On Irreducible Symplectic Varieties of K3[n]-type in Positive Characteristic Ziquan Yang Abstract We show that there is a good notion of irreducible sympelectic varieties of K3[n]-type over an arbitrary field of characteristic zero or p > n + 1. Then we construct mixed characteristic moduli spaces for these varieties. Our main result is a generalization of Ogus’ crystalline Torelli theorem for supersingular K3 surfaces. For 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. Contents 1 Introduction 1 2 General Properties of K3[n]-type Varieties 5 3 Shimura Varieties and Period Morphisms 14 4 Proofs of Theorems 21 5 Examples and Applications 31 A Formal Brauer Groups, Serre’s Witt Vector Cohomology, and Line Bundles 36 1 Introduction A complex irreducible symplectic manifold, or a hyperkähler manifold, is a compact simply connected 0 2 Kähler manifold M whose H (M; ΩM ) is generated by a nowhere degenerate 2-form. Similarly, a smooth projective variety X over C, or rather over any algebraically closed field of characteristic zero, is an 0 2 irreducible symplectic variety if its étale fundamental group is trivial, and its H (X; ΩX ) satisfies the same condition. One of the earliest known classes of irreducible symplectic manifolds are those of K3[n]-type, i.e., those obtained as deformations of the Hilbert scheme of n points on a K3 surface. The first goal of our paper is to show that there is a good notion of K3[n]-type varieties over a general field k, when char k = 0 or p > n + 1. In the following, whenever k denotes a perfect field of characteristic p > 0, we write the associated ring of Witt vectors W (k) as W . For any projective variety Y , Y [n] denotes the Hilbert scheme of n-points on Y . For any field k, k = k¯ means that k is algebraically closed. Definition 1. Let X be a smooth projective variety over a field k. (a) If k = k¯ and char k = 0, X is said to be of K3[n]-type if for some connected variety S over k there exists a smooth projective family X! S of irreducible symplectic varieties over k and points 0 ∼ [n] s; s 2 S(k) such that Xs = X and Xs0 is birational to Y for some K3 surface Y over k. (b) If k¯ = k and char k = p, we say X is a K3[n]-type variety if for some finite flat extension V of W , there exists a smooth projective scheme XV over V lifting X such that (i) XV carries a lifting of a [n] primitive polarization on X and (ii) some geometric generic fiber of XV is of K3 -type and has the same Hodge numbers as X. 1 (c) If k is not algebraically closed, X is said to be of K3[n]-type if for some algebraic closure k¯ of k, the [n] base change Xk¯ is of K3 -type. [n] [n] If k = C, X is of K3 -type if and only if its underlying manifold is of K3 -type (see (2.3.1)). Part (b) of our definition is motivated by Deligne’s result [15, Cor. 1.7] that every (polarized) K3 surface in characteristic p lifts to characteristic zero. Therefore, when n = 1, our definition is equivalent to the usual definition. We will verify that when p > n + 1, part (b) is independent of the choices involved: Theorem 2. Let X be a K3[n]-type variety over an algebraically closed field k with char k = p > n + 1. (a) For any line bundle ζ 2 Pic (X), there exists a finite flat extension W 0 of W together with a formal 0 deformation XbW 0 ! Spf W of X such that ζ extends to XbW 0 . 0 (b) If XV 0 is another deformation of X over a finite flat extension V of W such that some primitive [n] polarization on X extends to XV 0 , then every geometric generic fiber of XV 0 is of K3 -type. Under mild restrictions, the following constructions provide examples of K3[n]-type varieties (see §5.): (i) The Hilbert scheme of n points on a K3 surface. (ii) The moduli space stable sheaves on a K3 surface with a fixed Mukai vector. (iii) The Fano variety of lines on a cubic fourfold. Deformations of K3[n]-type varieties in reasonable families are still K3[n]-type varieties in an arithmetic setting (see §2.3). Next, we show that K3[n]-type varieties have a good mixed-characteristic moduli theory. We consider the functor Mn;d which sends each scheme S over Z(p) to the groupoid of families of primitively polarized K3[n]-type varieties of degree d over S. Our result is: Theorem 3. Assume that p > n + 1. Mn;d is a Deligne-Mumford