theory of B-fields and Mukai lattices in positive characteristic, which may be of independent interest. We also prove some results on lifting twisted Fourier-Mukai equivalences to characteristic 0, generalizing results of Lieblich and Olsson [29].

We generalize Mukai and Shafarevich's definition of isogenies between K3 surfaces over C to an arbitrary perfect field and describe how to construct isogenous K3 surfaces over Fp by prescribing linear algebraic data when p is large. The main step is to show that isogenies between Kuga-Satake abelian varieties induce isogenies between K3 surfaces, in the context of integral models of Shimura varieties. As a byproduct, we show that every K3 surface of finite height admits a CM lifting under a mild assumption on p.Surjectivity of the period map gives us a complex analytic K3 X ′ which corresponds to the induced Hodge structure on Λ. Then one can use [3, IV Theorem 6.2] to see that X ′ has to be algebraic.We give an analogue of Theorem 1.2 for quasi-polarized K3 surfaces over Fp , using K3 crystals and Z p -lattices to replace Z-lattices. K3 crystals were originally defined by Ogus to characterize the F-crystals that arise from K3 surfaces (c.f. [34, Definition 3.1]): Definition 1.3. Let k be a perfect field of characteristic p > 0 and let σ be the lift of Frobenius on W (k). A K3 crystal over k is a finitely generated free W (k)-module D equipped with a σ-linear injection ϕ : D → D and a perfect symmetric bilinear pairing −, − such that ϕ(x), ϕ(y) = p 2 σ( x, y ), p 2 D ⊂ ϕ(D), and rank ϕ⊗k = 1.Recall that a quasi-polarized K3 surface is a pair (X, ξ) where X is a K3 surface and ξ is a primitive big and nef line bundle on X. We call the self-intersection number of ξ the degree of (X, ξ). Customarily, we write P 2 * (X) := ch * (ξ) ⊥ ⊂ H 2 * (X) for the primitive cohomology of (X, ξ) ( * = dR, cris, ét, etc). Our main theorem states:

The Tate conjecture for squares of K3 surfaces over finite fields was recently proved by Ito-Ito-Koshikawa. We give a more geometric proof when the characteristic is at least 5. The main idea is to use twisted derived equivalences between K3 surfaces to link the Tate conjecture to finiteness results over finite fields, in the spirit of Tate.