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Twisted Derived Equivalences and Isogenies Between K3 Surfaces In TWISTED DERIVED EQUIVALENCES AND ISOGENIES BETWEEN K3 SURFACES IN POSITIVE CHARACTERISTIC DANIEL BRAGG AND ZIQUAN YANG Abstract. We study isogenies between K3 surfaces in positive characteristic. Our main result is a char- acterization 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]. Contents 1. Introduction 1 2. B–fields and the twisted Mukai lattice in positive characteristic 5 3. Twisted Chern characters and action on cohomology 15 4. Rational Chow motives and isogenies 17 5. Lifting derived isogenies to characteristic 0 18 6. Existence Theorems 21 7. Uniqueness Theorems 29 8. Isogenies and Hecke Orbits 33 References 37 1. Introduction ∼ An isogeny between two K3 surfaces X and X′ is an isomorphism h2(X′) −→ h2(X) of rational Chow motives which preserves the cohomological pairing (see Def. 4.1). An isogeny has cohomological realizations, 2 ′ ∼ 2 2 which are isomorphisms H (X )Q −→ H (X)Q, where H (−)Q denotes the second rational Betti, ℓ–adic, or crystalline cohomology group. A natural Torelli–type problem is to understand which isomorphisms of arXiv:2102.01193v1 [math.AG] 1 Feb 2021 cohomology groups are induced by isogenies. The purpose of this paper is to investigate this question for K3 surfaces in positive characteristic. This problem has been considered over the complex numbers by Huybrechts [15], who made the impor- tant observation that twisted Fourier-Mukai equivalences provide a natural source for isogenies between K3 surfaces. For a K3 surface X and a Brauer class α ∈ Br(X), we denote by Db(X, α) the bounded derived category of coherent α-twisted sheaves on X. Suppose given another K3 surface X′, a class α′ ∈ Br(X′), ∼ and a Fourier–Mukai equivalence Φ: Db(X′, α′) −→ Db(X, α). Up to some choices, Φ induces an isogeny ∼ h2(X′) −→ h2(X) (see section 4). We call isogenies which arise this way primitive derived isogenies, and compositions of such isogenies derived isogenies. Huybrechts showed the following structural theorem for derived isogenies between K3 surfaces over the complex numbers (refining earlier work of Buskin [8, Thm. 1.1]). Theorem 1.1. (Huybrechts, [15, Thm. 0.1]) Let X,X′ be two K3 surfaces over C. Every isomorphism of ∼ Hodge structures H2(X′, Q) → H2(X, Q) which preserves the Poicaré pairings is induced by a derived isogeny ∼ f : h2(X′) → h2(X). 1 2 DANIEL BRAGG AND ZIQUAN YANG Schematically, we may visualize this result as follows. Fix two K3 surfaces X and X′. We have maps derived isogenies isogenies Hodge isometries (1) ∼ ∼ ∼ h2(X′) −→ h2(X) h2(X′) −→ h2(X) H2(X′, Q) −→ H2(X, Q) Huybrecht’s result is that the composition is surjective. In particular, every isogeny is equivalent to a derived isogeny (see Def. 4.1). This result affirms Shafarevich’s conjecture that the right map of (1) should be surjective. Note that this latter statement only concerns isogenies; it was an insight of Huybrechts to prove this by considering derived isogenies. We now turn to positive characteristic. There is no hope of a direct translation of Thm. 1.1, as there is no suitably strong replacement for the Hodge structure on H2. We instead consider a variant of Huybrecht’s theorem: rather than fixing two K3 surfaces X and X′ as before, we fix one K3 surface X and consider ∼ pairs (X′,f : h2(X′) → h2(X)), where X′ is a K3 surface and f is an isogeny. Given such a pair, we get an 2 ′ 2 2 2 integral sublattice f∗(H (X , Z)) ⊂ H (X, Q), typically different from the sublattice H (X, Z) ⊂ H (X, Q). ⊕3 ⊕2 Let Λ= U ⊕ E8 be the K3 lattice, where U is the standard hyperbolic plane and E8 is the unique even unimodular negative definite Z-lattice of rank 8. We have the following maps. ′ f is a derived ′ f is an sublattices (2) (X ,f) (X ,f) 2 ∼ ( isogeny ) ( isogeny ) H ⊂ H (X, Q),H = Λ Using the surjectivity of the period map and the global Torelli theorem, one checks that Huybrecht’s theorem is equivalent to the surjectivity of the composition of the maps (2) (and hence implies the surjectivity of the right map). Note that this statement does not involve Hodge structures. This suggests a more flexible problem: we consider the