A Stronger Derived Torelli Theorem for K3 Surfaces

Abstract

In an earlier paper the notion of a filtered derived equivalence was introduced, and it was shown that if two K3 surfaces admit such an equivalence, then they are isomorphic. In this paper we study more refined aspects of filtered derived equivalences related to the action on the cohomological realizations of the Mukai motive. It is shown that if a filtered derived equivalence between K3 surfaces also preserves ample cones then one can find an isomorphism that induces the same map as the equivalence on the cohomological realizations.

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1 Introduction

1.1

Let k be an algebraically closed field of odd positive characteristic and let X and Y be K3 surfaces over k. Let

$$ \Phi :D(X)\rightarrow D(Y) $$

be an equivalence between their bounded triangulated categories of coherent sheaves given by a Fourier–Mukai kernel $P\in D(X\times Y)$, so $\Phi$ is the functor given by sending $M\in D(X)$ to

$$ R\text {pr}_{2*}(L\text {pr}_1^*M\otimes ^{\mathbb {L}}P). $$

As discussed in [16, 2.9] the kernel P also induces an isomorphism on rational Chow groups modulo numerical equivalence

$$ \Phi _P^{A^*}:A^*(X)_{\text {num}, \mathbb {Q}}\rightarrow A^*(Y)_{\text {num}, \mathbb {Q}}. $$

We can consider how a given equivalence $\Phi$ interacts with the codimension filtration on $A^*$, or how it acts on the ample cone of X inside $A^1(X)$. The underlying philosophy of this work is that tracking filtrations and ample cones (in ways we will make precise in Sect. 2) gives a semi-linear algebraic gadget that behaves a lot like a Hodge structure. In Sect. 2 we will define a notion of strongly filtered for an equivalence $\Phi$ that imposes conditions reminiscent of the classical Torelli theorem for K3 surfaces.

With this in mind, the purpose of this paper is to prove the following result.

Theorem 1.2

If $\Phi _P:D(X)\rightarrow D(Y)$ is a strongly filtered equivalence, then there exists an isomorphism $\sigma :X\rightarrow Y$ such that the maps on the crystalline and étale realizations of the Mukai motive induced by $\Phi _P$ and $\sigma$ agree.

For the definition of the realizations of the Mukai motive see [16, Sect. 2]. In [16, Proof of 6.2] it is shown that any filtered equivalence can be modified to be strongly filtered. As a consequence, we get a new proof of the following result.

Theorem 1.3

([16, 6.1]). If $\Phi _P^{A^*}$ preserves the codimension filtrations on $A^*(X)_{\text {num}, \mathbb {Q}}$ and $A^*(Y)_{\text {num}, \mathbb {Q}}$ then X and Y are isomorphic.

Whereas the original proof of Theorem 1.3 relied heavily on liftings to characteristic 0 and Hodge theory, the proof presented here works primarily in positive characteristic using algebraic methods.

In Sect. 8 we present a proof of Theorem 1.2 using certain results about “Kulikov models” in positive characteristic (see Sect. 5). This argument implicitly uses Hodge theory which is an ingredient in the proof of Theorem 5.3. In Sect. 9 we discuss a characteristic 0 variant of Theorem 1.2, and finally in the last Sect. 10 we explain how to bypass the use of the Hodge theory ingredient of Theorem 5.3. This makes the argument entirely algebraic, except for the Hodge theory aspects of the proof of the Tate conjecture. This also gives a different algebraic perspective on the statement that any Fourier–Mukai partner of a K3 surface is a moduli space of sheaves, essentially inverting the methods of [16].

The bulk of this paper is devoted to proving Theorem 1.2. The basic idea is to consider a certain moduli stack $\mathscr {S}_d$ classifying data $((X, \lambda ), Y, P)$ consisting of a primitively polarized K3 surface $(X, \lambda )$ with polarization of some degree d, a second K3 surface Y, and a complex $P\in D(X\times Y)$ defining a strongly filtered Fourier–Mukai equivalence $\Phi _P:D(X)\rightarrow D(Y)$. The precise definition is given in Sect. 3, where it is shown that $\mathscr {S}_d$ is an algebraic stack which is naturally a $\mathbb {G}_m$-gerbe over a Deligne–Mumford stack $\overline{\mathscr {S}}_d$ étale over the stack $\mathscr {M}_d$ classifying primitively polarized K3 surfaces of degree d. The map $\overline{\mathscr {S}}_d\rightarrow \mathscr {M}_d$ is induced by the map sending a collection $((X, \lambda ), Y, P)$ to $(X, \lambda )$. We then study the locus of points in $\mathscr {S}_d$ where Theorem 1.2 holds showing that it is stable under both generization and specialization. From this it follows that it suffices to consider the case when X and Y are supersingular where we can use Ogus’ crystalline Torelli theorem [24, Theorem I].

Remark 1.4

Our restriction to odd characteristic is because we appeal to the Tate conjecture for K3 surfaces, proven in odd characteristics by Charles, Maulik, and Madapusi Pera [7, 21, 26], which at present is not known in characteristic 2.

1.5

(Acknowledgments) Lieblich partially supported by NSF CAREER Grant DMS-1056129 and Olsson partially supported by NSF grant DMS-1303173 and a grant from The Simons Foundation. Olsson is grateful to F. Charles for inspiring conversations at the Simons Symposium “Geometry Over Nonclosed Fields” which led to results of this paper. We also thank E. Macrì, D. Maulik, A. Cǎldǎraru, and K. Madapusi Pera for useful correspondence, as well as the referee for a number of helpful comments. Finally, we thank D. Bragg, T. Srivastava, and S. Tirabassi for pointing out errors in an earlier version of this paper and suggesting corrections.

2 Strongly Filtered Equivalences

2.1

Let X and Y be K3 surfaces over an algebraically closed field k and let $P\in D(X\times Y)$ be an object defining an equivalence

$$ \Phi _P:D(X)\rightarrow D(Y), $$

and let

$$ \Phi ^{A^*_{\text {num}, \mathbb {Q}}}_P:A^*(X)_{\text {num}, \mathbb {Q}}\rightarrow A^*(Y)_{\text {num}, \mathbb {Q}} $$

denote the induced map on Chow groups modulo numerical equivalence and tensored with $\mathbb {Q}$. We say that $\Phi _P$ is filtered (resp. strongly filtered, resp. Torelli) if $\Phi ^{A^*_{\text {num}, \mathbb {Q}}}_P$ preserves the codimension filtration (resp. is filtered, sends (1, 0, 0) to (1, 0, 0), and sends the ample cone of X to plus or minus the ample cone of Y; resp. is filtered, sends (1, 0, 0) to $\pm (1, 0, 0)$, and sends the ample cone of X to the ample cone of Y).

Remark 2.2

Note that if P is strongly filtered then either P or P[1] is Torelli. If P is Torelli then either P or P[1] is strongly filtered.

Remark 2.3

Note that $A^1(X)$ is the orthogonal complement of $A^0(X)\oplus A^2(X)$ and similarly for Y. This implies that if $\Phi _P$ is filtered and sends (1, 0, 0) to $\pm (1, 0, 0)$ then $\Phi _P(A^1(X)_{\text {num}, \mathbb {Q}})\subset A^1(Y)_{\text {num}, \mathbb {Q}}$.

Remark 2.4

It is shown in [16, 6.2] that if $\Phi _P:D(X)\rightarrow D(Y)$ is a filtered equivalence, then there exists a strongly filtered equivalence $\Phi :D(X)\rightarrow D(Y)$. In fact it is shown there that $\Phi$ can be obtained from $\Phi _P$ by composing with a sequence of shifts, twists by line bundles, and spherical twists along $(-2)$-curves.

2.5

As noted in [16, 2.11] an equivalence $\Phi _P$ is filtered if and only if the induced map on Chow groups

$$ \Phi ^{A^*_{\text {num}, \mathbb {Q}}}_P:A^*(X)_{\text {num}, \mathbb {Q}}\rightarrow A^*(Y)_{\text {num}, \mathbb {Q}} $$

sends $A^2(X)_{\text {num}, \mathbb {Q}}$ to $A^2(X)_{\text {num}, \mathbb {Q}}$.

Lemma 2.6

Let $\ell$ be a prime invertible in k, let $\widetilde{H}(X, \mathbb {Q}_\ell )$ (resp. $\widetilde{H}(Y, \mathbb {Q}_\ell )$) denote the $\mathbb {Q}_\ell$-realization of the Mukai motive of X (resp. Y) as defined in [16, 2.4], and let

$$\Phi _P^{{\acute{\mathrm{{e}}}}\text {t }}:\widetilde{H}(X, \mathbb {Q}_\ell )\rightarrow \widetilde{H}(Y, \mathbb {Q}_\ell )$$

denote the isomorphism defined by P. Then $\Phi _P$ is filtered if and only if $\Phi _P^{\acute{\mathrm{{e}}}}\text {t }$ preserves the filtrations by degree on $\widetilde{H}(X, \mathbb {Q}_\ell )$ and $\widetilde{H}(Y, \mathbb {Q}_\ell )$.

Proof

By the same reasoning as in [16, 2.4] the map $\Phi _P^{{\acute{\mathrm{{e}}}}\text {t }}$ is filtered if and only if

$$\Phi _P^{\acute{\mathrm{{e}}}}\text {t }(H^4(X, \mathbb {Q}_\ell )) = H^4(Y, \mathbb {Q}_\ell ).$$

Since the cycle class maps

$$ A^2(X)_{\text {num}, \mathbb {Q}}\otimes _{\mathbb {Q}}\mathbb {Q}_\ell \rightarrow H^4(X, \mathbb {Q}_\ell ), \ \ A^2(Y)_{\text {num}, \mathbb {Q}}\otimes _{\mathbb {Q}}\mathbb {Q}_\ell \rightarrow H^4(Y, \mathbb {Q}_\ell ) $$

are isomorphisms and the maps $\Phi _P$ and $\Phi _P^{\acute{\mathrm{{e}}}}\text {t }$ are compatible in the sense of [16, 2.10] it follows that if $\Phi _P$ is filtered then so is $\Phi _P^{\acute{\mathrm{{e}}}}\text {t }$. Conversely if $\Phi _P^{\acute{\mathrm{{e}}}}\text {t }$ is filtered then since the cycle class maps

$$ A^*(X)_{\text {num}, \mathbb {Q}}\rightarrow \widetilde{H}(X, \mathbb {Q}_\ell ), \ \ A^*(Y)_{\text {num}, \mathbb {Q}}\rightarrow \widetilde{H}(Y, \mathbb {Q}_\ell ) $$

are injective it follows that $\Phi _P$ is also filtered. $\square$

Remark 2.7

The same proof as in Lemma 2.6 gives variant results for crystalline cohomology and in characteristic 0 de Rham cohomology.

The condition that $\Phi _P$ takes the ample cone to plus or minus the ample cone appears more subtle. A useful observation in this regard is the following.

