Derived equivalences of canonical covers of hyperelliptic and Enriques surfaces in positive characteristic

We prove that any Fourier--Mukai partner of an abelian surface over an algebraically closed field of positive characteristic is isomorphic to a moduli space of Gieseker-stable sheaves. We apply this fact to show that the Fourier--Mukai set of canonical covers of hyperelliptic and Enriques surfaces over an algebraically closed field of characteristic greater than three is trivial. These results extend to positive characteristic earlier results of Bridgeland--Maciocia and Sosna.


Introduction
The main motivation of this paper is the recent series of results in the study of equivalences of derived categories of sheaves of smooth projective varieties over fields other than the field of complex numbers. For instance, the first named author proves that the Zeta function of an abelian variety, as well as of smooth varieties of dimension at most three, is unaltered under equivalences of derived categories [Ho1,Ho3]. Moreover, Ward in his thesis [Wa] produces examples of genus one curves over Q admitting an arbitrary number of distinct Fourier-Mukai partners, revealing in this way consistent differences with the case of elliptic curves over C. Furthermore, Ward also studies arithmetic aspects of Calabi-Yau threefolds in positive characteristic. Finally, we mention that Lieblich and Olsson in [LibO1] extend to positive characteristic seminal works of Mukai and Orlov concerning derived equivalences of K3 surfaces. In particular, they prove that any Fourier-Mukai partner of a K3 surface X over an algebraically closed field of characteristic p = 2 is a moduli space of Gieseker-stable sheaves on X, and in addition X admits only a finite number of Fourier-Mukai partners. While Orlov's proof relies on Hodge theory, Lieblich-Olsson's proof relies on deformation theory of perfect complexes and on the theory of liftings to the Witt ring, that allows them to lift the whole problem in characteristic zero where Orlov's results can be applied.
Inspired by [LibO1], we seek to study the set of Fourier-Mukai partners of surfaces defined in positive characteristic, such as its finiteness and its members. In this paper we focus on a special class of abelian and K3 surfaces that arise as canonical covers of hyperelliptic and Enriques surfaces. Our first main result is that derived equivalent canonical covers of hyperelliptic surfaces are isomorphic. This extends to positive characteristic the work of Sosna [So,Theorem 1.1].
Theorem 1.1. Let S be an hyperelliptic surface over an algebraically closed field of characteristic p > 3 and let A be its canonical cover. Then any smooth projective surface that is derived equivalent to A is isomorphic to either A or its dual A.
We refer to Theorem 4.4 for a slightly stronger result. The work of Orlov in the study of derived equivalences of abelian varieties shows that any two abelian varieties A and B are derived equivalent if and only if there exists a symplectic isomorphism between the products A × A and B × B ( [Or]). In particular, derived equivalent abelian varieties are isogenous and admit only a finite number of partners, even in positive characteristic ([Ho2,Corollary 4.1.3]). However this concrete picture involving symplectic isomorphisms did not lead us too far towards the solution of our problem; in fact this was already observed by Sosna who grounds his proofs on Hodge theory and lattice theory. On the other hand our main ingredient is the characterization of Fourier-Mukai partners of abelian surfaces in positive characteristic as moduli spaces of Gieseker-stable sheaves. The following result extends to positive characteristic the result [BM,Theorem 5.1] of Bridgeland-Maciocia. In the following we denote by D(X) the bounded derived category of coherent sheaves on a smooth projective variety X.
Theorem 1.2. Let A be an abelian surface over an algebraically closed field k of positive characteristic and let Y be a smooth projective variety over k. Suppose furthermore that there is an equivalence of triangulated categories Φ : D(A) → D(Y ). Then Y is an abelian surface and A is isomorphic to a moduli space of Gieseker-stable sheaves on either Y or its dual Y .
The proof of the previous theorem is centered on the notion of filtered equivalences. As equivalences of derived categories attached to smooth projective varieties are of Fourier-Mukai type, a derived equivalence Φ : D(X) → D(Y ) of surfaces induces a homomorphism Φ CH : CH * (X) → CH * (Y ) between the numerical Chow rings which in general does not respect the grading. Then one says that Φ is filtered if Φ CH (0, 0, 1) = (0, 0, 1). In Proposition 3.1 we show that in the case of abelian surfaces, a filtered equivalence induces an isomorphism between the surfaces themselves. This mainly follows from the fact that derived equivalences of abelian varieties enjoy the property that skyscraper sheaves are sent to sheaves up to shift. Therefore in order to complete the proof of Theorem 1.2 we construct an equivalence as the composition of Φ with an autoequivalence Ψ of D(Y ) such that Ξ CH (0, 0, 1) is a vector v = (r, l, χ) for which the corresponding moduli space M Y (v) of Gieseker-stable sheaves is a smooth surface that admits a universal family U . This completes the proof as the composition of Ξ with the Fourier-Mukai functor associated to U is a filtered equivalence. However we have been able to carry out this plan only if the rank component of Φ CH (0, 0, 1) is non-zero, while in the remaining case we can perform the same strategy at the price of involving the Mukai's equivalence S Y : D(Y ) → D( Y ) in the construction of Ψ. This explains why the conclusions of Theorem 1.2 and [BM,Theorem 5.1] are not completely symmetric. Finally, we point out that filtered equivalences have been introduced in [LibO1], together with other stronger versions in [LibO2], to establish a derived version of Torelli theorem for K3 surfaces in characteristic p = 2. Now we go back to the proof of Theorem 1.1. The starting point is Bagnera-De Franchis' list that realizes a hyperelliptic surface S as a quotient of a product of two elliptic curves E × F by a finite group G. In particular one deduces that the dual of the canonical cover A of S is either a product of two elliptic curves, or anétale cover of degree two of two elliptic curves, or anétale cover of degree three of two elliptic curves of which one admits an automorphism of groups of order three. The strategy now is to lift to the Witt ring both the elliptic curves E and F , theétale cover A → E × F , and an arbitrary Fourier-Mukai partner B of A so that, upon restricting to the geometric generic fibers, one can involve Sosna's result in characteristic zero in order to get an isomorphism between the closed fibers of the lifts of B and A. We remark that the main difficulty in doing so is that lifts should be carefully chosen. In fact one needs to involve arguments similar to the ones of the proof of Theorem 1.2 in order to build a lift of B as a relative moduli space, and in addition, in the case of covers of degree three, one needs to lift an elliptic curve together with an automorphism and a principal polarization. This is possible because elliptic curves, both ordinary and supersingular, have good lifting properties. In particular we use the theory of canonical covers in the case of ordinary elliptic curves, and a result of Deuring (cf. [Oo]) that allows us to lift to a ramified extension of the Witt ring a supersingular elliptic curve together with an automorphism. Furthermore, another peculiar fact shared by supersingular elliptic curves that we need of, is that the graph of isogenies of supersingular curves is connected ( [Kohe,Corollary 78]).
