Descent of tautological sheaves from Hilbert schemes to Enriques manifolds

Let $X$ be a K3 surface which doubly covers an Enriques surface $S$. If $n\in\mathbb{N}$ is an odd number, then the Hilbert scheme of $n$-points $X^{[n]}$ admits a natural quotient $S_{[n]}$. This quotient is an Enriques manifold in the sense of Oguiso and Schr\"oer. In this paper we construct slope stable sheaves on $S_{[n]}$ and study some of their properties.


Introduction
In 1896 Federigo Enriques gave examples of smooth projective surfaces with irregularity q = 0 and geometric genus p g = 0 which are not rational.Therefore these surfaces were counterexamples to a conjecture by Max Noether, which stated that surfaces with q = p g = 0 are rational.Nowadays such a surface is called an Enriques surface.
The canonical bundle ω S of an Enriques surface S has order two in the Picard group of S. The induced double cover turns out to be a K3 surface (a two dimensional hyperkähler manifold), hence it is the universal cover of S. On the other hand, every K3 surface X which admits a fixed point free involution doubly covers an Enriques surface S.
Mimicking this correspondence Oguiso and Schröer defined higher dimensional analogues of Enriques surfaces, the so called Enriques manifolds in [19].To be precise a connected complex manifold that is not simply connected and whose universal cover is a hyperkäler manifold is called an Enriques manifold.
The following class of examples is of interest to us in this work: take an odd natural number n ∈ N and an Enriques surface S. We have the induced K3 surface X with a fixed point free involution ι such that S = X/ι.Since n is odd we get an induced fixed point free involution ι [n] on the Hilbert scheme of n-points X [n] .The quotient of X [n] by the involution ι [n] is an Enriques manifold S [n] of dimension 2n.We have an étale Galois cover ρ : X [n] → S [n] .
In this article we construct and study stable sheaves on Enriques manifolds of type S [n] .The main idea is to start with slope stable sheaves on X [n] and check if they descend to S [n] .Known examples of stable sheaves on X [n] are given by the tautological bundles E [n]  associated to slope stable locally free sheaves E on X.
For example, we prove that E [n] descends to S [n] if and only if E descends to S. If E [n] descends we have E [n] ∼ = ρ * F [n] for some locally free sheaf F [n] on S [n] .We then show that it is possible to find an ample divisor D ∈ Amp(S [n] ) such that F [n] is slope stable with respect to D. Finally using results from Kim [10] and Yoshioka [25], we are able to prove that, given certain conditions are satisfied, we have in fact a morphism between a moduli spaces of stable sheaves on S and moduli space of stable sheaves on S [n] .This morphism identifies the former moduli space as a smooth connected component in the latter.
This paper consists of four sections.In Section 1 we generalize some results concerning tautological bundles on Hilbert schemes of points.Section 2 contains results about the descent of tautological sheaves from X [n] to S [n] .We compute certain Ext-spaces in Section 3. In the final Section 4 we study the stability of sheaves on Enriques manilfolds of type S [n] .