Lemma 2.8

Let $P\in D(X\times Y)$ be an object defining a filtered equivalence $\Phi _P:D(X)\rightarrow D(Y)$ such that $\Phi _P^{A^*_{\text {num}}}$ sends (1, 0, 0) to (1, 0, 0). Then $\Phi _P$ is strongly filtered if and only if for some ample invertible sheaf L on X the class $\Phi _P^{A^*_{\text {num}}}(L)\in NS(Y)_\mathbb {Q}$ is plus or minus an ample class.

Proof

Following [24, p. 366] define

$$ V_X:= \{x\in NS(X)_\mathbb R|x^2>0, \ \text {and } \langle x, \delta \rangle \ne 0 \text { for all } \delta \in NS(X) \text { with } \delta ^2 = -2\}, $$

and define $V_Y$ similarly. Since $\Phi _P^{A^*_{\text {num}}}$ is an isometry it induces an isomorphism

$$ \sigma :V_X\rightarrow V_Y. $$

By [24, Proposition 1.10 and Remark 1.10.9] the ample cone $C_X$ (resp. $C_Y$) of X (resp. Y) is a connected component of $V_X$ (resp. $V_Y$) and therefore either $\sigma (C_X)\cap C_Y = \emptyset$ or $\sigma (C_X) = C_Y$, and similarly $\sigma (-C_X) \cap C_Y = \emptyset$ or $\sigma (-C_X) = C_Y$.$\square$

Proposition 2.9

Let X and Y be K3-surfaces over a scheme S and let $P\in D(X\times _SY)$ be a relatively perfect complex. Assume that X / S is projective. Then the set of points $s\in S$ for which the induced transformation on the derived category of the geometric fibers

$$ \Phi _{P_{\bar{s}}}:D(X_{\bar{s}})\rightarrow D(Y_{\bar{s}}) $$

is a strongly filtered equivalence is an open subset of S.

Proof

By a standard reduction we may assume that S is of finite type over $\mathbb {Z}$.

First note that the condition that $\Phi _{P_{\bar{s}}}$ is an equivalence is an open condition. Indeed by [12, 5.9] if $P^\vee$ denotes $\mathscr {R}Hom (P, \mathscr {O}_{X\times _SY})$ and Q denotes $P^\vee [2]$, then for every $s\in S$ the functor $\Phi _{Q_{\bar{s}}}$ is both left and right adjoint to $\Phi _{P_{\bar{s}}}$, and the adjunction maps are induced by morphisms of complexes

$$ Rp_{13*}(Lp_{12}^*P\otimes ^{\mathbb {L}}Lp_{23}^*Q)\rightarrow \Delta _{X*}\mathscr {O}_X, \ \ Rq_{13*}(Lq_{12}^*Q\otimes Lq_{23}^*P)\rightarrow \Delta _{Y*}\mathscr {O}_Y, $$

where we write $p_{ij}$ (resp. $q_{ij}$) for the projection from $X\times _SY\times _SX$ (resp. $Y\times _SX\times _SY$) to the product of the i-th and j-th factor. Since the condition that these adjunction maps are quasi-isomorphisms is clearly an open condition on S it follows that the condition that $\Phi _{P_{\bar{s}}}$ is an equivalence is open.

Replacing S by an open set we may therefore assume that $\Phi _{P_{\bar{s}}}$ is an equivalence in every fiber.

Next we show that the condition that $\Phi _P$ is filtered is an open and closed condition. For this we may assume we have a prime $\ell$ invertible in S. Let $f_X:X\rightarrow S$ (resp. $f_Y:Y\rightarrow S$) be the structure morphism. Define $\widetilde{\mathscr {H}}_{X/S}$ to be the lisse $\mathbb {Q}_\ell$-sheaf on S given by

$$ \widetilde{\mathscr {H}}_{X/S}:= (R^0f_{X*}\mathbb {Q}_\ell (-1))\oplus (R^2f_{X*}\mathbb {Q}_\ell )\oplus (R^4f_{X*}\mathbb {Q}_\ell )(1), $$

and define $\widetilde{\mathscr {H}}_{Y/S}$ similarly. The kernel P then induces a morphism of lisse sheaves

$$ \Phi _{P/S}^{{\acute{\mathrm{{e}}}}\text {t }, \ell }:\widetilde{\mathscr {H}}_{X/S}\rightarrow \widetilde{\mathscr {H}}_{Y/S} $$

whose restriction to each geometric fiber is the map on the $\mathbb {Q}_\ell$-realization of the Mukai motive as in [16, 2.4]. In particular, $\Phi _{P/S}^{{\acute{\mathrm{{e}}}}\text {t }, \ell }$ is an isomorphism. By Lemma 2.6 for every geometric point $\bar{s}\rightarrow S$ the map $\Phi _{P_{\bar{s}}}$ is filtered if and only if the stalk $\Phi _{P/S, \bar{s}}^{{\acute{\mathrm{{e}}}}\text {t }, \ell }$ preserves the filtrations on $\widetilde{\mathscr {H}}_{X/S}$ and $\widetilde{\mathscr {H}}_{Y/S}$. In particular this is an open and closed condition on S. Shrinking on S if necessary we may therefore further assume that $\Phi _{P_{\bar{s}}}$ is filtered for every geometric point $\bar{s}\rightarrow S$.

It remains to show that in this case the set of points s for which $\Phi _P$ takes the ample cone $C_{X_{\bar{s}}}$ of $X_{\bar{s}}$ to $\pm C_{Y_{\bar{s}}}$ is an open subset of S. For this we can choose, by our assumption that X / S is projective, a relatively ample invertible sheaf L on X. Define

$$ M:= \text {det}(R\text {pr}_{2*}(L\text {pr}_1^*(L)\otimes P)), $$

an invertible sheaf on Y. Then by Lemma 2.8 for a point $s\in S$ the transformation $\Phi _{P_{\bar{s}}}$ is strongly filtered if and only if the restriction of M to the fiber $Y_{\bar{s}}$ is plus or minus the class of an ample divisor. By openness of the ample locus [9, III, 4.7.1] we get that being strongly filtered is an open condition. $\square$

3 Moduli Spaces of K3 Surfaces

3.1

For an integer d invertible in k let $\mathscr {M}_{d}$ denote the stack over k whose fiber over a scheme T is the groupoid of pairs $(X, \lambda )$ where X / T is a proper smooth algebraic space all of whose geometric fibers are K3 surfaces and $\lambda :T\rightarrow \text {Pic}_{X/T}$ is a morphism to the relative Picard functor such that in every geometric fiber $\lambda$ is given by a primitive ample line bundle $L_\lambda$ whose self-intersection is 2d. The following theorem summarizes the properties of the stack $\mathscr {M}_d$ that we will need (see [17, 2.10] for a more detailed discussion considering also the case when the characteristic of k divides d).

Theorem 3.2

(i) $\mathscr {M}_d$ is a Deligne–Mumford stack, smooth over k of relative dimension 19.

(ii) The geometric fiber of $\mathscr {M}_d$ is irreducible (here we use that d is invertible in k).

(iii) The locus $\mathscr {M}_{d, \infty }\subset \mathscr {M}_d$ classifying supersingular K3 surfaces is closed of dimension $\ge 9$.

Proof

A review of (i) and (iii) can be found in [23, p. 1]. Statement (ii) can be found in [17, 2.10 (3)]. $\square$

Remark 3.3

The stack $\mathscr {M}_d$ is defined over $\mathbb {Z}$, and it follows from (ii) that the geometric generic fiber of $\mathscr {M}_d$ is irreducible (this follows also from the Torelli theorem over $\mathbb {C}$ and the resulting description of $\mathscr {M}_{d, \mathbb {C}}$ as a period space). Furthermore over $\mathbb {Z}[1/d]$ the stack $\mathscr {M}_d$ is smooth. In what follows we denote this stack over $\mathbb {Z}[1/d]$ by $\mathscr {M}_{d, \mathbb {Z}[1/d]}$ and reserve the notation $\mathscr {M}_d$ for its reduction to k.

Remark 3.4

Note that in the definition of $\mathscr {M}_d$ we consider ample invertible sheaves, and don’t allow contractions in the corresponding morphism to projective space.

3.5

Let $\mathscr {S}_d$ denote the fibered category over k whose fiber over a scheme S is the groupoid of collections of data

$$\begin{aligned} ((X, \lambda ), Y, P), \end{aligned}$$

(3.5.1)

where $(X, \lambda )\in \mathscr {M}_{d}(S)$ is a polarized K3 surface, Y / S is a second K3 surface over S, and $P\in D(X\times _SY)$ is an S-perfect complex such that for every geometric point $\bar{s}\rightarrow S$ the induced functor

$$ \Phi _{P_{\bar{s}}}:D(X_{\bar{s}})\rightarrow D(Y_{\bar{s}}) $$

is strongly filtered.

Theorem 3.6

The fibered category $\mathscr {S}_d$ is an algebraic stack locally of finite type over k.

Proof

By fppf descent for algebraic spaces we have descent for both polarized and unpolarized K3 surfaces.

To verify descent for the kernels P, consider an object (3.5.1) over a scheme S. Let $P^\vee$ denote $\mathscr {R}Hom (P, \mathscr {O}_X)$. Since P is a perfect complex we have $\mathscr {R}Hom (P, P)\simeq P^\vee \otimes P$. By [15, 2.1.10] it suffices to show that for all geometric points $\bar{s}\rightarrow S$ we have $H^i(X_{\bar{s}}\times Y_{\bar{s}}, P^\vee _{\bar{s}}\otimes P_{\bar{s}}) = 0$ for $i<0$. This follows from the following result (we discuss Hochschild cohomology further in Sect. 4 below):

Lemma 3.7

([28, 5.6], [11, 5.1.8]). Let X and Y be K3 surfaces over an algebraically closed field k, and let $P\in D(X\times Y)$ be a complex defining a Fourier–Mukai equivalence $\Phi _P:D(X)\rightarrow D(Y)$. Denote by $HH^*(X)$ the Hochschild cohomology of X defined as

$$ \text {RHom}_{X\times X}(\Delta _*\mathscr {O}_X, \Delta _*\mathscr {O}_X). $$

(i) There is a canonical isomorphism $\text {Ext}^*_{X\times Y}(P, P)\simeq HH^*(X)$.

(ii) $\text {Ext}^i_{X\times Y}(P, P) = 0$ for $i<0$ and $i=1$.

(iii) The natural map $k\rightarrow \text {Ext}^0_{X\times Y}(P, P)$ is an isomorphism.

Proof

Statement (i) is [28, 5.6]. Statements (ii) and (iii) follow immediately from this, since $HH^1(X) = 0$ for a K3 surface. $\square$

Next we show that for an object (3.5.1) the polarization $\lambda$ on X induces a polarization $\lambda _Y$ on Y. To define $\lambda _Y$ we may work étale locally on S so may assume there exists an ample invertible sheaf L on X defining $\lambda$. The complex

$$ \Phi _P(L):= R\text {pr}_{2*}(\text {pr}_1^*L\otimes ^{\mathbb {L}}P) $$

is S-perfect, and therefore a perfect complex on Y. Let M denote the determinant of $\Phi _P(L)$, so M is an invertible sheaf on Y. By our assumption that $\Phi ^{P_s}$ is strongly filtered for all $s\in S$, the restriction of M to any fiber is either ample or antiample. It follows that either M or $M^\vee$ is a relatively ample invertible sheaf and we define $\lambda _Y$ to be the resulting polarization on Y. Note that this does not depend on the choice of line bundle L representing $\lambda$ and therefore by descent $\lambda _Y$ is defined even when no such L exists.