In the last section we observe that one can push a little further the techniques of [LibO1] in order to prove that K3 surfaces that are canonical covers of Enriques surfaces in characteristic p > 3 do not admit any non-trivial Fourier-Mukai partner. This in particular extends to positive characteristic the second part of the result of Sosna [So,Theorem 1.1].
Theorem 1.3. Let S be an Enriques surface over an algebraically closed field of characteristic p > 3 and let X be its canonical cover. Then any smooth projective surface that is derived equivalent to X is isomorphic to X.
Notation. Unless otherwise specified we work over an algebraically closed field k of positive characteristic p. Further notation is introduced in Notation 2.2.

Background material
2.1. Fourier-Mukai transforms and Chow rings. Let k be an algebraically closed field of positive characteristic p. The bounded derived category of sheaves of a smooth projective variety X is defined as D(X) := D b Coh(X) . The category D(X) is k-linear and triangulated. If Y is another smooth projective variety, an object E in D(X × Y ) defines a Fourier-Mukai functor via the assignment: where p 1 and p 2 denote the projections from X ×Y onto the first and second factor respectively. An important theorem of Orlov tells us that any equivalence F : Finally we recall that the composition of Fourier-Mukai transforms is again of Fourier-Mukai type.
Consider now an abelian surface A over k. We denote by CH * (A) num = ⊕ i CH i (A) num the graded ring of algebraic cycles modulo numerical equivalence so that where NS(A) denotes the Néron-Severi group of A up to torsion. Moreover we set CH * (A) num,Q := CH * (A) num ⊗ Q. For an object F in D(A) we denote by v(F) ∈ CH * (A) num,Q its Mukai vector (see [Huy,§5.2] and [BBHP,p. 3]). Hence the Mukai vector of a locally free sheaf E on A is v(E) = rk(E), c 1 (E), χ(E) and the map Finally, we denote the Mukai pairing on CH * (A) num,Q by (r, l, χ) , (r ′ , l ′ , χ ′ ) A := l · l ′ − r χ ′ − χ r ′ , so that by the Grothendieck-Riemann-Roch Theorem there are equalities Some examples of (auto)equivalences. We denote by A an abelian surface and by A its dual variety. Moreover let P be the normalized Poincaré line bundle on A × A so that S A := Φ P : D(A) → D( A) is an equivalence of triangulated categories [Muk1]. The action of S CH A swaps the first and third entry of a vector, e.g.: Let now H be a line bundle on A and h be its class in CH 1 (A). The autoequivalence T A (H ⊗n ) : D(A) → D(A) (n ∈ Z) defined by F → F ⊗ H ⊗n acts on the numerical Chow rings as: (3) T A (H ⊗n ) CH (r, l, χ) = r, l + r n h, χ + n l · h + r n 2 h 2 2 .

Isogenies and exponents. If
A is an abelian variety over k, we denote by n A : A → A the multiplication-by-n-map on A and by A[n] the kernel of n A . We say that an elliptic curve E over k is ordinary if E[p](k) = Z/pZ and supersingular if E[p](k) = 0 (cf. [LiO]). Therefore E is supersingular if and only if p E is inseparable and the j-invariant is defined over F p 2 , the finite field with p 2 elements (cf. [Si,Theorem V.3.1]). The exponent exp ϕ of a separable isogeny ϕ : A → B of abelian varieties is the smallest positive integer that annihilates its kernel. Finally we recall that if ϕ : A → B is a separable isogeny of exponent e, then there exists a separable isogeny ψ : B → A of exponent e such that ψ • ϕ = e A and ϕ • ψ = e B (cf. [BL,Proposition 1.2.6]).
Proposition 2.3. Let ν : A → B be a separable isogeny of exponent e and denote by ν * : CH 1 (B) → CH 1 (A) the pull-back homomorphism. Then there is an inclusion of groups e 2 CH 1 (A) ⊂ Im(ν * ).
Proof. Let µ : B → A be the isogeny such that µ • ν = e A and note that Im e * A : CH 1 (A) → CH 1 (A) ⊂ Im(ν * ). We conclude by using [VdGM,Corollary 7.25] which shows that e * A is the multiplication-by-e 2 -map.
Proposition 2.4. If E and F are supersingular elliptic curves and l = p = char(k) is a prime, then there exist an integer r ≫ 0 and a separable isogeny ξ : F → E of degree l r .