Stability of tautological sheaves on Hilbert schemes of points
Let X be a smooth projective surface.The Hilbert scheme X [n] := Hilb n (X) classifies length n subschemes in X, that is In fact X [n] is smooth itself and has dimension 2n, see [6,Theorem 2.4].Moreover X [n]  is a fine moduli space for the classification of length n subschemes and comes with the universal length n subscheme The universal subscheme Z comes with two projections p : Z → X [n] and q : Z → X.Note that the morphism p is finite and flat of degree n.
To any locally free sheaf E of rank r on X one can associate the so called tautological vector bundle E [n] on X [n] via E [n] := p * q * E. As p is finite and flat of degree n the sheaf E [n] is indeed locally free and has rank nr.The fiber at [Z] ∈ X [n] can be computed to be Remark 1.1.Note that the definition of E [n] makes sense for E a coherent sheaf on X or even a complex E ∈ D b (X) in the derived category of X, see [12,Definition 2.4].
In [23, Theorem 1.4, Theorem 4.9] Stapleton proves that if h ∈ Amp(X) is an ample divisor on X and E ∼ = O X is a slope stable (with respect to h) locally free sheaf, then there is H ∈ Amp(X [n] ) such that the associated tautological bundle E [n] is slope stable with respect to H on X [n] .
In fact Stapleton's result remains true, if we drop the locally free condition and allow for torsion free sheaves, see for example [24,Proposition 2.4] for a first step toward the following observation: Lemma 1.2.Assume E is torsion free and slope stable with respect to h ∈ Amp(X) such that its double dual satisfies E * * = O X , then the associated tautological sheaf E [n] is slope stable with respect to some H ∈ Amp(X [n] ).
Proof.Since X is a smooth projective surface and E is torsion free we can canonically embed E into its double dual.This gives an exact sequence (1) 0 Here E * * is locally free and also slope stable with respect to h.Furthermore Q has support of codimension two.By [22,Corollary 6] the functor (−) [n] : Coh(X) → Coh(X [n] ) is exact.So we get an exact sequence on X [n]   0 By our assumptions (E * * ) [n] is slope stable with respect to some H ∈ Amp(X [n] ).But Q [n] has support of codimension two in X [n] so that E [n] is isomorphic to (E * * ) [n] in codimension one and thus must be also be slope stable with respect to H.
The previous lemma shows that for every slope stable E with E * * ∼ = O X there is H ∈ Amp(X [n] ) such that the tautological sheaf E [n] is slope stable with respect to H. Since E belongs to some moduli space M X,h (r, c 1 , c 2 ), one may ask how H varies if E varies in its moduli.We can answer this question in the case that all sheaves classified by M X,h (r, c 1 , c 2 ) are locally free.
the sheaf E is slope stable and locally free, then there is H ∈ Amp(X [n] ) such that E [n] is slope stable with respect to H for all Proof.By a result of Stapleton, see [23,Theorem 1.4], we know that for [E] ∈ M X,h (r, c 1 , c 2 ) the locally free sheaf E [n] is slope stable with respect to the induced nef divisor h n ∈ NS(X [n] ).It is also well known that the Hilbert -Chow morphism HC : X [n] → X (n) is semi-small and that q : Z → X is flat, see [14,Theorem 2.1].
The proof is now exactly the same as for tautological bundles on the generalized Kummer variety Kum n (A) associated to an abelian surface A, see [21, Proposition 2.9].
Remark 1.4.The condition that all sheaves in M X,h (r, c 1 , c 2 ) are slope stable can be achieved (for example) in the following two different ways: the first is by a special choice of the numerical invariants, see [9,Lemma 1.2.14].The second way is by choosing a special ample class h, see [9,Theorem 4.C.3].
To find a moduli space such that all sheaves are locally free, one can do the following: if the tuple (r, c 1 ) is fixed, then by Bogomolov's inequality the second Chern class is bounded from below, see [9,Theorem 3.4.1].Choose the minimal c 2 , then every sheaf in M X,h (r, c 1 , c 2 ) is locally free.Indeed, if such an E is not locally free, then E * * is locally free, stable with respect to h and has the same tuple (r, c 1 ), but it has smaller c 2 by exact sequence (1) as c 2 (Q) < 0, contradicting minimality.See also [9, Remark 6.1.9]for a similar argument.Now let X be a K3 surface.Denote the Mukai vectors of E and E [n] by v respectively v [n] ∈ H * (X [n] , Q).If E [n] is slope stable, then it belongs to the moduli space M X [n] ,H (v [n] ) of semistable sheaves on X [n] with Mukai vector v [n] .In fact we can generalize [24,Corollary 4.6] to get the following the sheaf E is slope stable, locally free and h i (X, E) = 0 for i = 1, 2, then the functor (−) [n]  induces a morphism Proof.First note that the map [E] → [E [n] ] is indeed a regular morphism, see for example [13,Proposition 2.1].Furthermore this morphism is injective on closed points, which follows immediately from [1, Theorem 1.1] (see also [14,Theorem 1.2] for a generalization).