The degree of $\lambda _Y$ is equal to d. Indeed if $s\in S$ is a point then since $\Phi ^{P_s}$ is strongly filtered the induced map $NS(X_{\bar{s}})\rightarrow NS(Y_{\bar{s}})$ is compatible with the intersection pairings and therefore $\lambda _Y^2 = \lambda ^2 = 2d$.

From this we deduce that $\mathscr {S}_d$ is algebraic as follows. We have a morphism

$$\begin{aligned} \mathscr {S}_d\rightarrow \mathscr {M}_d\times \mathscr {M}_d, \ \ ((X, \lambda ), Y, P)\mapsto ((X, \lambda ), (Y, \lambda _Y)), \end{aligned}$$

(3.7.1)

and $\mathscr {M}_d\times \mathscr {M}_d$ is an algebraic stack. Let $\mathscr {X}$ (resp. $\mathscr {Y}$) denote the pullback to $\mathscr {M}_d\times \mathscr {M}_d$ of the universal family over the first factor (resp. second factor). Sending a triple $((X, \lambda ), Y, P)$ to P then realizes $\mathscr {S}_d$ as an open substack of the stack over $\mathscr {M}_d\times \mathscr {M}_d$ of simple universally gluable complexes on $\mathscr {X}\times _{\mathscr {M}_d\times \mathscr {M}_d}\mathscr {Y}$ (see for example [16, Sect. 5]). $\square$

3.8

Observe that for any object $((X, \lambda ), Y, P)\in \mathscr {S}_d$ over a scheme S there is an inclusion

$$ \mathbb {G}_m\hookrightarrow \underline{\text {Aut}}_{\mathscr {S}_d}((X, \lambda ), Y, P) $$

giving by scalar multiplication by P. We can therefore form the rigidification of $\mathscr {S}_d$ with respect to $\mathbb {G}_m$ (see for example [2, Sect. 5]) to get a morphism

$$ g:\mathscr {S}_d\rightarrow \overline{\mathscr {S}}_d $$

realizing $\mathscr {S}_d$ as a $\mathbb {G}_m$-gerbe over another algebraic stack $\overline{\mathscr {S}}_d$. By the universal property of rigidification the map $\mathscr {S}_d\rightarrow \mathscr {M}_d$ sending $((X, \lambda ), Y, P)$ to $(X, \lambda )$ induces a morphism

$$\begin{aligned} \pi :\overline{\mathscr {S}}_d\rightarrow \mathscr {M}_d. \end{aligned}$$

(3.8.1)

Theorem 3.9

The stack $\overline{\mathscr {S}}_d$ is Deligne–Mumford and the map (3.8.1) is étale.

Proof

Consider the map (3.7.1). By the universal property of rigidification this induces a morphism

$$ q:\overline{\mathscr {S}}_d\rightarrow \mathscr {M}_d\times \mathscr {M}_d. $$

Since $\mathscr {M}_d\times \mathscr {M}_d$ is Deligne–Mumford, to prove that $\overline{\mathscr {S}}_d$ is a Deligne–Mumford stack it suffices to show that q is representable. This follows from Lemma 3.7 (iii) which implies that for any object $((X, \lambda ), Y, P)$ over a scheme S the automorphisms of this object which map under q to the identity are given by scalar multiplication on P by elements of $\mathscr {O}_S^*$.

It remains to show that the map (3.8.1) is étale, and for this it suffices to show that it is formally étale.

Let $A\rightarrow A_0$ be a surjective map of artinian local rings with kernel I annhilated by the maximal ideal of A, and let k denote the residue field of $A_0$ so I can be viewed as a k-vector space. Let $((X_0, \lambda _0), Y_0, P_0)\in \mathscr {S}_d(A_0)$ be an object and let $(X, \lambda )\in \mathscr {M}_d(A)$ be a lifting of $(X_0, \lambda _0)$ so we have a commutative diagram of solid arrows

Since $\mathscr {S}_d$ is a $\mathbb {G}_m$-gerbe over $\overline{\mathscr {S}}_d$, the obstruction to lifting a map $\bar{x}$ as indicated to a morphism x is given by a class in $H^2(\!\text { Spec }(A), \widetilde{I}) = 0$, and therefore any such map $\bar{x}$ can be lifted to a map x. Furthermore, the set of isomorphism classes of such liftings x of $\bar{x}$ is given by $H^1(\!\text { Spec }(A), \widetilde{I}) = 0$ so in fact the lifting x is unique up to isomorphism. The isomorphism is not unique but determined up to the action of

$$ \text {Ker}(A^*\rightarrow A_0^*) \simeq I. $$

From this it follows that it suffices to show the following:

• (i) The lifting $(X, \lambda )$ of $(X_0, \lambda _0)$ can be extended to a lifting $((X, \lambda ), Y, P)$ of $((X_0, \lambda _0), Y_0, P_0)$.

• (ii) This extension $((X, \lambda ), Y, P)$ of $(X, \lambda )$ is unique up to isomorphism.

• (iii) The automorphisms of the triple $((X, \lambda ), Y, P)$ which are the identity on $(X, \lambda )$ and reduce to the identity over $A_0$ are all given by scalar multiplication on P by elements of $1+I\subset A^*$.

Statement (i) is shown in [16, 6.3].

Next we prove the uniqueness statements in (ii) and (iii). Following the notation of [16, Discussion preceding 5.2], let $s\mathscr {D}_{X/A}$ denote the stack of simple, universally gluable, relatively perfect complexes on X, and let $sD_{X/A}$ denote its rigidification with respect to the $\mathbb {G}_m$-action given by scalar multiplication. The complex $P_0$ on $X_0\times _{A_0}Y_0$ defines a morphism

$$ Y_0\rightarrow sD_{X/A}\otimes _AA_0 $$

which by [16, 5.2 (ii)] is an open imbedding. Any extension of $(X, \lambda )$ to a lifting $((X, \lambda ), Y, P)$ defines an open imbedding $Y\hookrightarrow sD_{X/A}$. This implies that Y, viewed as a deformation of $Y_0$ for which there exists a lifting P of $P_0$ to $X\times _AY$, is unique up to unique isomorphism.

Let Y denote the unique lifting of $Y_0$ to an open subspace of $sD_{X/A}$. By [15, 3.1.1 (2)] the set of isomorphism classes of liftings of $P_0$ to $X\times _AY$ is a torsor under

$$ \text {Ext}^1_{X_k\times Y_k}(P_k, P_k)\otimes I, $$

which is 0 by Lemma 3.7 (ii). From this it follows that P is unique up to isomorphism, and also by Lemma 3.7 (iii) we get the statement that the only infinitesimal automorphisms of the triple $((X, \lambda ), Y, P)$ are given by scalar multiplication by elements of $1+I$. $\square$

3.10

There is an automorphism

$$ \sigma :\mathscr {S}_d\rightarrow \mathscr {S}_d $$

satisfying $\sigma ^2 = \text {id}$. This automorphism is defined by sending a triple $((X, \lambda ), Y, P)$ to $((Y, \lambda _Y), X, P^\vee [2])$. This automorphism induces an involution $\bar{\sigma }:\overline{\mathscr {S}}_d\rightarrow \overline{\mathscr {S}}_d$ over the involution $\gamma :\mathscr {M}_d\times \mathscr {M}_d\rightarrow \mathscr {M}_d\times \mathscr {M}_d$ switching the factors.

Remark 3.11

In fact the stack $\mathscr {S}_d$ is defined over $\mathbb {Z}[1/d]$ and Theorems 3.6 and 3.9 also hold over $\mathbb {Z}[1/d]$. In what follows we write $\mathscr {S}_{d, \mathbb {Z}[1/d]}$ for this stack over $\mathbb {Z}[1/d]$.

4 Deformations of Autoequivalences

In this section, we describe the obstructions to deforming Fourier–Mukai equivalences. The requisite technical machinery for this is worked out in [13, 14]. The results of this section will play a crucial role in Sect. 6.

Throughout this section let k be a perfect field of positive characteristic p and ring of Witt vectors W. For an integer n let $R_n$ denote the ring $k[t]/(t^{n+1})$, and let R denote the ring k[[t]].

4.1

Let $X_{n+1}/R_{n+1}$ be a smooth proper scheme over $R_{n+1}$ with reduction$X_n$ to $R_n$. We then have the associated relative Kodaira–Spencer class, defined in [13, p. 486], which is the morphism in $D(X_n)$

$$ \kappa _{X_n/X_{n+1}}:\Omega ^1_{X_n/R_n}\rightarrow \mathscr {O}_{X_n}[1] $$

defined as the morphism corresponding to the short exact sequence

4.2

We also have the relative universal Atiyah class which is a morphism

$$ \alpha _n:\mathscr {O}_{\Delta _n}\rightarrow i_{n*}\Omega ^1_{X_n/R_n}[1] $$

in $D(X_n\times _{R_n}X_n)$, where $i_n:X_n\rightarrow X_n\times _{R_n}X_n$ is the diagonal morphism and $\mathscr {O}_{\Delta _n}$ denotes $i_{n*}\mathscr {O}_{X_n}$.

This map $\alpha _n$ is given by the class of the short exact sequence

$$ 0\rightarrow I/I^2\rightarrow \mathscr {O}_{X_n\times _{R_n}X_n}/I^2\rightarrow \mathscr {O}_{\Delta _n}\rightarrow 0, $$

where $I\subset \mathscr {O}_{X_n\times _{R_n}X_n}$ is the ideal of the diagonal. Note that to get the morphism $\alpha _n$ we need to make a choice of isomorphism $I/I^2\simeq \Omega ^1_{X_n/R_n}$, which implies that the relative universal Atiyah class is not invariant under the map switching the factors, but rather changes by $-1$.

4.3

Define the relative Hochschild cohomology of $X_n/R_n$ by

$$ HH^*(X_n/R_n):= \text {Ext}^*_{X_n\times _{R_n}X_n}(\mathscr {O}_{\Delta _n}, \mathscr {O}_{\Delta _n}). $$

The composition

is a class

$$ \nu _{X_n/X_{n+1}}\in HH^2(X_n/R_n). $$

4.4

Suppose now that $Y_{n}/R_n$ is a second smooth proper scheme with a smooth lifting $Y_{n+1}/R_{n+1}$ and that $E_n\in D(X_n\times _{R_n}Y_n)$ is a $R_n$-perfect complex.