Proof. Since E and F are supersingular, their j-invariants are defined over F p 2 . Moreover by [Kohe,Corollary 78] there exists an isogeny ξ ′ : F (F p 2 ) → E(F p 2 ) of degree l r for some positive integer r ≫ 0. Therefore we obtain our desired isogeny from ξ ′ by extension of scalars. Finally we observe that ξ is separable as the degree of every non-separable isogeny is divisible by char(k) ([Si, Corollary 2.12]).
2.4. Line bundles on a product of two elliptic curves. Let (E, O E ) be an elliptic curve over k. We denote the Mumford bundle on E × E by is the diagonal divisor and pr 1 , pr 2 are the projections of E × E onto the first and second factor respectively. Given another elliptic curve (F, O F ), line bundles L E and L F on E and F respectively, and a morphism ϕ : F → E, we define a line bundle on the product E × F where pr E and pr F are the projections onto E and F respectively.
Proposition 2.5. If ϕ : F → E and ψ : F → E are isogenies, then Therefore for any choice of line bundles M E and N E on E, and line bundles M F and N F on F , there are isomorphisms Proof. The proof is a simple application of the see-saw principle.
If L E , L ′ E and L F , L ′ F are line bundles on E and F respectively such that a E : . Corollary 2.6. With notation as in Proposition 2.5, in CH 1 (E × F ) there are equalities of classes Finally we show that any line bundle L ∈ Pic(E × F ) can be realized as a line bundle of the form (4).
Proposition 2.7. For any line bundle L ∈ Pic(E × F ) there exists a morphism ϕ : F → E and line bundles L E ∈ Pic(E) and L F ∈ Pic(F ) such that L ≃ L(ϕ, L E , L F ).

Proof. Denote by
We note that the restriction of L ′ to {O E }× F is trivial, while the restrictions L ′ | E×{y} lie in Pic 0 (E) for all y ∈ F . Thus by the universal property of the dual variety ([Mum, Theorem on p. 117]), there exists a unique morphism ϕ : * M E and the conclusion follows by setting ϕ = η −1 ϕ.

2.5.
Lifting results. Let k be a perfect field of positive characteristic p and let W = W (k) be the ring of Witt vectors with quotient field K. We recall that W is a complete discrete valuation ring such that K is of characteristic zero (see for instance [Lie,§11.1]). With W we will also denote a finite ramified extension of the ring of Witt vectors W (k). If X is a smooth projective scheme over k, we say that ψ : X → W is a projective lift of X if X is a projective scheme, the morphism ψ is flat, and the closed fiber X k is isomorphic to X. Grothendieck's existence theorem establishes that smooth curves always lift, as well as the line bundles on them. Moreover ordinary abelian varieties admit a canonical lift over W characterized by the fact that the absolute Frobenious lifts sideways with the abelian variety (we recall that an abelian variety A is ordinary if We refer to [MSN,Appendix,Theorem 1] for the proof of the following result.
Theorem 2.8. Let A be an ordinary abelian variety over a perfect field k of positive characteristic

lift and is unique up to a unique isomorphism inducing the identity on A. Moreover, the restriction morphism Pic(A) → Pic(A) is surjective and
. Finally, if ϕ : A → B is a morphism between ordinary abelian varieties, then there exists a unique morphism ϕ : A → B of canonical liftings such that F B • ϕ = ϕ • F A and ϕ |A = ϕ.
Another result we will need in the sequel is the existence of liftings ofétale covers. A reference for the following theorem is [SGA, §IX, 1.10].
Theorem 2.9. Let S be the spectrum of a complete local Noetherian ring, and let X → S be a proper S-scheme. Moreover denote by X 0 the closed fiber over the unique closed point of S. Then the assignment X ′ → X ′ × X X 0 yields an equivalence between the category ofétale coverings of X and the category ofétale coverings of X 0 .
2.6. Moduli spaces. Let A be an abelian surface over an algebraically closed field k and let h ∈ NS(A) be the class of an ample line bundle. Given a vector v = (r, l, χ) ∈ CH * (A) with integral coefficients, we consider the moduli space M h (v) of Gieseker-semistable sheaves with Mukai vector v, where stability is computed with respect to h.
Theorem 2.10. If r > 0 and χ are coprime integers, then every Gieseker-semistable sheaf on Proof. The first assertion follows by [HL,Remark 4.6.8] (cf. also [HL,Remark 6.1.9]), while the second follows by [Muk2,Corollary 0.2]. The existence of a universal sheaf follows by [Muk3,Theorem A.6]. Now we show that the functor Φ U : D(M ) → D(A) induces an equivalence of derived categories for any irreducible component M of M h (v). By [Muk3,Proposition 3.12] the sheaf U is strongly simple, and hence it yields a fully faithful Fourier-Mukai functor Φ U (cf. [BBHP,Theorem 1.33]). However as both A and M are two-dimensional smooth varieties with trivial canonical bundles, we conclude by [Huy,Corollary 7.8] that Φ U is an equivalence. 2.7. Relative Moduli Spaces. We also need to consider relative moduli spaces of Giesekersemistable sheaves on a projective lift f : A → W of A over the ring of Witt vectors. Let h as before be the class of an ample line bundle, and let h be a lifting of h to A. Let v = (r, l, χ) ∈ CH * (A) be a vector with integral coefficients such that l is the class of a line bundle L that lifts to a line bundle L on A. Moreover set v = (r, l, χ) where l is the class of L. By [Ma,Theorem 0.7 that is a coarse moduli space for the subfunctor of families of pure Gieseker-stable sheaves. Thus, if (r, χ) = 1 (i.e. every Gieseker-semistable sheaf on any geometric fiber of f is Gieseker-stable), then M s . Moreover, if we denote by A k the closed fiber of f and by A η the geometric generic fiber, then there are isomorphisms
In [LibO1,Theorem 6.1] the authors prove that a filtered equivalence of K3 surfaces induces an isomorphism between them. The proof of this statement is quite involved and uses deformation theory in order to lift the derived equivalence between K3 surfaces in positive characteristic, to an equivalence of K3 surfaces in characteristic zero. Here we notice that a filtered equivalence of abelian surfaces still induces an isomorphism. As the kernel of an equivalence of abelian varieties is a sheaf up to shift, its proof turns out to be rather simple. Proof. Equivalences of derived categories of abelian varieties send (up to shift) structure sheaves of points O x to sheaves. This is proved in [Br,Lemma 10.2.6] in characteristic zero, but its proof extends to positive characteristic without any change. Hence we can suppose that Φ(O x ) is a sheaf with Mukai vector (0, 0, 1), so it is itself a skyscraper sheaf. Since the argument holds for all points x in A, the proposition follows by [Huy,Corollary 5.23].
