Stability and certain P n -functors

Let X be a K3 surface. We prove that Addington’s P n -functor between the derived categories of X and the Hilbert scheme of points X [ k ] maps stable vector bundles on X to stable vector bundles on X [ k ] , given some numerical conditions are satisﬁed.


Introduction
Moduli spaces of stable sheaves on K3 surfaces are examples of hyperkähler manifolds, also known as compact irreducible holomorphic symplectic manifolds.These arise in the classification of compact Kähler manifolds with trivial first Chern class and are therefore interesting objects to study.It is natural to wonder if moduli spaces of stable sheaves on hyperkähler manifolds of higher dimension also have interesting properties.
Unfortunately moduli spaces of stable sheaves on higher dimensional varieties behave badly in general.Furthermore there are not many explicit examples of stable sheaves on higher dimensional hyperkähler manifolds.There are the tautological bundles on Hilbert schemes of points on a K3 surface and on generalized Kummer varities, see [15], [16], [18] and [19].A second class is given by the "wrong-way" fibers of a universal family of stable vector bundles on a moduli space of stable vector bundles, see [13], [14] and [20].
In this article we want to give a new class of stable sheaves on Hilbert schemes of points.For this we use a result by Addington in [1], saying that the integral functor Φ : D b (X) → D b (X [k] ) with kernel the universal ideal sheaf I Z on X × X [k] is a P k−1 -functor.
Our main results can be summarized as follows: Theorem.Let X be a K3 surface with NS(X) = Zh and assume E is a µ h -stable locally free sheaf with Mukai vector v = (r, h, s) such that χ(E) v(E) 2  2 + (r + 1)k + 1.
Then the image of E under the integral functor Φ : D b (X) → D b (X [k] ) is a µ H -stable locally free sheaf on X [k] (for some ample class H ∈ NS(X [k] )).
Choosing v = (r, h, s) such that v 2 + 2 < 2r implies that all sheaves classified by the moduli space M X,h (v) are locally free.If furthermore M X,h (v) is a fine moduli space, then the integral functor Φ restricts to a morphism which identifies M X,h (v) with a smooth connected component of a certain moduli space M of stable sheaves on X [k] .
The main theorem is not void, that is there are Mukai vectors on certain K3 surfaces that satisfy all conditions stated in the theorem.We will give two examples in 3.5.
This result is in the same vein as Yoshioka's results in [22].There he starts with an arbitrary K3 surface together with an isotropic Mukai vector v such that M X,h (v) is also a fine moduli space with universal family E. It is well known that Y = M X,h (v) is again a K3 surface.He then proves that the Fourier-Mukai transform Φ E : D b (X) → D b (Y ), an equivalence in this case, preserves stability (even S-equivalence classes), given that some numerical conditions are satisfied.
All objects in this text are defined over the field of complex numbers C.
Acknowledgement.I thank Ziyu Zhang for many useful discussions.I am also grateful to the anonymous referees who helped to improve the presentation of the manuscript greatly.

Background on P n -functors
We start by recalling some basic facts about the P n -functors we are interested in.For the general definition of P n -functors we refer to Addington's paper, see [1,Section 4].For similar results see also [12,Theorem 1.1] and [3].Definition 1.1.Let X be a K3 surface.Define the integral functor Φ by Thus Φ has the universal ideal sheaf I Z on X ×X [k] as kernel.Here p : X ×X [k] → X [k] and q : X × X [k] → X are the projections.We note that Φ is a P k−1 -functor with associated autoequivalence Remark 1.2.The fact that the integral functor Φ is a P k−1 -functor with associated autoequivalence H = [−2] has the following helpful consequence: for any two elements E, F ∈ D b (X) there is an isomorphism of graded vector spaces ) are isomorphisms for i = 0 and 1.This especially applies to the case that E and F are in fact sheaves (interpreted as complexes concentrated in degree zero in D b (X)).