Consider the class

$$ \nu := \nu _{X_n\times _{R_n}Y_n/X_{n+1}\times _{R_{n+1}}Y_{n+1}}:\mathscr {O}_{\Delta _{n, X_n\times _{R_n}Y_n}}\rightarrow \mathscr {O}_{\Delta _{n, X_n\times _{R_n}Y_n}}[2]. $$

Viewing this is a morphism of Fourier–Mukai kernels

$$ D(X_n\times _{R_n}Y_n)\rightarrow D(X_n\times _{R_n}Y_n) $$

and applying it to $E_n$ we get a class

$$ \omega (E_n)\in \text {Ext}^2_{X_n\times _{R_n}Y_n}(E_n, E_n). $$

In the case when

$$ \text {Ext}^1_{X_0\times Y_0}(E_0, E_0) = 0, $$

which will hold in the cases of interest in this paper, we know by [13, Lemma 3.2] that the class $\omega (E_n)$ is 0 if and only if $E_n$ lifts to a perfect complex on $X_{n+1}\times _{R_{n+1}}Y_{n+1}$.

4.5

To analyze the class $\omega (E_n)$ it is useful to translate it into a statement about classes in $HH^2(Y_n/R_n)$. This is done using Toda’s argument [28, Proof of 5.6]. Denote by $p_{ij}$ the various projections from $X_n\times _{R_n}X_n\times _{R_n}Y_n$, and let

$$ E_n\circ :D(X_n\times _{R_n}X_n)\rightarrow D(X_n\times _{R_n}Y_n) $$

denote the Fourier–Mukai functor defined by the pushforward of $p_{23}^*E_n$ along the morphism

$$ \text {id}_{X_n}\times \Delta _{X_n}\times \text {id}_{Y_n}:X_n\times _{R_n}\times X_n\times _{R_n}Y_n\rightarrow (X_n\times _{R_n}X_n)\times _{R_n}(X_n\times _{R_n}Y_n). $$

So for an object $K\in D(X_n\times _{R_n}X_n)$ the complex $E_n\circ K\in D(X_n\times _{R_n}Y_n)$ represents the Fourier–Mukai transform $\Phi _{E_n}\circ \Phi _K$, and is given explicitly by the complex

$$ p_{13*}(p_{12}^*K\otimes p_{23}^*E_n). $$

As in loc. cit. the diagram

commutes.

In particular we get a morphism

$$ \eta _{X}^*:HH^*(X_n/R_n)\rightarrow \text {Ext}^*_{X_n\times _{R_n}Y_n}(E_n, E_n). $$

Now assume that both $X_n$ and $Y_n$ have relative dimension d over $R_n$ and that the relative canonical sheaves of $X_n$ and $Y_n$ over $R_n$ are trivial. Let $E_n^\vee$ denote $\mathscr {R}Hom (E_n, \mathscr {O}_{X_n\times _{R_n}Y_n})$ viewed as an object of $D(Y_n\times _{R_n}X_n)$. In this case the functor

$$ \Phi _{E_n^\vee [d]}:D(Y_n)\rightarrow D(X_n) $$

is both a right and left adjoint of $\Phi _{E_n}$ [4, 4.5]. By the same argument, the functor

$$ \circ E_n^\vee [d]:D(X_n\times _{R_n}Y_n)\rightarrow D(Y_n\times _{R_n}Y_n), $$

defined in the same manner as $E_n\circ$ has left and right adjoint given by

$$ \circ E_n:D(Y_n\times _{R_n}Y_n)\rightarrow D(X_n\times _{R_n}Y_n). $$

Composing with the adjunction maps

$$\begin{aligned} \alpha :\text {id}\rightarrow \circ E_n\circ E_n^\vee [d], \ \ \beta :\circ E_n\circ E_n^\vee [d]\rightarrow \text {id} \end{aligned}$$

(4.5.1)

applied to the diagonal $\mathscr {O}_{\Delta _{Y_n}}$ we get a morphism

$$ \eta _{Y*}:\text {Ext}^*_{X_n\times _{R_n}Y_n}(E_n, E_n)\rightarrow HH^*(Y_n/R_n). $$

We denote the composition

$$ \eta _{Y*}\eta _X^*:HH^*(X_n/R_n)\rightarrow HH^*(Y_n/R_n) $$

by $\Phi _{E_n}^{HH^*}$. In the case when $E_n$ defines a Fourier–Mukai equivalence this agrees with the standard definition (see for example [28]).

4.6

Evaluating the adjunction maps (4.5.1) on $\mathscr {O}_{\Delta _{Y_n}}$ we get a morphism

(4.6.1)

We say that $E_n$ is admissible if this composition is the identity map.

If $E_n$ is a Fourier–Mukai equivalence, then it is clear that $E_n$ is admissible. Another example is if there exists a lifting $(\mathscr {X}, \mathscr {Y}, \mathscr {E})$ of $(X_n, Y_n, E_n)$ to R, where $\mathscr {X}$ and $\mathscr {Y}$ are smooth proper R-schemes with trivial relative canonical bundles and $\mathscr {E}$ is a R-perfect complex on $\mathscr {X}\times _R\mathscr {Y}$, such that the restriction $\mathscr {E}$ to the generic fiber defines a Fourier–Mukai equivalence. Indeed in this case the map (4.6.1) is the reduction of the corresponding map $\mathscr {O}_{\Delta _{\mathscr {Y}}}\rightarrow \mathscr {O}_{\Delta _{\mathscr {Y}}}$ defined over R, which in turn is determined by its restriction to the generic fiber.

4.7

Consider Hochschild homology

$$ HH_i(X_n/R_n):= H^{-i}(X_n, Li_n^*\mathscr {O}_{\Delta _n}). $$

As explained for example in [5, Sect. 5] we also get a map

$$ \Phi _{E_n}^{HH_*}:HH_*(X_n/R_n)\rightarrow HH_*(Y_n/R_n). $$

Hochschild homology is a module over Hochschild cohomology, and an exercise (that we do not write out here) shows that the following diagram

commutes.

Remark 4.8

We thank the referee for pointing out that the necessary functoriality of Hochschild homology was settled by Keller, Swan, and Weibel by the mid-1990s. See for example [10], [30, 9.8.19], and [27]. The Ref. [5] provides a very readable account of the results we need here.

4.9

Using this we can describe the obstruction $\omega (E_n)$ in a different way, assuming that $E_n$ is admissible. First note that viewing the relative Atiyah class of $X_n\times _{R_n}Y_n$ as a morphism of Fourier–Mukai kernels we get the Atiyah class of $E_n$ which is a morphism

$$ A(E_n):E_n\rightarrow E_n\otimes \Omega ^1_{X_n\times _{R_n}Y_n/R_n}[1] $$

in $D(X_n\times _{R_n}Y_n)$. There is a natural decomposition

$$ \Omega ^1_{X_n\times _{R_n}Y_n/R_n}\simeq p_1^*\Omega ^1_{X_n/R_n}\oplus p_2^*\Omega ^1_{Y_n/R_n}, $$

so we can write $A(E_n)$ as a sum of two maps

$$ A(E_n)_X:E_n\rightarrow E_n\otimes p_1^*\Omega ^1_{X_n/R_n}[1], \ \ A(E_n)_Y:E_n\rightarrow E_n\otimes p_2^*\Omega ^1_{Y_n/R_n}[1]. $$

Similarly the Kodaira–Spencer class of $X_n\times _{R_n}Y_n$ can be written as the sum of the two pullbacks

$$ p_1^*\kappa _{X_n/X_{n+1}}:p_1^*\Omega ^1_{X_n/R_n}\rightarrow p_1^*\mathscr {O}_{X_n}[1], \ \ p_2^*\kappa _{Y_n/Y_{n+1}}:p_2^*\Omega ^1_{Y_n/R_n}\rightarrow p_2^*\mathscr {O}_{Y_n}[1]. $$

It follows that the obstruction $\omega (E_n)$ can be written as a sum

$$ \omega (E_n) = (p_1^*\kappa _{X_n/X_{n+1}}\circ A(E_n)_X)+ (p_2^*\kappa (Y_n/Y_{n+1})\circ A(E_n)_Y). $$

Now by construction we have

$$ \eta _{X_n}^*(\nu _{X_n/X_{n+1}}) = p_1^*\kappa _{X_n/X_{n+1}}\circ A(E_n)_X, $$

and

$$ \eta _{Y_n*}(p_2^*\kappa (Y_n/Y_{n+1})\circ A(E_n)_Y) = -\nu _{Y_n/Y_{n+1}}, $$

the sign coming from the asymmetry in the definition of the relative Atiyah class (it is in the verification of this second formula that we use the assumption that $E_n$ is admissible). Summarizing we find the formula

$$\begin{aligned} \eta _{Y_n*}(\omega (E_n)) = \Phi _{E_n}^{HH^*}(\nu _{X_n/X_{n+1}})-\nu _{Y_n/Y_{n+1}}. \end{aligned}$$

(4.9.1)

In the case when $\Phi _{E_n}$ is an equivalence the maps $\eta _{Y_n*}$ and $\eta _{X_n}^*$ are isomorphisms, so the obstruction $\omega (E_n)$ vanishes if and only if we have

$$ \Phi _{E_n}^{HH^*}(\nu _{X_n/X_{n+1}})-\nu _{Y_n/Y_{n+1}} = 0. $$

Remark 4.10

By [13, Remark 2.3 (iii)], the functor $\Phi _{E_n}$ is an equivalence if and only if $\Phi _{E_0}:D(X_0)\rightarrow D(Y_0)$ is an equivalence.

Corollary 4.11

Suppose $F_n\in D(X_n\times _{R_n}Y_n)$ defines a Fourier–Mukai equivalence, and that $E_n\in D(X_n\times _{R_n}Y_n)$ is another admissible $R_n$-perfect complex such that $\Phi _{F_n}^{HH^*} = \Phi _{E_n}^{HH^*}$. If $E_n$ lifts to a $R_{n+1}$-perfect complex $E_{n+1}\in D(X_{n+1}\times _{R_{n+1}}Y_{n+1})$ then so does $F_n$.

Proof

Indeed the condition that $\Phi _{F_n}^{HH^*} = \Phi _{E_n}^{HH^*}$ ensures that

$$ \eta _{Y_n*}(\omega (E_n) ) = \eta _{Y_n*}(\omega (F_n)), $$

and since $\omega (E_n) = 0$ we conclude that $\omega (F_n) = 0$. $\square$

4.12

The next step is to understand the relationship between $\Phi _{E_n}^{HH^*}$ and the action of $\Phi _{E_n}$ on the cohomological realizations of the Mukai motive.