Descent of tautological sheaves to Enriques manifolds
Let G be a finite group.Consider an étale Galois cover ϕ : Y → Z with Galois group G, that is there is a free G-action on Y such that Z = Y /G and ϕ is the quotient map.In this situation there is an equivalence between the categories Coh(Z) of coherent sheaves on Z and Coh G (Y ) of G-equivariant coherent sheaves on Y given by the functors A coherent sheaf E on X is said to be G-invariant, if there are isomorphisms E ∼ = g * E for every g ∈ G.A G-equivariant coherent sheaf is G-invariant, but the converse is not true.For our purposes the following will suffice, see [20, Lemma 1]: Remark 2.2.Recall that if (X, ι) is a pair consisting of a K3 surface and a fixed point free involution, then G = ι ∼ = Z/2Z acts freely on X and the quotient S is an Enriques surface.The morphism π : X → S is an étale Z/2Z-Galois cover.
On the other hand if S is an Enriques surface, then its canonical bundle ω S is 2-torsion.One can consider the induced canonical cover φ : S := Spec(O S ⊕ω S ) → S. The morphism φ is an étale Z/2Z-Galois cover and S is a K3 surface with fixed point free involution, the covering involution of φ.Furthermore φ * O S ∼ = O S ⊕ω S .
In [19] Oguiso and Schröer generalized the notion of an Enriques surface to that of an Enriques manifold by mimicking the above constructions: Definition 2.3.A manifold Y is called an Enriques manifold if it is a connected complex manifold that is not simply connected and whose universal cover is a hyperkähler manifold.
Remark 2.4.In [2] the authors also gave a definition of higher dimensional Enriques varieties, which slightly differs from the one of Enriques manifolds in [19].Example 2.6.Let (X, ι) be a pair consisting of a K3 surface together with a fixed point free involution ι on X.Then X covers the Enriques surface S = X/ι.If n ∈ N is odd, then (X, ι) induces the pair (X [n] , ι [n] ) of the Hilbert scheme of n-points on X and the induced fixed point free involution ι [n] on X [n] .Thus G = ι [n] ∼ = Z/2Z acts freely on X [n] and the quotient S [n] is an Enriques manifold with index d = 2 coming with an étale Z/2Z-cover ρ : We want to study the descent of sheaves from X to S respectively from X [n] to S [n] .
To do this we need the following lemma: Lemma 2.7.There is an isomorphism of functors from Coh(X) to Coh(X [n] ): Proof.Recall that (−) [n] = FM O Z (−) can be written as the Fourier -Mukai transform with kernel the structure sheaf of universal family Z in X × X [n] , see for example [14,Section 2.3].Define a group isomorphism and note that this is a so-called c-isomorphism, see [15,Definitions 3.1 and 3.3].By the definition of the universal family we see that there is an isomorphism by [15,Lemma 3.6 (iii)].
We can now prove the main result of this section: Theorem 2.8.Assume (X, ι) is a K3 surface together with a fixed point free involution and let n ∈ N be an odd number.If a torsion free sheaf E on X is simple, then the associated tautological sheaf E [n] on X [n] descends to S [n] if and only if E descends to S.
Proof.First we note that if E is simple then E [n] is also simple.Indeed by [12,Corollary 4.2 (11)] there is an isomorphism Since E is simple the second summand must vanish, since otherwise E would have an endomorphism, which has image of rank one and thus is no homothety.
Proposition 2.1 shows By Lemma 2.7 we get (ι The theorem shows that given a simple ι-invariant torsion free sheaf E on X then there is F ∈ Coh(S) and G ∈ Coh(S [n] ) such that In fact, there is a close relationship between the sheaves F and G: as O Z is µ-invariant on X × X [n] , the structure sheaf O Z is naturally µ-linearizable on Z, hence so is O Z as a sheaf on X × X [n] .

For
Here For is the functor forgetting the linearizations.
That is if we start with a simple sheaf E on X, which descends to S i.e.E ∼ = π * F , then Remark 2.9.As O Z has two choices of a µ-linearization (differing by the non-trivial character), there are actually two choices of the descent (−) We end this section by giving a more explicit description of (−) [n] similar to (−) [n] .For this recall that by [13, 2.4] we have , where p X : Z → X and p X [n] : Z → X [n] are the projections.
The group G = Z/2Z acts freely on X via ι with quotient S, freely on X [n] via ι [n] with quotient S [n] and thus also freely on X × X [n] via ι × ι [n] .As the universal family Z ֒→ X × X [n] is G-invariant, we get a closed subvariety Z/G ֒→ (X × X [n] )/G.Furthermore the projections p X and p X Theorem 2.10.