Preservation of slope-stability for vector bundles
Throughout this article we assume that X is a K3 surface with NS(X) = Zh, where h is a primitive ample class.Lemma 2.1.Let E be a µ h -stable vector bundle on X with Mukai vector v = (r, h, s) such that Using [21,Lemma 2.1] shows that G is also µ h -stable with Mukai vector The Euler characteristic of a pair (F 1 , F 2 ) of coherent sheaves can be expressed as We can now compute Using the inequality (1) shows that we have Corollary 2.2.Let E be a locally free µ h -stable sheaf with Mukai vector v = (r, h, s) satisfying the inequality (1), then We are now ready to study the image of a µ h -stable locally free sheaf with Mukai vector v = (r, h, s) under the P k−1 -functor Φ.A priori this object is just a complex in D b (X [k] ) but in our situation we have: Lemma 2.3.Let E be a locally free µ h -stable sheaf with Mukai vector v = (r, h, s) satisfying the inequality (1), then Φ(E) is a locally free sheaf of rank r + s − rk on X [k] .
Proof.Using cohomology and base change results, the vanishing in Lemma 2.1 implies Consequently Φ(E) = p * (q * E ⊗ I Z ) is a sheaf.Furthermore the map , is indeed a locally free sheaf on X [k] .As the fiber at a point [Z] is just H 0 (X, E ⊗ I Z ) the rank follows from Corollary 2.2.
Next we want to study the slope-stability of Φ(E).For this we recall that in our situation we have NS(X [k] ) = Zh k ⊕ Zδ, where h k is the divisor on X [k] induced by the divisor h on X and 2δ is the exceptional divisor of the Hilbert-Chow morphism X [k] → X (k) .For any coherent sheaf F on X we denote the associated coherent tautological sheaf by Proof.By Lemma 2.3 we have R 1 p * (q * E ⊗ I Z ) = 0, thus there is an exact sequence: and the sheaf p * (q * E ⊗ O Z ) is by definition the tautological bundle E [k] .The exact sequence (2) can be rewritten as Using [19, Lemma 1.5] we get We also recall the notations introduced by Stapleton in [18, Section 1].The ample divisor h on X induces the ample divisor , where q i is the i-th projection from X k , as well as a semi-ample divisor h k on X [k] .We denote by X k • , S k X • and X [k] • the loci of the relevant spaces parametrizing distinct points.The natural map σ is an étale cover and j : • , and define (F ) Proof.Note that (−) • and σ * • (−) are exact, and j * (−) is left exact.Applying these functors to (3) we obtain an exact sequence of S k -invariant reflexive sheaves on X k : X k ϕ where ϕ may not be surjective.Certainly we have and also by [18, Lemma 1.1] The above sequence can be written as More accurately, ϕ is the evaluation map on X k • : for any k-tuple (x 1 , . . ., x k ) ∈ X k of closed points with x i = x j , the morphism of fibers can be written as Since for a non-trivial section s ∈ H 0 (E), one can always choose a k-tuple of distinct points (x 1 , . . .x k ) ∈ X k with (s(x 1 ), . . ., s(x k )) = (0, . . ., 0), we see that the induced map is injective.It follows by exact sequence ( 5) that (Φ(E)) X k has no global sections.
Proposition 2.6.Let E be a locally free µ h -stable sheaf with Mukai vector v = (r, h, s) satisfying the inequality (1), then the locally free sheaf Φ(E) defined in Lemma 2.3 is slope stable with respect to h k .
Proof.We follow the proof of [18,Theorem 1.4] and start with the exact sequence and note that ϕ is surjective on X k • , hence its cokernel coker(ϕ) is supported on the big diagonal of X k , which is of codimension 2. We get Now assume G is a reflexive subsheaf of Φ(E).Then (G) X k is an S k -invariant reflexive subsheaf of (Φ(E)) X k .By [18, Lemma 1.2] we have It is therefore enough to prove that (Φ(E)) X k has no S k -invariant destabilizing subsheaf (with respect to h X k ).Assume F is an S k -invariant subsheaf, then we find: If a = 0, we pick a (not necessarily S k -invariant) non-zero stable subsheaf F ′ ⊆ F that has maximal slope with respect to h X k (for example one could take a stable factor in the first Harder-Narasimhan factor of F ). Without loss of generality, we may assume F and F ′ are both reflexive.Since F ′ is also a subsheaf of H 0 (E) ⊗ O X k , there is a projection from H 0 (E) ⊗ O X k to a certain direct summand of it, such that the composition of the embedding and projection and O X k is also stable with respect to h X k , the map F ′ → O X k must be injective, and its cokernel is supported in codimension at least 2. Since both sheaves are reflexive, we must have F ′ = O X k .As a result F , and consequently (Φ(E)) X k , have non-trivial global sections, a contradiction to Lemma 2.5.
If a 1, F would be a subsheaf of the trivial bundle H 0 (E) ⊗ O X k of positive slope which is not possible since a trivial bundle is semistable of slope zero.
Theorem 2.7.Let E be a locally free µ h -stable sheaf with Mukai vector v = (r, h, s) satisfying the inequality (1), then Φ(E) is a locally free µ H -stable sheaf for some ample class H ∈ NS(X [k] ) near h k .
Proof.Proposition 2.6 and [5, Theorem 2.3.1]guarantee that the assumptions in [18,Proposition 4.8] are satisfied for Φ(E), hence Φ(E) is slope stable with respect to some ample class H near h k by [18,Proposition 4.8].
Remark 2.8.a) Here we understand slope stability with respect to the non-ample divisor h k as slope stability with respect to a movable curve class.This stability was studied in detail by Greb, Kebekus and Peternell in [6, Section 2.2].They show that all elementary properties satisfied by sheaves that are stable with respect to an ample polarization also hold for this notion of stability.b) We restrict to locally free sheaves in this paper for two reasons: the first is to pass freely between H * (E ⊗ I Z ) and Ext 2− * X (I Z , E * ) ∨ in Lemma 2.1.The second reason is that we use Stapleton's description of (E [k] ) X k , see [18,Lemma 1.1].This result uses the fact that (E [k] ) X k and E ⊞k are reflexive (since they are locally free).But this fails if we start with a torsion free but not locally free sheaf E.