Assuming that the characteristic p is bigger than the dimension of $X_0$ (which in our case will be a K3 surface so we just need $p>2$) we can exponentiate the relative Atiyah class to get a map

$$ \exp (\alpha _n):\mathscr {O}_{\Delta _n}\rightarrow \oplus _ii_{n*}\Omega ^i_{X_n/R_n} $$

which by adjunction defines a morphism

$$\begin{aligned} Li_n^*\mathscr {O}_{\Delta _n}\rightarrow \oplus _i\Omega ^i_{X_n/R_n} \end{aligned}$$

(4.12.1)

in $D(X_n)$. By [1, Theorem 0.7], which also holds in positive characteristic subject to the bounds on dimension, this map is an isomorphism. We therefore get an isomorphism

$$ I^{HKR}:HH^*(X_n/R_n)\rightarrow HT^*(X_n/R_n), $$

where we write

$$ HT^*(X_n/R_n) := \oplus _{p+q=*}H^p(X_n, \bigwedge ^qT_{X_n/R_n}). $$

We write

$$ I^K_{X_n}:HH^*(X_n/R_n)\rightarrow HT^*(X_n/R_n) $$

for the composition of $I^{HKR}$ with multiplication by the inverse square root of the Todd class of $X_n/R_n$, as in [6, 1.7].

The isomorphism (4.12.1) also defines an isomorphism

$$ I_{HKR}:HH_*(X_n/R_n)\rightarrow H\Omega _*(X_n/R_n), $$

where

$$ H\Omega _*(X_n/R_n):= \oplus _{q-p=*}H^p(X_n, \Omega ^q_{X_n/R_n}). $$

We write

$$ I_{K}^{X_n}:HH_*(X_n/R_n)\rightarrow H\Omega _*(X_n/R_n) $$

for the composition of $I_{HKR}$ with multiplication by the square root of the Todd class of $X_n/R_n$.

We will consider the following condition $(\star )$ on a $R_n$-perfect complex $E_n\in D(X_n\times _{R_n}Y_n)$:

$(\star )$ :

The diagram

commutes.

Remark 4.13

We expect this condition to hold in general. Over a field of characteristic 0 this is shown in [19, 1.2]. We expect that a careful analysis of denominators occurring of their proof will verify $(\star )$ quite generally with some conditions on the characteristic relative to the dimension of the schemes. However, we will not discuss this further in this paper.

4.14

There are two cases we will consider in this paper were $(\star )$ is known to hold:

• (i) If $E_n = \mathscr {O}_{\Gamma _n}$ is the structure sheaf of the graph of an isomorphism $\gamma _n:X_n\rightarrow Y_n$. In this case the induced maps on Hochschild cohomology and $H\Omega _*$ are simply given by pushforward $\gamma _{n*}$ and condition $(\star )$ immediately holds.

• (ii) Suppose $B\rightarrow R_n$ is a morphism from an integral domain B which is flat over W and that there exists a lifting $(\mathscr {X}, \mathscr {Y}, \mathscr {E})$ of $(X_n, Y_n, E_n)$ to B, where $\mathscr {X}$ and $\mathscr {Y}$ are proper and smooth over B and $\mathscr {E}\in D(\mathscr {X}\times _B\mathscr {Y})$ is a B-perfect complex pulling back to $E_n$. Suppose further that the groups $HH^*(\mathscr {X}/B)$ and $HH^*(\mathscr {Y}/B)$ are flat over B and their formation commutes with base change (this holds for example if $\mathscr {X}$ and $\mathscr {Y}$ are K3 surfaces). Then $(\star )$ holds. Indeed it suffices to verify the commutativity of the corresponding diagram over B, and this in turn can be verified after passing to the field of fractions of B. In this case the result holds by [19, 1.2].

Lemma 4.15

Let $E_n, F_n\in D(X_n\times _{R_n}Y_n)$ be two $R_n$-perfect complexes satisfying condition $(\star )$. Suppose further that the maps $\Phi _{E_0}^{\text {crys} }$ and $\Phi _{F_0}^\text {crys}$ on the crystalline realizations $\widetilde{H}(X_0/W)\rightarrow \widetilde{H}(Y_0/W)$ of the Mukai motive are equal. Then the maps $\Phi _{E_n}^{HH_*}$ and $\Phi _{F_n}^{HH_*}$ are also equal. Furthermore if the maps on the crystalline realizations are isomorphisms then $\Phi _{E_n}^{HH_*}$ and $\Phi _{F_n}^{HH_*}$ are also isomorphisms.

Proof

Since $H\Omega _*(X_n/R_n)$ (resp. $H\Omega _*(Y_n/R_n)$) is obtained from the de Rham realization $\widetilde{H}_{\text {dR}}(X_n/R_n)$ (resp. $\widetilde{H}_{\text {dR}}(X_n/R_n)$) of the Mukai motive of $X_n/R_n$ (resp. $Y_n/R_n$) by passing to the associated graded, it suffices to show that the two maps

$$ \Phi _{E_n}^{\text {dR}}, \Phi _{F_n}^{\text {dR}}: \widetilde{H}_{\text {dR}}(X_n/R_n)\rightarrow \widetilde{H}_{\text {dR}}(Y_n/R_n) $$

are equal, and isomorphisms when the crystalline realizations are isomorphisms. By the comparison between crystalline and de Rham cohomology it suffices in turn to show that the two maps on the crystalline realizations

$$ \Phi _{E_n}^{\text {crys}}, \Phi _{F_n}^{\text {crys}}: \widetilde{H}_{\text {crys}}(X_n/W[t]/(t^{n+1}))\rightarrow \widetilde{H}_{\text {crys}}(Y_n/W[t]/(t^{n+1})) $$

are equal. Via the Berthelot-Ogus isomorphism [3, 2.2], which is compatible with Chern classes, these maps are identified after tensoring with $\mathbb {Q}$ with the two maps obtained by base change from

$$ \Phi _{E_0}^{\text {crys}}, \Phi _{F_0}^{\text {crys}}: \widetilde{H}_{\text {crys}}(X_0/W)\rightarrow \widetilde{H}_{\text {crys}}(Y_0/W). $$

The result follows. $\square$

4.16

In the case when $X_n$ and $Y_n$ are K3 surfaces the action of $HH^*(X_n/R_n)$ on $HH_*(X_n/R_n)$ is faithful. Therefore from Lemma 4.15 we obtain the following.

Corollary 4.17

Assume that $X_n$ and $Y_n$ are K3 surfaces and that $E_n, F_n\in D(X_n\times _{R_n}Y_n)$ are two $R_n$-perfect complexes satisfying condition $(\star )$. Suppose further that $\Phi ^{\text {crys}}_{E_0}$ and $\Phi ^{\text {crys}}_{F_0}$ are equal on the crystalline realizations of the Mukai motives of the reductions. Then $\Phi _{E_n}^{HH^*}$ and $\Phi _{F_n}^{HH^*}$ are equal.

Proof

Indeed since homology is a faithful module over cohomology the maps $\Phi _{E_n}^{HH^*}$ and $\Phi _{F_n}^{HH^*}$ are determined by the maps on Hochschild homology which are equal by Lemma 4.15. $\square$

Corollary 4.18

Let $X_{n+1}$ and $Y_{n+1}$ be K3 surfaces over $R_{n+1}$ and assume given an admissible $R_{n+1}$-perfect complex $E_{n+1}$ on $X_{n+1}\times _{R_{n+1}}Y_{n+1}$ such that $E_n$ satisfies condition $(\star )$. Assume given an isomorphism $\sigma _n:X_n\rightarrow Y_n$ over $R_n$ such that the induced map $\sigma _0:X_0\rightarrow Y_0$ defines the same map on crystalline realizations of the Mukai motive as $E_0$. Then $\sigma _n$ lifts to an isomorphism $\sigma _{n+1}:X_{n+1}\rightarrow Y_{n+1}$.

Proof

Indeed by (4.9.1) and the fact that $\Phi ^{HH^*}_{E_n}$ and $\Phi ^{HH^*}_{\Gamma _{\sigma _n}}$ are equal by Corollary 4.17, we see that the obstruction to lifting $\sigma _{n}$ is equal to the obstruction to lifting $E_n$, which is zero by assumption. $\square$

Corollary 4.19

Assume the hypotheses of Corollary 4.18 with the following exception: $\Phi _{E_0}$ is strongly filtered and $\sigma _0$ and $\Phi _{E_0}$ are only assumed to induce the same map

$$H^2_{\mathrm{crys}}(X_0/W)\rightarrow H^2_{\mathrm{crys}}(Y_0/W).$$

(In other words, we do not require the same action on all crystalline cohomology, just on the middle.) Then the same conclusion holds: $\sigma _n$ lifts to some $\sigma _{n+1}$.

Proof

The assumption that $\Phi _{E_0}$ is strongly filtered implies that the map$I_{Y_n}^K\circ \Phi _{E_n}^{HH^*}\circ (I_{X_n}^{K})^{-1}$ sends $H^1(X_n, T_{X_n})$ to $H^1(Y_n, T_{Y_n})$. By construction we have $I_{X_n}^K(\nu _{X_n/X_{n+1}})\in H^1(X_n, T_{X_n})$; in fact, this class is given by the relative Kodaira-Spencer class multiplied by the square root of the Todd class. Now by looking at the module structure of $H\Omega _*$ over $HT^*$ one sees that the map$H^1(X_n, T_{X_n})\rightarrow H^1(Y_n, T_{Y_n})$ is determined by the action on $H^2_{\text {crys}}$. From this the conclusion follows.$\square$

Remark 4.20

Applying the argument of Corollary 4.19 with $E_n$[1] we see that if $\sigma _0$ induces $\pm \Phi _{E_0}$ on $H^{2}_\mathrm{crys}$ then $\sigma _n$ can be lifted to $R_{n+1}$.

5 A Remark on Reduction Types

5.1

In the proof of Theorem 1.2 we need the following Theorem 5.3, whose proof relies on known characteristic 0 results obtained from Hodge theory. In Sect. 10 below we give a different algebraic argument for Theorem 5.3 in a special case which suffices for the proof of Theorem 1.2.

5.2

Let V be a complete discrete valuation ring with field of fractions K and residue field k. Let X / V be a projective K3 surface with generic fiber $X_K$, and let $Y_K$ be a second K3 surface over K such that the geometric fibers $X_{\overline{K}}$ and $Y_{\overline{K}}$ are derived equivalent.

Theorem 5.3

Under these assumptions the K3 surface $Y_K$ has potentially good reduction.

Remark 5.4

Here potentially good reduction means that after possibly replacing V be a finite extension there exists a K3 surface Y / V whose generic fiber is $Y_K$.

Proof of Theorem 5.3 We use [16, 1.1 (1)] which implies that after replacing V by a finite extension $Y_K$ is isomorphic to a moduli space of sheaves on $X_K$.