We now prove Theorem 1.2 of the Introduction which relies on the following technical proposition. (i). r is positive; (ii). l is the class of an ample line bundle on C; (iii). r is coprime with χ.
Therefore the first entry of v 2 is positive and moreover there is no divisor of r 1 that divides both I and χ ′ (see (5)). Suppose now that there is a prime divisor q of r 1 that divides χ 1 but not I. In this case we can choose the integer n to be coprime with q that in turns makes χ 2 coprime with q. Since there are infinitely many choices for such n, it follows that χ 2 is coprime with r 2 = r 1 . Suppose now that there is a prime divisor q of r 1 that divides I but not χ 1 . Then it is easy to see that χ 2 is again coprime with r 2 (in fact we can choose any integer n). Suppose now that both χ 1 and I are coprime with r 2 . We can choose n so that χ 1 + nI = 0 modulo any prime divisor q of r 1 . With this choice of n, it is easy to verify that χ 2 is relatively prime with r 2 . Therefore we set Ψ 2 := T C (B ⊗n 1 ) • Ψ 1 . Let now Θ be an ample line bundle with class θ ∈ CH 1 (C) and consider the equivalence Ψ 3 := T C (Θ ⊗(r 1 d) ) • Ψ 2 where d is a sufficiently large integer. Then Ψ CH 3 sends (0, 0, 1) to v 3 := r 1 , l 1 + r 1 n b 1 + r 2 1 d θ, χ 1 + n I + r 1 n 2 b 2 1 2 + r 1 d θ · (l 1 + r 1 n b 1 ) + r 3 1 d 2 θ 2 2 .
We notice that, for d large enough, the second component of v 3 is an ample class, and that the third component of v 3 is congruent to χ 2 modulo r 1 . Therefore it is itself coprime with r 1 . In conclusion, the equivalence we are looking for is simply Ψ 3 . Proof. By general theory Y is a smooth surface with trivial canonical bundle. Let l = p be a prime and consider the l-adic cohomology groups H í et (Y, Q l ). By [Ho1,Lemma 3.1] the equivalence Φ induces an isomorphism In positive characteristic the fact that a smooth surface has trivial canonical bundle and b 1 = 4 imply that Y is isomorphic to an abelian surface (see [Lie,Section 7]  the rank component r is positive, the class l is ample, and χ is coprime with r. Moreover, by looking at the proof of Proposition 3.2, it is the case that C = Y if the rank component of Φ CH (0, 0, 1) is non-zero, and C = Y otherwise. Now let L be a line bundle on C whose class is l and set v l = (r, l, χ) ∈ CH * (C). Then by Theorem 2.10 there is a universal family of derived categories satisfying Φ CH U (0, 0, 1) = v. As the composition Φ −1 U • Ψ CH sends (0, 0, 1) to (0, 0, 1), by Proposition 3.1 we get A ≃ M l (v l ).

FM partners of canonical covers of hyperelliptic surfaces
We denote the set of Fourier-Mukai partners of a smooth projective variety X by We say that FM(X) is trivial if FM(X) = {X}. In the case of an abelian variety A, we say that its set of Fourier-Mukai partners is trivial if FM(A) ⊂ {A, A}.
A hyperelliptic surface over an algebraically closed field k of positive characteristic p > 3 is a smooth projective minimal surface X with K X ≡ 0, b 2 (X) = 2, and such that each fiber of the Albanese map is a smooth elliptic curve (cf. [Ba,§10]). These surfaces can be described as quotients (E × F )/G of two elliptic curves E and F by a finite group G. The group G acts on E by translations, and on F in a way such that F/G ≃ P 1 . Moreover, there are only a finite number of possibilities for the action of G on E × F , which have been classified by Bagnera-De Franchis [Ba,10.27].
By [Ba,§9.3] the order n of the canonical bundle of X is finite with n = 2, 3, 4, 6. Therefore we can consider the canonical cover π : X → X of the surface X which is theétale cyclic cover associated to the canonical bundle ω X . The degree of π is the order n of ω X , and in addition π comes equipped with an action of the cyclic group that realizes X as the quotient X/(Z/nZ).
By looking at the Bagnera-De Franchis' list [Ba,10.27], the canonical cover X of an arbitrary hyperelliptic surface X = (E × F )/G is an abelian surface that sits inside a tower of surfaces where π ′ is anétale cyclic cover of degree one, two, or three. Moreover, if π ′ has degree three, then F admits an automorphism of groups of order three and has j-invariant equals to zero. Therefore the dual morphism π ′ realizes the dual of X either as the product E × F , or as a degree twó etale cyclic cover of E × F , or as a degree threeétale cyclic cover of E × F such that F has an automorphism of groups of order three.