analog of (2) with the singular cohomology group H2(X, Q) replaced with an appropriate rational cohomology group attached to X, and the right hand set replaced with the set of embeddings of suitable integral lattices. We then ask whether either the composition or the right hand map is surjective, or more generally for a characterization of their images. For example, over any algebraically closed field of characteristic 0, one can formulate a purely algebraic statement using ℓ–adic cohomology; see Rem. 6.11 and Thm. 6.12. With this motivation, we formulate a result which we view as a positive characteristic analogue of Huy- brecht’s theorem. Let k be an algebraically closed field of characteristic p > 0. Write W for the ring of Witt vectors W (k) and let K = W [p−1] denote its field of fractions. For a variety Y over k, we set ∗ p ∗ ∗ p ∗ p ′ ∗ H (Y, Z ) := ℓ=6 p H (Y, Zℓ) and H (Y, Af ) := H (Y, Z ) ⊗Z Q = ℓ=6 p H (Y, Qℓ), where the restricted ∗ product consists of tuples {xℓ} such that for all but finitely many ℓ we have xℓ ∈ H (Y, Zℓ), and as usual ∗ Q∗ Q H (Y, Qb ℓ) := H (Y, Zℓ) ⊗Zℓ Qℓ. We set b ∗ ∗ ∗ p ∗ ∗ ∗ ∗ p H (Y ) := Hcris(Y/W ) × H´et(Y, Z ) and H (Y )Q := H (Y ) ⊗Z Q = Hcris(Y/K) × H (Y, Af ) where H∗ (Y/K) := H∗ (Y/W ) ⊗ K. We shall also use Ogus’ notion of K3 crystals ([42, Def. 3.1]). Using cris cris Wb these notations, we have: Theorem 1.2. Let X be a K3 surface over an algebraically closed field k of characteristic p ≥ 5. Let p p p 2 Hp = Λ ⊗ W and H = Λ ⊗ Z . Let ι : Hp × H ֒→ H (X)Q be an isometric embedding such that the 2 restriction of the crystalline Frobenius on Hcris(X/K) endows Hp with the structure of a K3 crystal. ′ 2 ′ ∼ 2 2 ′ There exists a K3 surface Xb and a derived isogeny f : h (X ) → h (X) such that f∗(H (X )) = im(ι) if 2 and only if ι maps the slope < 1 part of Hp isomorphically to the slope < 1 part of Hcris(X/W ). Here, an isometric embedding is a map of W × Zp-modules which is compatible with the pairings (see also Rem. 6.6). The extra restriction in Thm. 1.2 on the image under ι of the slope < 1 part of crystalline cohomology is trivial if X is supersingular, but in theb finite height case gives a nontrivial restriction on the cohomological realizations of derived isogenies. This restriction may be seen as arising from the fact that the p-primary part of the Brauer group of a finite height K3 surface is smaller than expected (see Rem. 2.9). As a consequence of this restriction, not every isogeny in characteristic p is equivalent to a derived isogeny, unlike in characteristic 0. In terms of the obvious analog of the diagram (2), this shows that the composition is not surjective, instead having image equal to the subset of embeddings obeying this condition at p. Given this, it is of interest to also characterize the image of the right hand map. It seems likely that it should DERIVED ISOGENY THEORY FOR K3 SURFACES IN POSITIVE CHARACTERISTIC 3 nevertheless be surjective; that is, that the restriction on ι in Thm. 1.2 can be removed if we instead consider all isogenies. We prove this under some technical assumptions in the following result. Theorem 1.3. Assume that k = F¯ p and that p ≥ 5. Let X be a K3 surface over k of finite height, and let p Hp,H , and ι be as in Thm. 1.2. Suppose that at least one of the following conditions holds. (a) Pic(X) has rank ≥ 12 or contains a standard hyperbolic plane. (b) Pic(X) contains an ample line bundle L of degree L2 <p − 4. (c) X is ordinary, and contains a line bundle of prime-to-p degree. ′ 2 ′ ∼ 2 2 ′ There exists another K3 surface X over k and an isogeny f : h (X ) → h (X) such that f∗(H (X )) = im(ι). This above result is a strengthening of [52, Thm. 1.4]. The isogenies produced in the conclusion of Thm. 1.3 are obtained as specializations of derived isogenies in characteristic 0, but need not themselves be derived isogenies. We mention that a byproduct in the course of proving the above is a generalization (Thm. 6.22) of Taelman’s characterization of the canonical liftings of ordinary K3 surfaces [50, Thm. C] to all non- supersingular K3 surfaces, for which canonical liftings are generalized by “Nygaard-Ogus liftings”.
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