After replacing V by a finite extension we may assume that we have a complex $P\in D(X\times Y)$ defining an equivalence

$$ \Phi _P:D(X_{\overline{K}})\rightarrow D(Y_{\overline{K}}). $$

Let $E\in D(Y\times X)$ be the complex defining the inverse equivalence

$$ \Phi _E:D(Y_{\overline{K}})\rightarrow D(X_{\overline{K}}) $$

to $\Phi _P$. Let $\nu := \Phi _E(0,0,1)\in A^*(X_{\overline{K}})_{\text {num}, \mathbb {Q}}$ be the Mukai vector of a fiber of E at a closed point $y\in Y_{\overline{K}}$ and write

$$ \nu = (r, [L_X], s)\in A^0(X_{\overline{K}})_{\text {num}, \mathbb {Q}}\oplus A^1(X_{\overline{K}})_{\text {num}, \mathbb {Q}}\oplus A^2(X_{\overline{K}})_{\text {num}, \mathbb {Q}}. $$

By [16, 8.1] we may after possibly changing our choice of P, which may involve another extension of V, assume that r is prime to p and that $L_X$ is very ample. Making another extension of V we may assume that $\nu$ is defined over K, and therefore by specialization also defines an element, which we denote by the same letter,

$$ \nu = (r, [L_X], s)\in \mathbb {Z}\oplus \text {Pic}(X)\oplus \mathbb {Z}. $$

This class has the property that r is prime to p and that there exists another class $\nu '$ such that $\langle \nu , \nu '\rangle = 1$. This implies in particular that $\nu$ restricts to a primitive class on the closed fiber. Fix an ample class h on X, and let $\mathscr {M}_h(\nu )$ denote the moduli space of semistable sheaves on X with Mukai vector $\nu$. By [16, 3.16] the stack $\mathscr {M}_h(\nu )$ is a $\mu _r$-gerbe over a relative K3 surface $M_h(\nu )/V$, and by [16, 8.2] we have $Y_{\overline{K}}\simeq M_h(\nu )_{\overline{K}}$. In particular, Y has potentially good reduction. $\square$

Remark 5.5

As discussed in [18, p. 2], to obtain Theorem 5.3 it suffices to know that every K3 surface $Z_K$ over K has potentially semistable reduction and this would follow from standard conjectures on resolution of singularities and toroidization of morphisms in mixed and positive characteristic. In the setting of Theorem 5.3, once we know that $Y_K$ has potentially semistable reduction then by [18, Theorem on bottom of p. 2] we obtain that $Y_K$ has good reduction since the Galois representation $H^2(Y_{\overline{K}}, \mathbb {Q}_\ell )$ is unramified being isomorphic to direct summand of the $\ell$-adic realization $\widetilde{H}(X_{\overline{K}}, \mathbb {Q}_\ell )$ of the Mukai motive of $X_K$.

5.6

One can also consider the problem of extending $Y_K$ over a higher dimensional base. Let B denote a normal finite type k-scheme with a point $s\in B(k)$ and let X / B be a projective family of K3 surfaces. Let K be the function field of B and let $Y_K$ be a second K3 surface over K Fourier–Mukai equivalent to $X_K$. Dominating $\mathscr {O}_{B, s}$ by a suitable complete discrete valuation ring V we can find a morphism

$$ \rho :\text { Spec }(V)\rightarrow B $$

sending the closed point of $\text { Spec }(V)$ to s and an extension $Y_V$ of $\rho ^*Y_K$ to a smooth projective K3 surface over V. In particular, after replacing B by its normalization in a finite extension of K we can find a primitive polarization $\lambda _K$ on $Y_K$ of degree prime to the characteristic such that $\rho ^*\lambda _K$ extends to a polarization on $Y_V$. We then have a commutative diagram of solid arrows

for a suitable integer d. Base changing to a suitable étale neighborhood $U\rightarrow \mathscr {M}_d$ of the image of the closed point of $\text { Spec }(V)$, with U an affine scheme, we can after shrinking and possibly replacing B by an alteration find a commutative diagram

where j is a dense open imbedding and $\overline{U}$ is projective over k. It follows that the image of s in $\overline{U}$ in fact lands in U which gives an extension of $Y_K$ to a neighborhood of s. This discussion implies the following:

Corollary 5.7

In the setup of Paragraph 5.6, we can, after replacing (B, s) by a neighborhood of a point in the preimage of s in an alteration of B, find an extension of $Y_K$ to a K3 surface over B.

6 Supersingular Reduction

6.1

Let B be a normal scheme of finite type over an algebraically closed field k of characteristic p at least 5, and let W denote the ring of Witt vectors of k. Let K denote the function field of B and let $s\in B$ be a closed point. Let $f:X\rightarrow B$ be a projective K3 surface over B and let $Y_K/K$ be a second K3 surface over K such that there exists a strongly filtered Fourier–Mukai equivalence

$$ \Phi _P:D(X_K)\rightarrow D(Y_K) $$

defined by an object $P\in D(X_K\times _KY_K)$. Assume further that the fiber $X_s$ of X over s is a supersingular K3 surface.

6.2

Using Corollary 5.7 we can, after possibly replacing B by a neighborhood of a point over s in an alteration, assume that we have a smooth K3 surface Y / B extending $Y_K$ and an extension of the complex P to a B-perfect complex $\mathscr {P}$ on $X\times _BY$, and furthermore that the complex Q defining the inverse of $\Phi _P$ also extends to a complex $\mathscr {Q}$ on $X\times _BY$. Let $f_{\mathscr {X}}:\mathscr {X}\rightarrow B$ (resp. $f_{\mathscr {Y}}:\mathscr {Y}\rightarrow B$) be the structure morphism, and let $\mathscr {H}^i_{\text {crys}}(X/B)$ (resp. $\mathscr {H}^i_{\text {crys}}(Y/B)$) denote the F-crystal $R^if_{\mathscr {X}*}\mathscr {O}_{\mathscr {X}/W}$ (resp. $R^if_{\mathscr {Y}*}\mathscr {O}_{\mathscr {Y}/W}$) on B / W obtained by forming the i-th higher direct image of the structure sheaf on the crystalline site of $\mathscr {X}/W$ (resp. $\mathscr {Y}/W$). Because $\Phi _{\mathscr {P}}$ is strongly filtered, it induces an isomorphism of F-crystals

$$ \Phi ^{\text {crys}, i}_{\mathscr {P}}:\mathscr {H}^i_{\text {crys}}(X/B)\rightarrow \mathscr {H}^i_{\text {crys}}(Y/B) $$

for all i, with inverse defined by $\Phi _{\mathscr {Q}}$. Note that since we are working here with K3 surfaces these morphisms are defined integrally.

We also have the de Rham realizations $\mathscr {H}^i_{\text {dR}}(X/B)$ and $\mathscr {H}^i_\text {dR}(Y/B)$ which are filtered modules with integrable connection on B equipped with filtered isomorphisms compatible with the connections

$$\begin{aligned} \Phi ^{\text {dR}, i}_{\mathscr {P}}:\mathscr {H}^i_{\text {dR}}(X/B)\rightarrow \mathscr {H}^i_{\text {dR}}(Y/B). \end{aligned}$$

(6.2.1)

as well as étale realizations $\mathscr {H}^i_{\acute{\mathrm{{e}}}}\text {t }(X/B)$ and $\mathscr {H}^i_{\acute{\mathrm{{e}}}}\text {t }(Y/B)$ equipped with isomorphisms

$$\begin{aligned} \Phi ^{{\acute{\mathrm{{e}}}}\text {t }, i}_{\mathscr {P}}:\mathscr {H}^i_{\acute{\mathrm{{e}}}}\text {t }(X/B)\rightarrow \mathscr {H}^i_{\acute{\mathrm{{e}}}}\text {t }(Y/B). \end{aligned}$$

(6.2.2)

6.3

Let $H^i_{\text {crys}}(X_s/W)$ (resp. $H^i_{\text {crys}}(Y_s/W)$) denote the crystalline cohomology of the fibers over s. The isomorphism $\Phi ^{\text {crys}, 2}_{\mathscr {P}}$ induces an isomorphism

$$\theta _2:H^2_{\text {crys}}(X_s/W)\rightarrow H^2_{\text {crys}}(Y_s/W) $$

of F-crystals. By [24, Theorem I] this implies that $X_s$ and $Y_s$ are isomorphic. However, we may not necessarily have an isomorphism which induces $\theta _2$ on cohomology.

6.4

Recall that as discussed in [12, 10.9 (iii)] if $C\subset X_K$ is a $(-2)$-curve then we can perform a spherical twist

$$ T_{\mathscr {O}_{C}}:D(X_K)\rightarrow D(X_K) $$

whose action on $NS(X_K)$ is the reflection

$$ r_C(a) := a+\langle a, C\rangle C. $$

Proposition 6.5

After possibly changing our choice of model Y for $Y_K$, replacing (B, s) by a neighborhood of a point in an alteration over s, and composing with a sequence of spherical twists $T_{\mathscr {O}_C}$ along $(-2)$-curves in the generic fiber $Y_K$, there exists an isomorphism $\sigma :X_s\rightarrow Y_s$ inducing $\pm \theta _2$ on the second crystalline cohomology group. If $\theta _2$ preserves the ample cone of the generic fiber then we can find an isomorphism $\sigma$ inducing $\theta _2$.

Proof

By [25, 4.4 and 4.5] there exists an isomorphism $\theta _0:NS(X_s)\rightarrow NS(Y_s)$ compatible with $\theta _2$ in the sense that the diagram

(6.5.1)

commutes. Note that as discussed in [17, 4.8] the map $\theta _0$ determines $\theta _2$ by the Tate conjecture for K3 surfaces, proven by Charles, Maulik, and Pera [7, 21, 26]. In particular, if we have an isomorphism $\sigma :X_s\rightarrow Y_s$ inducing $\pm \theta _0$ on Néron-Severi groups then $\sigma$ also induces $\pm \theta _2$ on crystalline cohomology. We therefore have to study the problem of finding an isomorphism $\sigma$ compatible with $\theta _0$.

Ogus shows in [24, Theorem II] that there exists such an isomorphism $\sigma$ if and only if the map $\theta _0$ takes the ample cone to the ample cone. So our problem is to choose a model of Y in such a way that $\pm \theta _0$ preserves ample cones. Set

$$ V_{X_s}:= \{x\in NS(X_s)_{\mathbb R}| x^2>0 \text { and } \langle x, \delta \rangle \ne 0 \text { for all } \delta \in NS(X_s) \text { with } \delta ^2 = -2\}, $$

and define $V_{Y_s}$ similarly. Being an isometry the map $\theta _0$ then induces an isomorphism $V_{X_s}\rightarrow V_{Y_s}$, which we again denote by $\theta _0$. Let $R_{Y_s}$ denote the group of automorphisms of $V_{Y_s}$ generated by reflections in $(-2)$-curves and multiplication by $-1$. By [24, Proposition 1.10 and Remark 1.10.9] the ample cone of $Y_s$ is a connected component of $V_{Y_s}$ and the group $R_{Y_s}$ acts simply transitively on the set of connected components of $V_{Y_s}$.

Let us show how to change model to account for reflections by $(-2)$-curves in $Y_s$. We show that after replacing (P, Y) by a new pair $(P', Y')$ consisting of the complex $P\in D(X_K\times _KY_K)$ obtained by composing $\Phi _P$ with a sequence of spherical twists along $(-2)$-curves in $Y_K$ and replacing Y by a new model $Y'$ there exists an isomorphism $\gamma :Y'_s\rightarrow Y_s$ such that the composition

is equal to the map $\theta '_0$ defined as for $\theta _0$ but using the model $Y'$.