Computation of certain extension spaces
In [11,Theorem 3.17] Krug gave explicit formulas for homological invariants of tautological objects in D b (X [n] ) in terms of those in D b (X), for example for E, F ∈ D b (X) there is an isomorphism of graded vector spaces: See also [12,Section 4] for a considerably simplified proof of this formula.
In this section we want to find homological invariants of sheaves of the form G [n] on S [n] in terms of the sheaf G on S. It is certainly possible to find a general formula similar to Krug's result, but to keep formulas and proofs short and readable and since it is enough for our purposes, we will restrict our attention to Hom-and Ext 1 -spaces as well as sheaves without higher cohomology.We will use the notations and results from [12].
We start by studying how Krug's result behaves with respect to the group actions by Z/2Z on X [n] via ι [n] and on X via ι.We will denote the various versions of the group G = Z/2Z in the following by their nontrivial element, that is by ι or ι [n] etc. Lemma 3.1.Assume (X, ι) is a K3 surface together with a fixed point free involution.For ι-equivariant coherent sheaves E, F ∈ Coh ι (X) there is an isomorphism of graded vector spaces: Proof.Note that on the right hand side of the formula we take invariants with respect to the actions induced by the linearizations of E, F and O X .On the left hand side we take invariants with respect to the induced linearizations on E [n] and F [n] .The existence of the induced linearizations follows from the right-hand side of diagram (2).By [12,Theorem 3.6] there is an isomorphism of functors ) is the Fourier -Mukai transform with kernel the structure sheaf of the isospectral Hilbert scheme I n X.Here the isosprectral Hilbert scheme is the reduced fiber product I n X := (X [n] × S n X X n ) red of the quotient map ν : X n → S n X to the symmetric power and the Hilbert -Chow morphism µ : X [n] → S n X.This Fourier -Mukai transform is an equivalence, see [12, Proposition 2.8] and satisfies ( 5) see for example [15,Section 5.6].Here ι ×n is the induced involution on X n .We have the following chain of isomorphisms: Here the first isomorphism is (4).The second isomorphism uses that Ψ is an equivalence and ( 5).The last isomorphism can be extracted from [12,Proposition 4.1].We look at the first summand, the second working similarly.First note that is simply by the pullback via ι on each factor in the box product.Since the action of Z/2Z via ι ×n and the S n action commute we finally see that: Theorem 3.2.Let (X, ι) be a K3 surface together with a fixed point free involution and let n ∈ N be an odd number.If G, H ∈ Coh(S) are such that π * G and π * H have no higher cohomology (here S = X/ι is the associated Enriques surface), then Proof.Define E := π * G and and . We therefore have an isomorphism .
By Lemma 3.1 the last space is isomorphic to We begin investigating the first summand.
and ι acts as +1 on the constants and as −1 on t.
We can now compute the invariants and find Looking at the in degree zero and one sees H).Next we study the second summand in (6): since E and F have no higher cohomology we have Ext ) which already lives in degree two.As we also have we see that the second summand in (6) can possibly have nontrivial components starting in degrees at least two.Especially for k ∈ {0, 1} we find