4.1. The work of Sosna. In [So, Theorem 1.1] the author proves that the set of Fourier-Mukai partners of the canonical cover of a complex hyperelliptic surface is trivial. By using Bagnera-De Franchis' classification, Sosna's theorem boils down to proving the following result concerning derived equivalences of special abelian surfaces.

Theorem 4.1 (Sosna). Let E and F be complex elliptic curves and A be a complex abelian surface. Then FM(E × F ) is trivial. Moreover, if E × F → A is a degree twoétale cyclic cover, then FM(A) is trivial. Finally, the same conclusion holds if E × F → A is a degree threeétale cyclic cover and rk NS(A) ∈ {2, 4}.
In view of Theorem 2.9 we prefer to work withétale covers instead of quotients. Thus we reformulate Sosna's theorem in the following version.

Proposition 4.2. Let E and F be two complex elliptic curves. Then FM(E × F ) is trivial. Moreover, if A is a degree twoétale cyclic cover of E × F , then FM(A) is trivial as well. Finally, the same conclusion holds if A is a degree threeétale cyclic cover of E × F and rk NS(A) ∈ {2, 4}.
Proof. If A → E × F is a cover of degree one, two, or three, then the dual isogeny E × F → A realizes A as a quotient of two elliptic curves. Then by Theorem 4.1 we conclude that FM( A) is trivial. As FM(A) = FM( A) and rk NS(A) = rk NS( A), the proposition follows at once.
As an application of Proposition 4.2, we deduce some further finitiness results that will be useful towards the proof of Theorem 1.1. Proof. We show that the dual abelian variety A satisfies the hypotheses of Proposition 4.2. The result will follows as FM(A) = FM( A). Let q be either 2 or 3 and consider an isogeny ψ : E ×F → A of exponent q such that ψ • ϕ = q A . As deg q A = q 4 and deg ϕ = q 3 , we deduce that deg ψ = q. Hence the dual isogeny ψ is a cyclic cover of E × F of order q. The second statement follows as rk NS(A) = rk NS( A).

4.2.
Strategy of the proof of Theorem 1.1. Since an abelian surface and its dual have the same Fourier-Mukai partners, the following theorem in particular proves Theorem 1.1.
Theorem 4.4. Let E and F be elliptic curves over an algebraically closed field of characteristic p > 0. Then FM(E × F ) is trivial. Moreover, if A is a degree twoétale cyclic cover over E × F and p > 2, then FM(A) is trivial as well. Finally, if F admits an automorphism of groups of order three, A is a degree threeétale cyclic cover over E × F and p > 3, then FM(A) is again trivial.
In order to prove the previous result, we will consider the following set of hypotheses Setting 4.5. We denote by E and F two elliptic curves over an algebraically closed field k of characteristic p > 0. Moreover we set ν : A → E × F to be either an isomorphism of abelian surfaces, or anétale cyclic cover of degree d ν = 2, 3 (as in the hypotheses of Theorem 4.4). Finally assume that p > deg ν.
Remark 4.6. Since the exponent of an isogeny divides its degree, the exponent of the isogeny ν of Setting 4.5 is either one if ν is an isomorphism, or d ν otherwise. Let now µ : E × F → A be an isogeny of exponent d ν such that µ • ν = (d ν ) A . Then the dual isogeny µ : A → E × F is either an isomorphism, or else its degree and exponent satisfy (deg µ, exp µ) = (d 3 ν , d ν ).
As an application of Theorem 2.9 we deduce that both the isogenies ν and µ of Setting 4.5 and Remark 4.6 lift to the ring of Witt vectors. In the following result we check that their degrees and exponents remain unchanged when passing from the special fiber to the general fiber.
Proposition 4.7. Let E and F be elliptic curves and ϕ : A → E × F be anétale isogeny of abelian surfaces. Let q = p be a prime integer and assume that either ϕ is an isomorphism, or deg ϕ = q, or deg ϕ = q 3 and exp ϕ = q. If E → W and F → W are projective lifts of E and F over a finite ramified extension of the ring of Witt vectors respectively, then there exist a projective lift A → W of A and an isogeny ϕ W : A → E × W F such that ϕ W lifts ϕ and its restriction ϕ η : A η → E η × F η to the geometric general fibers is an isogeny with deg ϕ η = deg ϕ and exp ϕ η = exp ϕ Proof. By Theorem 2.9 there is a projective lift A → W of A and anétale cover ϕ W : A → E × W F that specializes to ϕ. Up to composing with a translation of E × W F, we can suppose that ϕ W is a homomorphism of groups. Hence the restriction of ϕ W to the geometric generic fiber of A is an isogeny ϕ η : To see this we notice that the kernel K of ϕ W is a finiteétale group over W and moreover, as ϕ is separable, we have where K k is the closed fiber and K η is the geometric generic fiber. In particular this takes care of the case when the degree ϕ is either one or q.
Suppose now that deg ϕ = q 3 and exp ϕ = q. We only need to show that exp ϕ η = q. Let ψ : E×F → A be an isogeny such that ϕ•ψ = q E×F and let ψ W : X → A be a lift of ψ as in Theorem 2.9 so that ϕ W • ψ W |X k = ϕ • ψ = q E×F . As the multiplication-by-q map q W : E × W F → E × W F lifts q E×F as well, we conclude that X ≃ E × W F and ϕ W • ψ W = q W By restricting to the geometric general fibers, we find that ϕ η • ψ η = q (E×F )η is the multiplication-by-q-map on the geometric general fiber (E × F ) η . Therefore the exponent of ϕ η is either one or q, but if exp ϕ η = 1, then both ϕ η and ϕ are isomorphisms, which is excluded by hypothesis.