Let $C\subset Y_s$ be a $(-2)$-curve, and let

$$ r_C:NS(Y_s)\rightarrow NS(Y_s)$$

be the reflection in the $(-2)$-curve. If C lifts to a curve in the family Y we get a $(-2)$-curve in the generic fiber and so by replacing our P by the complex $P'$ obtained by composition with the spherical twist by this curve in $Y_K$ (see [12, 10.9 (iii)]) and setting $Y' = Y$ we get the desired new pair. If C does not lift to Y, then we take $P' = P$ but now replace Y by the flop of Y along C as explained in [24, 2.8].

Thus after making a sequence of replacements $(P, Y)\mapsto (P', Y')$ we can arrange that $\theta _0$ sends the ample cone of $X_s$ to plus or minus the ample cone of $Y_s$, and therefore we get our desired isomorphism $\sigma$.

To see the last statement, note that we have modified the generic fiber by composing with reflections along $(-2)$-curves. Therefore if $\lambda$ is an ample class on X with restriction $\lambda _K$ to $X_K$, and for a general ample divisor H we have $\langle \Phi _P(\lambda ), H\rangle >0$, then the same holds on the closed fiber. This implies that the ample cone of $X_s$ gets sent to the ample cone of $Y_s$ and not its negative. $\square$

Remark 6.6

One can also consider étale or de Rham cohomology in Proposition 6.5. Assume we have applied suitable spherical twists and chosen a model Y such that we have an isomorphism $\sigma :X_s\rightarrow Y_s$ inducing $\pm \theta _0$. We claim that the maps

$$ \theta _\text {dR}:H_\text {dR}^i(X_s/k)\rightarrow H_\text {dR}^i(Y_s/k), \ \ \theta _{\acute{\mathrm{{e}}}}\text {t }:H_{\acute{\mathrm{{e}}}}\text {t }^i(X_s, \mathbb {Q}_\ell )\rightarrow H_{\acute{\mathrm{{e}}}}\text {t }^i(Y_s, \mathbb {Q}_\ell ) $$

induced by the maps (6.2.1) and (6.2.2) also agree with the maps defined by $\pm \sigma$. For de Rham cohomology this is clear using the comparison with crystalline cohomology, and for the étale cohomology it follows from compatibility with the cycle class map.

6.7

With notation and assumptions as in Proposition 6.5 assume further that B is a curve or a complete discrete valuation ring, and that we have chosen a model Y such that each of the reductions satisfies the condition $(\star )$ of Paragraph 4.12 and such that the map $\theta _0$ in (6.5.1) preserves plus or minus the ample cones. Let $\sigma :X_s\rightarrow Y_s$ be an isomorphism inducing $\pm \theta _0$.

Lemma 6.8

The isomorphism $\sigma$ lifts to an isomorphism $\tilde{\sigma }:X\rightarrow Y$ over the completion $\widehat{B}$ at s inducing the maps defined by $\pm \Phi ^{\text {crys}, i} _{\mathscr {P}}$.

Proof

By Remark 4.20 $\sigma$ lifts uniquely to each infinitesimal neighborhood of s in B, and therefore by the Grothendieck existence theorem we get a lifting $\tilde{\sigma }$ over $\widehat{B}$. That the realization of $\tilde{\sigma }$ on cohomology agrees with $\pm \Phi ^{\text { crys}, i}_{\mathscr {P}}$ can be verified on the closed fiber where it holds by assumption. $\square$

Lemma 6.9

With notation and assumptions as in Lemma 6.8 the map $\Phi _P^{A^*_{\text {num}, \mathbb {Q}}}$ preserves the ample cones of the generic fibers.

Proof

The statement can be verified after making a field extension of the function field of B. If $\Phi _P$ does not preserve the ample cone, then by Lemma 6.8 we get an isomorphism $\sigma : X_K \rightarrow Y_K$ over some field extension K of k(B) such that $\sigma ^* \circ \Phi _P$ acts by $\text {id}_{H^{0}}\bigoplus -\text {id}_{H^{2}} \bigoplus \text {id}_{H^4}$ on any cohomological realization. Lifting the associated derived equivalence to characteristic 0 (for example, using Deligne’s liftability theorem [8] and Theorem 3.9) and applying [13, 4.1] we get a contradiction. $\square$

Remark 6.10

In the case when the original $\Phi _P$ preserves the ample cones of the geometric generic fibers, no reflections along $(-2)$–curves in the generic fiber are needed. Indeed, by the above argument we get an isomorphism $\sigma _K:X_K\rightarrow Y_K$ such that the induced map on crystalline and étale cohomology agrees with $\Phi _P\circ \alpha$ for some sequence $\alpha$ of spherical twists along $(-2)$-curves in $X_K$ (also using Lemma 6.9). Since both $\sigma$ and $\Phi _P$ preserve ample cones it follows that $\alpha$ also preserves the ample cone of $X_{\overline{K}}$. By [24, 1.10] it follows that $\alpha$ acts trivially on the Néron-Severi group of $X_{\overline{K}}$. We claim that this implies that $\alpha$ also acts trivially on any of the cohomological realizations. We give the proof in the case of étale cohomology $H^2(X_{\overline{K}}, \mathbb {Q}_\ell )$ (for a prime $\ell$ invertible in k) leaving slight modifications for the other cohomology theories to the reader. Let $\widetilde{R}_X$ denote the subgroup of $GL(H^2(X_{\overline{K}},\mathbb {Q}_\ell ))$ generated by $-1$ and the action induced by spherical twists along $(-2)$-curves in $X_{\overline{K}}$, and consider the inclusion of $\mathbb {Q}_\ell$-vector spaces with inner products

$$ NS(X_{\overline{K}})_{\mathbb {Q}_\ell }\hookrightarrow H^2(X_{\overline{K}}, \mathbb {Q}_\ell ). $$

By [12, Lemma 8.12] the action of the spherical twists along $(-2)$-curves in $X_{\overline{K}}$ on $H^2(X_{\overline{K}}, \mathbb {Q}_\ell )$ is by reflection across classes in the image of $NS(X_{\overline{K}})_{\mathbb {Q}_\ell }$. From this (and Gram-Schmidt!) it follows that the the group $\widetilde{R}_X$ preserves $NS(X_{\overline{K}})_{\mathbb {Q}_\ell }$, acts trivially on the quotient of $H^2(X_{\overline{K}}, \mathbb {Q}_\ell )$ by $NS(X_{\overline{K}})_{\mathbb {Q}_\ell }$, and that the restriction map

$$ \widetilde{R}_X\rightarrow GL(NS(X_{\overline{K}})_{\mathbb {Q}_\ell }) $$

is injective. In particular, if an element $\alpha \in \widetilde{R}_X$ acts trivially on $NS(X_{\overline{K}})$ then it also acts trivially on étale cohomology. It follows that $\sigma$ and $\Phi _{P}$ induce the same map on realizations.

7 Specialization

7.1

We consider again the setup of Paragraph 6.1, but now we don’t assume that the closed fiber $X_s$ is supersingular. Further we restrict attention to the case when B is a smooth curve, and assume we are given a smooth model Y / B of $Y_K$ and a B-perfect complex $\mathscr {P}\in D(X\times _BY)$ such that for all geometric points $\bar{z} \rightarrow B$ the induced complex $\mathscr {P}_{\bar{z}}$ on $X_{\bar{z}}\times Y_{\bar{z}}$ defines a strongly filtered equivalence $D(X_{\bar{z}})\rightarrow D(Y_{\bar{z}})$.

Let $\mathscr {H}^i(X/B)$ (resp. $\mathscr {H}^i(Y/B)$) denote either $\mathscr {H}^i_{\acute{\mathrm{{e}}}}\text {t }(X/B)$ (resp. $\mathscr {H}^i_{\acute{\mathrm{{e}}}}\text {t }(Y/B)$) for some prime $\ell \ne p$ or $\mathscr {H}^i_\text { crys} (X/B)$ (resp. $\mathscr {H}^i_\text {crys} (Y/B)$). Assume further given an isomorphism

$$ \sigma _K:X_K\rightarrow Y_K $$

inducing the map given by restricting

$$ \Phi ^{ i}_{\mathscr {P}}:\mathscr {H}^i(X/B)\rightarrow \mathscr {H}^i(Y/B) $$

to the generic point.

Remark 7.2

If we work with étale cohomology in this setup we could also consider the spectrum of a complete discrete valuation ring instead of B, and in particular also a mixed characteristic discrete valuation ring.

Proposition 7.3

The isomorphism $\sigma _K$ extends to an isomorphism $\sigma :X\rightarrow Y$.

Proof

We give the argument here for étale cohomology in the case when B is the spectrum of a discrete valuation ring. The other cases require minor modifications which we do not include here.

Let $Z\subset X\times _BY$ be the closure of the graph of $\sigma _K$, so Z is an irreducible flat V-scheme of dimension 3 and we have a correspondence

Fix an ample line bundle L on X and consider the line bundle $M:= \text {det}(Rq_*p^*L)$ on Y. The restriction of M to $Y_K$ is simply $\sigma _{K*}L$, and in particular the étale cohomology class of M is equal to the class of $\Phi _{\mathscr {P}}(L)$. By our assumption that $\Phi _{\mathscr {P}}$ is strongly filtered in the fibers the line bundle M is ample on Y. Note also that by our assumption that $\Phi _{\mathscr {P}}$ is strongly filtered in every fiber we have

$$ \Phi _{\mathscr {P}}(L^{\otimes n})\simeq \Phi ^{\mathscr {P}}(L)^{\otimes n}. $$

In particular we can choose L very ample in such a way that M is also very ample. The result then follows from Matsusaka-Mumford [20, Theorem 2]. $\square$

Remark 7.4

One can also prove a variant of Proposition 7.3 when k has characteristic 0 using de Rham cohomology instead of étale cohomology and similar techniques. However, we will not discuss this further here.

8 Proof of Theorem 1.2

In this section we prove Theorem 1.2 when the characteristic is $\ge 5$. Characteristic 3 is treated in Remark 8.5

8.1

Let K be an algebraically closed field extension of k and let X and Y be K3 surfaces over K equipped with a complex $P\in D(X\times _KY)$ defining a strongly filtered Fourier–Mukai equivalence

$$ \Phi _P:D(X)\rightarrow D(Y). $$

We can then choose a primitive polarization $\lambda$ on X of degree prime to p such that the triple $((X, \lambda ), Y, P)$ defines a K-point of $\mathscr {S}_d$. In this way the proof of Theorem 1.2 is reformulated into showing the following: For any algebraically closed field K and point $((X, \lambda ), Y, P)\in \mathscr {S}_d(K)$ there exists an isomorphism $\sigma :X\rightarrow Y$ such that the maps on crystalline and étale realizations defined by $\sigma$ and $\Phi _P$ agree.