Stable sheaves on Enriques manifolds
In this section we want to study the slope stability of sheaves of the form . For this we first recall the following fact: let ϕ : Y → Z be an étale Galois cover with finite Galois group G then there is the following relationship between slopes with respect to h ∈ Amp(Z): Using this fact we can prove the following lemma: Lemma 4.1.Let E be a torsion free coherent sheaf on Y , slope stable with respect to ϕ * h for some h ∈ Amp(Z).If E descends to Z, that is E ∼ = ϕ * F , then F is slope stable with respect to h.
Proof.Let H ⊂ F be a subsheaf of F .Then ϕ * H is a subsheaf of ϕ * F ∼ = E. Since E is slope stable with respect to ϕ * h we have which by (7) implies µ h (H) < µ h (F ).Hence F is slope stable with respect to h.
For the rest of this section we let (X, ι) be a K3 surface together with a fixed point free involution ι.We denote the associated Enriques surface by S.
To prove the main theorem in this section we need the following isomorphism: Remark 4.2.The summand NS(X) n is constructed as follows: take d ∈ NS(X) and consider the element This element is S n -invariant and thus descends to the symmetric product S n X by [5, Lemma 6.1].More exactly, there is an element D n ∈ NS(S n X) such that ν * D n = D n for the quotient map ν : X n → S n X.Then we define d n := µ * D n , where µ : By [3, Section 3] the involution ι [n] acts on NS(X) n via: We are now ready to prove the main result of this section: Proof.By the results of Stapleton in [23] and in Section 1 we know that for a given slope stable torsion free sheaf E on X with E * * = O X , the associated tautological sheaf E [n] is slope stable on X [n] .By Theorem 2.8 the sheaf E [n] descends to S [n] if and only if E descends to S. In this case . Now by Theorem 4.1 the sheaf F [n] is slope stable with respect to some D ∈ Amp(S [n] ) if E [n] is slope stable with respect to H ∈ Amp(X [n] ) of the form H = ρ * D for some D ∈ Amp(S [n] ).
To see that we find such a D ∈ Amp(S [n] ), we note that the divisor H is described quite explicitly in [23,Proposition 4.8]: it is of the form for an arbitrary ample divisor A on X [n] and ǫ sufficiently small.We choose A of the form A = ρ * C for some C ∈ Amp(S [n] ).By (8) we also have In the rest of this section we want to study the moduli spaces containing the slope stable sheaves F on S and F [n] on S [n] .For this we let v ∈ H * alg (S, Z) be a Mukai vector on S, that is v = ch(F ) td(S) for some F ∈ Coh(S).Here where ξ S denotes the fundamental class of S.
We begin with the following result: Proof.The assumptions imply that F is simple and that Hom S (F, F ⊗ ω S ) = 0. Hence E := π * F is is simple due to the formula By [9, Lemma 3.2.3], the sheaf E is polystable with respect to h = π * d.Being simple and polystable, E is stable.Since E * * ∼ = O X the sheaf E [n] is slope stable with respect to some H ∈ Amp(X [n] ) and descends to S Assume from now on, that S is an unnodal Enriques surface, that is S contains no smooth rational curves (that is no (−2)-curves).Note that in the moduli space of Enriques surfaces, a very general element will be unnodal by [17,Corollary 5.7].
Denote the moduli space of slope semistable sheaves (with respect to d ∈ Amp(S)) with Mukai vector v on S by M S,d (v).Assume that v is primitive and chosen such that every slope semistable sheaf is slope stable and the rank of v is odd.Then for a generic choice Furthermore in this situation there is a decomposition . By [25, Theorem 4.6.(ii)]for a general choice of d ∈ Amp(S) the moduli space M S,d (v, L) is irreducible, that is a smooth projective variety.
We also assume that the Mukai vector is chosen such that for all [F ] ∈ M S,d (v, L) the sheaf F is locally free on S and does not have higher cohomology.Denote the Mukai vector of the associated sheaf F Proof.Since all sheaves classified by M S,d (v, L) are locally free on S, so are all the E = π * F on X. Proposition 1.3 shows that there is one H ∈ Amp(X [n] ) such that all E [n] are slope stable with respect to H since E ∼ = O X .But then by the construction of D ∈ Amp(S [n] ) with H = ρ * D in Theorem 4.3, it follows that there is one such desired D.
We have the following corollary: Proof.We use the explicit description (−)  Denote the Mukai vector of E = π * F on X by w, that is w = π * v.In the rest of this section we want to study the fixed loci of ι * in M X,h (w) and ι [n] * in M X [n] ,H (w [n] ).In our situation we have a well defined morphism As the morphism π * : M S,d (v) → Fix(ι * ) is an étale 2:1-morphism, the decomposition (9) shows that π * induces an isomorphism M S,d (v, L) ∼ = Fix(ι * ).As M S,d (v, L) is irreducible, so is Fix(ι * ).
Proof.The fixed locus Fix(ι * ) is smooth and projective since M X,h (w) is smooth and projective.Since it is also irreducible, it is a smooth projective variety.By Lemma 2.7 the morphism (−) [n] restricts to a morphism between the fixed loci.Since (−) [n] is injective on closed points, so is its restriction to Fix(ι * ).
To identify Fix(ι * ) as a smooth connected component it is therefore enough to prove dim T [E] Fix(ι * ) = dim T [E [n] ] Fix( ι [n] * ) But a general fact says that the tangent space of the fixed locus satisfies see for example [4,Proposition 3.2].As we have E ∼ = π * F for some sheaf F on S, this shows π * ρ * .