stack of finite type over Z(p). Moreover, Mn;d has a finite étale cover which is representable by a regular quasi-projective scheme. More precisely, when equipped with orientations and appropriate level structures, primitively polarized K3[n]-type varieties of degree d have a fine moduli space which is representable by a scheme and admits a finite étale forgetful morphism to Mn;d (see (4.2.13) for details). The n = 1 case was treated thoroughly in [51], [36] and [42]. When we take n > 1, the above theorem in particular addresses a question asked by F. Charles in [12] on the boundedness of moduli spaces of stable sheaves on K3 surfaces (see (5.1.4)). [n] Thanks to the foundational work by Verbitsky [57], K3 -type varieties over C, as hyperkähler manifolds, satisfy a global Torelli theorem. In positive characteristics, Ogus proved a crystalline Torelli theorem [49, Thm II] for supersingular K3 surfaces. We generalize Ogus’ theorem to supersingular K3[n]-type varieties: Theorem 4. Assume p > n + 1 and let k be an algebraically closed field of characteristic p. Let X and X0 be two supersingular K3[n]-type varieties over k and let : NS(X) !∼ NS(X0) be an isomorphism which respects the top intersection numbers and sends some ample class to another ample class. Then is induced by an isomorphism f : X0 !∼ X if and only if fits into commutative diagrams NS(X) NS(X0) NS(X) NS(X0) c1 c1 c1 c1 2 cris 2 0 2 p ´et 2 0 p Hcris(X=W ) Hcris(X =W ) H´et(X; Zb ) H´et(X ; Zb ) with an isomorphism cris of W -modules and a primitively polarizable étale parallel transport operator ´et. If n−1 = 0; 1; or a prime power, then is induced by an isomorphism f if and only if fits into a diagram as the one on the left with some isomorphism cris. Clearly this requires some explanation. In general, if k be a perfect field of characteristic p > 0, a [n] 2 K3 -type variety X over k is said to be supersingular if the Newton polygon of Hcris(X=W ) is a straight line. Just as Ogus’ theorem was modeled on the classical Hodge theoretic Torelli theorem for K3 surfaces, the above theorem is modeled on Verbitsky’s theorem ([57, Thm 1.18], see also [39, Thm 1.3]): 2 Theorem 5. (Verbitsky) Let X; X0 be complex irreducible symplectic varieties. An isomorphism of Hodge ∼ structures :H2(X; Z) ! H2(X0; Z) is induced by an isomorphism f : X0 ! X if and only if is a polarizable parallel transport operator. As the name suggests, is a parallel transport operator if and only if there exists a complex analytic family X! S of irreducible symplectic manifolds which contains X; X0 as fibers and realizes via parallel transport along a topological path γ : [0; 1] ! S. We define étale parallel transport operators for the étale cohomology of K3[n]-type varieties over k in the same way except that S is replaced by a connected W -scheme and topological paths are replaced by étale paths (see (4.3.1)). In either case, is said to be (primitively) polarizable if the family X =S can be chosen to admit a relative (primitive) polarization. We will show that deformations of line bundles in families of supersingular K3[n]-type varieties naturally give rise to étale parallel transport operators (see (4.4.4)). The notion of parallel transport operators play a fundamental role in the study of complex hyperkähler manifolds or varieties. It is the author’s hope to illustrate by example of K3[n]-type varieties how one can adapt and work with this notion in an arithmetic setting. We study cubic fourfolds as applications. Let k be a perfect field with char k = p > 0. A cubic 5 i hypersurface Y ⊂ P , or cubic fourfold, is said to be supersingular if the Newton polygon of Hcris(X=W ) is j a straight line for every i. Below we denote by Ch (−)num the Chow group of codimension j cycles modulo ¯ numerical equivalence. Let Fp denote an algebraic closure of Fp. 0 ¯ 0 Theorem 6. Assume p ≥ 7. Let Y and Y be two supersingular cubic fourfolds over Fp. Let h and h 2 0 ∼ 2 denote their hyperplane sections. An isomorphism : Ch (Y )num ! Ch (Y )num is induced by a projective isomorphism f : Y !∼ Y 0 if and only if sends h02 to h2 and fits into a commutative diagram 2 0 2 Ch (Y )num Ch (Y )num cl cl 4 0 cris 4 Hcris(Y =W ) Hcris(Y=W ) with an isomorphism of W -modules cris.