8.2

To prove this it suffices to show that there exists such an isomorphism after replacing K by a field extension. To see this let I denote the scheme of isomorphisms between X and Y, which is a locally closed subscheme of the Hilbert scheme of $X\times _KY$, and is thus locally of finite type over K. Over I we have a tautological isomorphism $\sigma ^u:X_I\rightarrow Y_I$. The condition that the induced action on $\ell$-adic étale cohomology agrees with $\Phi _P$ is an open and closed condition on I. It follows that there exists a locally closed subscheme $I'\subset I$ classifying isomorphisms $\sigma$ as in the theorem, and this $I'$ is thus also locally of finite type over K. This implies that if we can find an isomorphism $\sigma$ over a field extension of K then such an isomorphism also exists over K, since K is algebraically closed and any scheme locally of finite type over such a field K has a K-point if and only if it is nonempty (i.e., has a point over some extension of K).

8.3

By Proposition 7.3 it suffices to show that the result holds for each generic point of $\mathscr {S}_d$. By Theorem 3.9 any such generic point maps to a generic point of $\mathscr {M}_d$ which by Theorem 3.2 admits a specialization to a supersingular point $x\in \mathscr {M}_d(k)$ given by a family $(X_R, \lambda _R)/R$, where R is a complete discrete valuation ring over k with residue field $\Omega$, for some algebraically closed field $\Omega$. By Theorem 5.3 the point $(Y, \lambda _Y)\in \mathscr {M}_d(K)$ also has a limit $y\in \mathscr {M}_d(\Omega )$ given by a second family $(Y_R, \lambda _R)/R$. Let $P'$ be the complex on $X\times Y$ giving the composition of $\Phi _P$ with suitable twists by $(-2)$-curves such that after replacing $Y_R$ by a sequence of flops the map $\Phi _{P'}$ induces an isomorphism on crystalline cohomology on the closed fiber preserving plus or minus the ample cone. By the Cohen structure theorem we have $R\simeq \Omega [[t]]$, and $((X, \lambda ), Y, P')$ defines a point of $\mathscr {S}_d(\Omega ((t)))$.

Let B denote the completion of the strict henselization of $\mathscr {M}_{\mathbb {Z}[1/d]}\times \mathscr {M}_{\mathbb {Z}[1/d]}$ at the point (x, y). So B is a regular complete local ring with residue field $\Omega$. Let $B'$ denote the formal completion of the strict henselization of $\overline{\mathscr {S}}_{d, \mathbb {Z}[1/d]}$ at the $\Omega ((t))$-point given by $((X, \lambda ), Y, P')$. So we obtain a commutative diagram

(8.3.1)

Over B we have a universal families $\mathscr {X}_B$ and $\mathscr {Y}_B$, and over the base changes to $B'$ we have, after trivializing the pullback of the gerbe $\mathscr {S}_{d, \mathbb {Z}[1/d]}\rightarrow \overline{\mathscr {S}}_{d, \mathbb {Z}[1/d]}$, a complex $\mathscr {P}_{B'}'$ on $\mathscr {X}_{B'}\times _{B'}\mathscr {Y}_{B'}$, which reduces to the triple $(X, Y, P')$ over $\Omega ((t))$. The map $B\rightarrow B'$ is a filtering direct limit of étale morphisms. We can therefore replace $B'$ by a finite type étale B-subalgebra over which all the data is defined and we still have the diagram (8.3.1). Let $\overline{B}$ denote the integral closure of B in $B'$ so we have a commutative diagram

where $\overline{B}$ is flat over $\mathbb {Z}[1/d]$ and normal. Let $Y\rightarrow \text { Spec }(\overline{B})$ be an alteration with Y regular and flat over $\mathbb {Z}[1/d]$, and let $Y'\subset Y$ be the preimage of $\text { Spec }(B')$. Lifting the map $B\rightarrow \Omega [[t]]$ to a map $\text { Spec }(\widetilde{R})\rightarrow Y$ for some finite extension of complete discrete valuation rings $\widetilde{R}/R$ and letting C denote the completion of the local ring of Y at the image of the closed point of $\text { Spec }(\widetilde{R})$ we obtain a commutative diagram

where $C\rightarrow C'$ is a localization, we have K3-surfaces $\mathscr {X}_C$ and $\mathscr {Y}_C$ over C and a perfect complex $\mathscr {P}_{C'}'$ on $\mathscr {X}_{C'}\times _{C'}\mathscr {Y}_{C'}$ defining a Fourier–Mukai equivalence and the triple $(\mathscr {X}_{C'}, \mathscr {Y}_{C'}, \mathscr {P}_{C'}')$ reducing to (X, Y, P) over $\Omega ((t))$. By [29, 5.2.2] we can extend the complex $\mathscr {P}'_{C'}$ to a C-perfect complex $\mathscr {P}'_C$ on $\mathscr {X}_C\times _C\mathscr {Y}_C$ (here we use that C is regular). It follows that the base change $(X_{\Omega [[t]]}, Y_{\Omega [[t]]}, P'_{\Omega [[t]]})$ gives an extension of (X, Y, P) to $\Omega [[t]]$ all of whose reductions satisfy our condition $(\star )$ of Paragraph 4.12.

This puts us in the setting of Lemma 6.8, and we conclude that there exists an isomorphism $\sigma :X\rightarrow Y$ (over $\Omega ((t))$, but as noted above we are allowed to make a field extension of K) such that the induced map on crystalline and étale cohomology agrees with $\Phi _P\circ \alpha$ for some sequence $\alpha$ of spherical twists along $(-2)$-curves in X (using also Lemma 6.9). By the same argument as in Remark 6.10 it follows that $\sigma$ and $\Phi _{P}$ induce the same map on realizations which concludes the proof of Theorem 1.2. $\square$

Remark 8.4

One consequence of the proof is that in fact any strongly filtered equivalence automatically takes the ample cone to the ample cone, and not its negative. This is closely related to [13, 4.1].

Remark 8.5

Given (X, Y, P) as in Theorem 1.2 and an isomorphism $\sigma : X \rightarrow Y$ inducing the same action as P on the $\ell$-adic realization of the Mukai motive for a fixed prime $\ell$ invertible in k, we have that in fact $\sigma$ and P define the same action on all the étale and crystalline realizations. This follows from the same specialization argument to supersingular K3s and [25, 3.20].

This observation can be used, in particular, to prove Theorem 1.2 in characteristic 3: With notation as in Theorem 1.2 fix a prime $\ell \ne 3$ and lift the triple (X, Y, P) to a triple $({\mathscr {X}}, {\mathscr {Y}}, {\mathscr {P}})$ over a finite extension of $\mathbb {Z}_3$. By the specialization argument of Sect. 7 it then suffices to exhibit an isomorphism between the generic fibers inducing the same map as P on the $\ell$-adic realizations. This reduces the result in characteristic 3 to the characteristic 0 result proven in Theorem 9.1 below.

9 Characteristic 0

From our discussion of positive characteristic results one can also deduce the following result in characteristic 0.

Theorem 9.1

Let K be an algebraically closed field of characteristic 0, let X and Y be K3 surfaces over K, and let $\Phi _P:D(X)\rightarrow D(Y)$ be a strongly filtered Fourier–Mukai equivalence defined by an object $P\in D(X\times Y)$. Then there exists an isomorphism $\sigma :X\rightarrow Y$ whose action on $\ell$-adic and de Rham cohomology agrees with the action of $\Phi _P$.

Proof

It suffices to show that we can find an isomorphism $\sigma$ which induces the same map on $\ell$-adic cohomology as $\Phi _P$ for a single prime $\ell$. For then by compatibility of the comparison isomorphisms with $\Phi _P$, discussed in [16, Sect. 2], it follows that $\sigma$ and $\Phi _P$ also define the same action on the other realizations of the Mukai motive.

Furthermore as in Paragraph 8.2 it suffices to prove the existence of $\sigma$ after making a field extension of K.

As in Paragraph 8.2 let $I'$ denote the scheme of isomorphisms $\sigma :X\rightarrow Y$ as in the theorem. Note that since the action of such $\sigma$ on the ample cone is fixed, the scheme $I'$ is in fact of finite type.

Since X, Y, and P are all locally finitely presented over K we can find a finite type integral $\mathbb {Z}$-algebra A, K3 surfaces $X_A$ and $Y_A$ over A, and an A-perfect complex $P_A\in D(X_A\times _AY_A)$ defining a strongly filtered Fourier–Mukai equivalence in every fiber, and such that (X, Y, P) is obtained from $(X_A, Y_A, P_A)$ by base change along a map $A\rightarrow K$. The scheme $I'$ then also extends to a finite type A-scheme $I'_A$ over A. Since $I'$ is of finite type over A to prove that $I'$ is nonempty it suffices to show that $I'_A$ has nonempty fiber over $\mathbb {F}_p$ for infinitely many primes p. This holds by Theorem 1.2. $\square$

10 Bypassing Hodge Theory

10.1

The appeal to analytic techniques implicit in the results of Sect. 5, where characteristic 0 results based on Hodge theory are used to deduce Theorem 5.3, can be bypassed in the following way using results of [18, 21].

10.2

Let R be a complete discrete valuation ring of equicharacteristic $p>0$ with residue field k and fraction field K. Let X / R be a smooth K3 surface with supersingular closed fiber. Let $Y_K$ be a K3 surface over K and $P_K\in D(X_K\times Y_K)$ a perfect complex defining a Fourier–Mukai equivalence $\Phi _{P_K}:D(X_{\overline{K}})\rightarrow D(Y_{\overline{K}})$.

Theorem 10.3

Assume that X admits an ample invertible sheaf L such that $p>L^2+4$. Then after replacing R by a finite extension there exists a smooth projective K3 surface Y / R with generic fiber $Y_K$.

Proof

Changing our choice of Fourier–Mukai equivalence $P_K$, we may assume that $P_K$ is strongly filtered. Setting $M_K$ equal to $\text {det}(\Phi _{P_K}(L))$ or its dual, depending on whether $\Phi _{P_K}$ preserves ample cones, we get an ample invertible sheaf on $Y_K$ of degree $L^2$. By [18, 2.2], building on Maulik’s work [21, Discussion preceding 4.9] we get a smooth K3 surface Y / R with Y an algebraic space. Now after replacing $P_K$ by the composition with twists along $(-2)$-curves and the model Y by a sequence of flops, we can arrange that the map on crystalline cohomology of the closed fibers induced by $\Phi _{P_K}$ preserves ample cones. Let $P\in D(X\times _RY)$ be an extension of $P_K$ and let M denote $\text {det}(\Phi _P(L))$. Then M is a line bundle on Y whose reduction is ample on the closed fiber. It follows that M is also ample on Y so Y is a projective scheme. $\square$

10.4

We use this to prove Theorem 1.2 in the case of étale realization in the following way. First observe that using the same argument as in Sect. 8, but now replacing the appeal to Theorem 5.3 by the above Theorem 10.3, we get Theorem 1.2 under the additional assumption that X admits an ample invertible sheaf L with $p>L^2+4$. By the argument of Sect. 9 this suffices to get Theorem 1.2 in characteristic 0, and by the specialization argument of Sect. 7 we then get also the result in arbitrary characteristic.