Acknowledgement
I thank Andreas Krug for answering my (many) questions regarding [12].I also thank the referee for a very detailed report which improved the paper greatly.

Remark 2 . 5 .
An Enriques manifold is of even dimension, say dim(Y ) = 2n.The fundamental group π 1 (Y ) is finite of order d with d | n + 1.This number d is called the index of Y .In addition Y is projective and the canonical bundle ω Y has finite order d and generates the torsion group of Pic(Y ), see [19, Section 2].We will work with the following class of Enriques manifolds, see [19, Proposition 4.1]: −)).Here we used the commutativity of diagram (3), the fact that G acts trivially on Z/G and S [n] hence by [12, Equation (5)] we have (−) G p S [n] * = p S [n] * (−) G and the G-equivariant projection formula.
h n which implies that we must have that h n = ρ * B for some divisor B on S [n] .Putting both facts together shows H = ρ * D for D = B + ǫC.It remains to see that D is ample.But since ρ is finite and surjective D is ample if and only if ρ * D = H is ample, see [7, Proposition I.4.4].

Theorem 4 . 4 .
Let F be a torsion free coherent sheaf with F ∼ = F ⊗ ω S .If F is slope stable with respect to d ∈ Amp(S), F * * ∼ = O S and F * * ∼ = ω S , then F [n] is a slope stable torsion free coherent sheaf on S [n] .

Remark 4 . 5 .
[n] via E [n] ∼ = ρ * F [n] .Now Theorem 4.3 implies that F [n]is slope stable with respect to some D ∈ Amp(S [n] ) satisfying ρ * D = H.Every torsion free coherent sheaf F of odd rank satisfies the condition F ∼ = F ⊗ ω S .
of d ∈ the moduli space M S,d (v) is smooth of dimension v 2 + 1 and M S,d (v) = ∅ if and only if v 2 −1, see [25, Proposition 4.2, Theorem 4.6 (i)].
[n] = p S [n] * (p * S (−)) given by Theorem 2.10.Since p X and p X [n] are flat we know by faithfully flat descent for π resp.ρ that the induced projections p S and p S [n] are flat.Similarly since p X [n] is a finite morphism so is p S [n] .Using these facts together with Theorem 4.4 and Proposition 4.6 shows that Krug's argument in the proof of [13, Proposition 2.1] also works in this case.Hence [F ] → [F [n] ] is a regular morphism.Similar to Theorem 1.5 it follows from Theorem 3.2 that (−) [n] is injective on closed points as Hom S [n] (F [n] , G [n] ) ∼ = Hom S (F, G).By Theorem 3.2 we also have dim(Ext 1 S [n] (F [n] , F [n] )) = dim(Ext 1 S (F, F )).Both facts together imply that (−) [n] identifies M S,d (v, L) with a smooth connected component of M S [n] ,D (v [n] ).

1
But the group Z/2Z acts on sheaves of the form pr * 1 E by definition of ι ×n as ι ×n * pr * 1 The natural Z/2-linearization of O X induces an Z/2-linearization on π * O X ∼ = O S ⊕ω S given by the generator of Z/2 acting by +1 on O S and by −1 on ω S , see for example [16, Remarks on p.72].Hence ι acts as +1 on H 0 (X, O X ) ∼ = H 0 (S, O S ) and by −1 on H 2 (X, O X ) ∼ = H 2 (S, ω S ).Furthermore, by the adjunction between π * and π * together with the projection formula, we get a splitting is torsion free and slope stable with respect to h = π * d for some d ∈ Amp(S).If E descends to S, that is E ∼ = π * F for some F ∈ Coh(S), then the induced torsion free sheaf F [n] is slope stable with respect to some ample divisor D on S [n] .