We will deduce Theorem 4.4 from the following technical proposition. (E 2 ). the class l ∈ CH 1 (C) is is ample; (E 3 ). χ is coprime with r.
Set now λ = ν if C = A, and λ = µ otherwise (see Remark 4.6). In addition assume that there exist projective lifts E → W and F → W of E and F over a finite ramified extension of the ring of Witt vectors respectively such that the following conditions hold: Proof.
Step 1. We first prove that there exists a projective lift B → W of B such that the geometric generic fiber B η is derived equivalent to C η . Let L be an ample line bundle on C with class l. Then Theorem 3.3 implies that B is isomorphic to a moduli space M l (v l ) of Gieseker-stable shaves with Mukai vector v l = (r, l, χ) ∈ CH * (C). Consider now a preimage L of L under ρ as in (A 1 ) and the relative moduli space where v l = (r, l, χ) and l is the class of L. As discussed in §2.7, this is a projective lift of B and the geometric generic fiber M η is a moduli space of Gieseker-stable sheaves on C η with Mukai vector v η = (r, l |Cη , χ). Therefore as discussed in Theorem 2.10, the condition (E3) implies that there exists a universal family U η on M η × C η inducing an equivalence Φ U : Step 2. The equivalence Φ U shows that M η is an abelian surface. Now we prove that under the assumptions of Theorem 4.4 the abelian surface C η is isomorphic to either M η or its dual M η . By Lefschetz's principle we can suppose that the abelian surface C is defined over a subfield of the complex numbers C and therefore that C η is defined over C. Suppose first that ν : A → E × F is an isomorphism. Then both λ and λ W are isomorphisms and therefore so is the restriction λ η : C η → E η × F η of λ W to the geometric generic fibers. As a product of elliptic curves has no non-trivial Fourier-Mukai partners (cf. Theorem 4.1), we deduce then an isomorphism C η ≃ M η .
Step 3. To conclude the proof we use the argument of [LibO1, Lemma 6.5] (based on a result of Matsusaka-Mumford) in order to prove that the isomorphism C η ≃ M η (resp. C η ≃ M η ) between the geometric generic fibers of the two liftings induces an isomorphism C ≃ B (resp. C ≃ B) between the closed fibers. This immediately yields that either B ≃ A or B ≃ A, and hence that FM(A) is trivial.

Finding a suitable equivalence
In this section we finish the proof of Theorem 1.1. According to Proposition 4.8, we only need to verify its hypotheses. We work under the hypotheses of Setting 4.5 and assume that the abelian surface B is a Fourier-Mukai partner of A. In the following we will distinguish two cases: (a) at least one of the two elliptic curves E or F is ordinary, and (b) both E and F are supersingular. 5.1. The case where one of the two curves is ordinary. The following two propositions show the existence of an equivalence Ψ : D(B) → D(C) satisfying the hypotheses (E 1 ), (E 2 ) and (E 3 ) of Proposition 4.8. (i). r is relatively prime with both p 1 and p 2 ; (ii). either p 1 divides r but not χ and p 2 divides χ but not r, or viceversa.
We divide the proof in five cases.
Case I: Suppose that neither p 1 nor p 2 divides χ 0 . In this case the equivalence Ψ is given by the composition S A • Φ : D(B) → D( A).
Case II: Suppose that both p 1 and p 2 divides both r 0 and χ 0 . By (6) we see that I is relatively prime with p 1 and p 2 as well. Choose now a positive integer n coprime with both p 1 and p 2 . Therefore by looking at the definition (7) of χ n , this immediately implies that χ n is relatively prime to both p 1 and p 2 . We conclude then as in Case I.
Case III: Suppose that both p 1 and p 2 divides r 0 , and that precisely one of them, say p 1 , divides χ 0 . We choose a generic positive integer n such that n is relatively prime to both p 1 and p 2 . By (6) I is relatively prime to p 1 , and by (7) p 1 does not divide χ n . Moreover, again by (7) and the fact that n is general, we can suppose that p 2 does not divide χ n as well. We then set Ψ := S A • Φ n .
Case IV: Suppose that both p 1 and p 2 divides χ 0 and that precisely one of them, say p 1 , divides r 0 . In this case we proceed as in Case III by considering the composition S A • Φ in place of Φ.
Case V: Suppose that one of the primes, say p 1 , divides both r 0 and χ 0 , but p 2 divide nor r 0 neither χ 0 . Let n = p 2 and consider Φ n . By (6) p 1 does not divide I, and hence p 1 does not divide χ n . Moreover, by our choice of n, we have that p 2 does not divide χ n as well. We conclude then as in Case I. (i). r is positive and relative prime with p; (ii). the class l ∈ CH 1 (C) is ample; (iii). χ is relative prime with r.
Set now λ = ν if C = A, and λ = µ otherwise. Then the class l is the pull-back of some ample class in CH 1 (E × F ) via λ.
We note that r 2 is not divisible by p, and moreover that by (9) χ 2 is not divisible by any prime divisor q = p 2 of r 1 that does not divide I. On the other hand, if a prime divisor q = p 2 divided both r 1 and I, then by (8) it does not divide χ 1 , and hence neither χ 2 . Finally we prove that χ 2 is not divisible by p 2 in case p 2 divides r 1 . But this follows by the construction of Ξ 1 , and by noting that in the case (b) earlier discussed, p 2 does not divide r 0 .