A morphism of moduli spaces
In the last section we saw that given a locally free µ h -stable sheaf E with Mukai vector v satisfying the inequality (1), then there is an ample class H ∈ NS(X [k] ) such that Φ(E) is a locally free µ H -stable sheaf on X [k] .
In this section we want to see that in certain cases we get a morphism for some moduli space M of stable sheaves on X [k] .Let v = (r, h, s) be a Mukai vector satisfying the following conditions: (1) v satisfies the inequality (1), (2) all sheaves classified by M X,h (v) are locally free and (3) the moduli space M X,h (v) is fine.Let v = (r, h, s) be a Mukai vector that satisfies the conditions (1) and (2).Then for every [E] ∈ M X,h (v) we know that there is an ample class H such that Φ(E) is µ H -stable.One may ask how H depends on [E].This question is answered in the following theorem.
Theorem 3.1.Let v be a Mukai vector that satisfies conditions (1) and (2), then there is an ample class H ∈ NS(X [k] ) such that Φ(E) is µ H -stable for all [E] ∈ M X,h (v) simultaneously.
Proof.The proof is same as for [14,Theorem 2.8].We just have to replace the sheaf E x by Φ(E) and the surface X by the moduli space M X,h (v).Then we note that the value of The finiteness of the set can be seen as follows: using Corollary 2.2 and exact sequence (3) we see that every such that µ β (F ) c}.But S ′ is finite due to [6, Theorem 2.29], so S is also finite.The rest of the proof works unaltered.
From now on we fix a Mukai vector v that satisfies conditions (1) − (3) and an ample class H ∈ NS(X [k] ) that satisfies Theorem 3.1.We denote by M X [k] ,H (Φ C (v)) the moduli space of µ H -stable sheaves on X [k] with Mukai vector Φ C (v).Here is the induced cohomological Fourier-Mukai transform, see [8, 5.28, 5.29].
In the following we want to give an explicit construction of the morphism We start by constructing a classifying family for the Φ(E).

Lemma 3.2.
There is a family F of sheaves on X v) the restriction to the fiber over [E] is given by Proof.As M X,h (v) is fine, there is a universal family U on X × M X,h (v), flat over M X,h (v).Lemma 2.3 shows that the complex Φ(U The family F induces a classifying morphism The morphism f has the following property: Proof.Using the family F as the kernel of an integral functor, we get an induced morphism ) which is the Kodaira-Spencer map of the family F at the point [E] ∈ M X,h (v) by [4,Proposition 4.4].Using the canonical identifications This implies that the classifying map f : M X,h (v) −→ M X [k] ,H (Φ C (v)) is étale and surjective onto a smooth connected component.But using again Remark 1.3 shows that f has to be of degree one since for [E] = [F ] ∈ M X,h (v) we have Hom X [k] (Φ(E), Φ(F )) ∼ = Hom X (E, F ) = 0.So f must be an isomorphism to a smooth connected component.
Remark 3.4.This argument uses essentially the same arguments as the proof in [10, Section 1] where a component in a moduli space of stables bundles on a moduli space of bundles on a curve is constructed.2 + (r + 1)k + 1.A further computation shows v 2 + 2 = 4 < 6 = 2r so that all sheaves in M X,h (v) are locally free by [20,Lemma 4.4.2].As gcd(3, 8) = 1 we see that M X,h (v) is a fine moduli space by [9,Corollary 4.6.7].In this case we have dim(M X,h (v)) = 4 which gives a 4-dimensional component in M X [2] ,H (Φ C (v)).b) Let X be a K3 surface with NS(X) = Zh such that h 2 = 186.Then a similar computation shows that for k = 3 then the Mukai vector v = (5, h, 18) satisfies the conditions (1) − (3).In this example dim(M X,h (v)) = 8 which gives an 8-dimensional component in M X [3] ,H (Φ C (v)).
v).The existence of the family F now follows from [4, Proposition 4.2].