By Lemma 2.3 and (9) we can write l 2 = λ * l 3 for some class l 3 ∈ CH 1 (E × F ), and hence as the isogeny λ has degree either p c 2 with c = 1 if C = A, and c = 3 otherwise. Therefore the first component of v 3 is still positive and relatively prime with p, while the second component is a pull-back of an ample class in CH 1 (E × F ) (for d ≫ 0). Moreover χ 3 ≡ χ 2 (mod r 2 ) and hence it is still relative prime with r 2 = r 1 . Our desired equivalence is hence given by Ξ := Ξ 3 .
Remark 5.3. In Proposition 5.2 the conclusion that r is coprime with p does not play any role towards the verification of the hypotheses of Proposition 4.8. However it will used in the supersingular case in order to construct an equivalence satisfying the hypotheses of Proposition 4.8.
In order to prove that the hypotheses (A 1 ) and (A 2 ) of Proposition 4.8 hold as well, we first prove a couple of auxiliary results.
Proposition 5.4. Let E and F be ordinary elliptic curves. Then there exist projective liftings E → W and F → W of E and F over the ring of Witt vectors respectively such that the restriction morphism Pic(E × W F) → Pic(E × F ) is surjective.
Proof. The product E × F is an ordinary abelian surface so that we can consider its canonical lift (Y → W, F Y ), by virtue of Theorem 2.8. Moreover the restriction morphism Pic(Y) → Pic(E × F ) is surjective. However by using the universal property of the fiber product, it is immediate to show that Y ≃ E × W F where E → W and F → W are the canonical lifts of E and F respectively.
Proposition 5.5. If E is an ordinary elliptic curve and F is supersingular, then any projective lifts E → W and F → W of E and F respectively are such that the restriction morphism Pic(E × W F) → Pic(E × F ) is surjective.
Proof. As Hom(E, F ) = 0 we obtain an isomorphism Pic(E × F ) ≃ Pic(E) × Pic(F ). Let now pr * E L E ⊗ pr * F L F be an arbitrary line bundle on E × F . Since line bundles on curves lift, we can consider lifts L E and L F of L E and L F respectively. Hence the line bundle pr * E L E ⊗ pr * F L F on E × W F is a lift of pr * E L E ⊗ pr * F L F . Proposition 5.6. Assume the assumptions of Setting 4.5 and let Φ : D(B) → D(A) be an equivalence of derived categories of abelian surfaces. If both E and F are ordinary elliptic curves, then the hypotheses of Proposition 4.8 hold.
Proof. By Propositions 3.2 and 5.2 there exists an equivalence Ξ : D(B) → D(C) with C ∈ {A, A} such that the vector Ξ CH (0, 0, 1) = (r, λ * l, χ) satisfies the hypotheses (E 1 ), (E 2 ) and (E 3 ) of Proposition 4.8. Let L be a line bundle on E × F with class l ∈ CH 1 (E × F ). By Propositions 5.4 there exist projective lifts E → W and F → W of E and F to the ring of Witt vectors W respectively, and a lift L ∈ Pic(E × W F) of L. Let λ W : C → E × W F be the lift of λ : C → E × F defined by Proposition 4.7. It follows that λ * W L lifts λ * L which proves condition (A 1 ). In order to prove (A 2 ) we can assume that F has an automorphism of groups of order three. Then by Theorem 2.8 the geometric generic fiber F η admits an automorphism of groups of order three and hence rk CH 1 (E η × F η ) ∈ {2, 4} where E η is the geometric generic fiber of the lift E → W .
Proposition 5.7. Assume the assumptions of Setting 4.5 and let Φ : D(B) → D(A) be an equivalence of derived categories of abelian surfaces. If E is ordinary and F is supersingular or viceversa, then the hypotheses of Proposition 4.8 hold.
Proof. By Propositions 3.2 and 5.2 there exists an equivalence Ξ : D(B) → D(C) with C ∈ {A, A} such that the vector Ξ CH (0, 0, 1) = (r, λ * l, χ) satisfies the hypotheses (E 1 ), (E 2 ) and (E 3 ) of Proposition 4.8. By Proposition 5.5 we can choose general lifts E and F of E and F to W respectively and a line bundle L ∈ Pic(C) such that the condition (A 1 ) holds (see the argument of Proposition 5.6). In fact we choose E and F so that there are no non-trivial morphisms between the geometric generic fibers E η and F η of E and F respectively. It follows that rk NS(E η × F η ) = 2 independently from the fact that F admits an automorphism of groups of order three or not. As in Proposition 5.6 this immediately implies that rk NS(C η ) = 2. 5.2. Supersingular case. We are going to prove that the hypotheses of Proposition 4.8 hold when the elliptic curves E and F are both supersingular. The following proposition proves the conditions (E 1 ), (E 2 ) and (E 3 ). (ii). the class l ∈ N S(C) is ample; (iii). χ is relatively prime with r.
To conclude the proof we need to analyze the case when ϕ is separable. But in this case the proof is simpler as there is no need to introduce the isogeny ξ and the equivalence Ψ 2 . Then it is enough to set γ = ϕ and proceed as in the inseparable case.
Proposition 5.9. Assume the assumptions of Setting 4.5 and let Φ : D(B) → D(A) be an equivalence of derived categories of abelian surfaces. Moreover assume that the elliptic curves E and F are both supersingular. Then the hypotheses of Proposition 4.8 hold.
Proof. By Propositions 3.2 and 5.8 we can find an equivalence Ξ : D(B) → D(C) with C ∈ {A, A} such that the vector v := r, λ * l(ϕ, m 1 , m 2 ), χ = Ξ CH (0, 0, 1) satisfies the assumptions (E 1 ), (E 2 ), and (E 3 ) of Proposition 4.8. The class l(ϕ, m 1 , m 2 ) is the class of an ample line bundle L(ϕ, M 1 , M 2 ) on E × F where M 1 and M 2 are line bundles on E and F respectively, and ϕ : F → E is either the constant morphism O E in case ν is an isomorphism, or anétale isogeny otherwise. Let E → W be a lift of E to the ring of Witt vectors. If ϕ = O E is the constant morphism, then on any lift F → W of F we can construct a lift of L = M 1 ⊠ M 2 by simply lifting M 1 and M 2 to W and taking their exterior product. Assume now that ϕ is a separable isogeny. Then Theorem 2.9 determines a projective lift F → W of F and anétale cover ϕ : F → E that specializes to ϕ. It is not difficult to check that L(ϕ, M 1 , M 2 ) lifts to a line bundle L on E × W F. This goes as follows. As line bundles on curves lift, we can find a lift of O E (O E ) which in turn allows us to construct a lift M E,W to W of the Mumford line bundle M E (cf. §2.4). By denoting by M 1,W and M 2,W the liftings of M 1 and M 2 on E and F respectively, we see that the line bundle L := (1 E × W ϕ) * M E,W ⊗ pr * E M 1,W ⊗ pr * F M 2,W is a lift of L(ϕ, M 1 , M 2 ), where pr E and pr F are the obvious projections. In conclusion the line bundle λ * W L lifts λ * L(ϕ, M 1 , M 2 ) where λ W : C → E × W F is the lift of Proposition 4.8. This proves the condition (A 1 ) in the case deg = 1, 2. Now suppose deg ν = 3 and that F has an automorphism of groups ρ : F → F of order three. By a result of Deuring (cf. [Oo,p. 189 or p. 172]) it is possible to lift a supersingular elliptic curve together with an endomorphism over a finite ramified extension W of the ring of Witt vectors. Thus we denote by F 0 → W and ρ W : F 0 → F 0 the lifts of F and ρ to W respectively. Theétale cover ϕ : E → F determines a lift E 0 of E together with a lift ϕ 0 of ϕ over W . Denote now by λ W,0 : C 0 → E 0 × W F 0 the lift of λ as defined in Proposition 4.8. After noting the isomorphism (1 E ×ϕ) * M E ≃ ( ϕ×1 F ) * M F , and hence that L(ϕ, M 1 , M 2 ) ≃ ( ϕ×1 F ) * M F ⊗pr * E M 1 ⊗pr * F M 2 , the previous argument shows that the line bundle λ * L(ϕ, M 1 , M 2 ) lifts to C 0 . Moreover the restriction of ρ W to the geometric generic fiber (F 0 ) η is an automorphism that is not proportional to the identity. Hence rk CH 1 (E 0 ) η × (F 0 ) η = {2, 4} according to whether or not there is an isogeny between (E 0 ) η and (F 0 ) η . As the rank of the Néron-Severi group does not change under separable isogenies, we have that rk CH 1 ((C 0 ) η ) ∈ {2, 4} where (C 0 ) η is the geometric generic fiber of C 0 . In particular the assumption (A 2 ) holds.

Canonical Covers of Enriques Surfaces
An Enriques surface S over an algebraically closed field k of characteristic p = 2 is a smooth projective minimal surface with canonical bundle ω S of order two, χ(O S ) = 1, and b 2 (S) = 10. The canonical cover π : X → S induced by ω S is a doubleétale cover with X a K3 surface. Let Φ : D(Y ) → D(X) be a derived equivalence so that Y is itself a K3 surface by [LibO1,Proposition 3.9]. We aim to prove Theorem 1.3, namely that X ≃ Y .
In [LibO1] the authors prove that Shioda-supersingular K3 surfaces do not admit any nontrivial Fourier-Mukai partners, thus we can assume that X has Picard rank less than 22. Since in odd characteristic Shioda-supersingularity is equivalent to Artin-supersingularity (cf. for instance [Pe]), we can assume that the formal Brauer group of X is of finite height. In particular there exists a lift X → W of X over the ring of Witt vectors such that NS(X) ≃ NS(X η ), where as usual X η is the geometric generic fiber ( [NygO,p. 505]). By the work of Jang ([Ja, Theorem 2.5]) there is a primitive embedding of the Enriques lattice Γ(2) := U (2) ⊕ E 8 (−2) of S into NS(X) such that the orthogonal complement of the embedding does not contain any vector of self intersection −2. Using the isomorphism NS(X) ≃ NS(X η ), together with the characterization of Enriques-K3 surfaces in terms of its periods ([Nam, Theorem 1.14]), we deduce that X η is a K3 surface arising as the canonical cover of an Enriques surface. In particular X η does not have any non-trivial Fourier-Mukai partner thanks to [So,Theorem 1.1]. On the other hand, [LibO1,Proposition 8.2] implies that Y is isomorphic to a moduli space M h (v) of h-Gieseker-stable sheaves for some Mukai vector v = (r, l, χ). Since all the line bundles on X deform to line bundles over X → W , we can consider the relative moduli space M X /W (r, l, χ) → W where l is a lift of l. The geometric generic fiber of M X /W (r,l, χ) → W is by construction a Fourier-Mukai partner of X η and therefore, by the previous considerations, it is isomorphic to X η . We conclude by using [LibO1,Lemma 6.5] saying that the isomorphism between the geometric generic fibers of relative K3 surfaces induces an isomorphism between the closed fibers.