The wall-crossing behavior for Bridgeland’s stability conditions on the derived category of coherent sheaves on K3 or abelian surface is studied. We introduce two types of walls. One is called the wall for categories, where the t-structure encoded by stability condition is changed. The other is the wall for stabilities, where stable objects with prescribed Mukai vector may get destabilized. Some fundamental properties of walls and chambers are studied, including the behavior under Fourier–Mukai transforms. A wall-crossing formula of the counting of stable objects will also be derived. As an application, we will explain previous results on the birational maps induced by Fourier–Mukai transforms on abelian surfaces. These transformations turns out to coincide with crossing walls of certain property.
Funding statement: The second author is supported by JSPS Fellowships for Young Scientists (No. 21-2241, 24-4759) and the Grant-in-aid for Scientific Research (No. 25800014), JSPS. The third author is supported by the Grant-in-aid for Scientific Research (No. 22340010, No. 24224001, No. 26287007), JSPS.
A Appendix: Positive characteristic case
A.1 Stability condition and base change
Here we discuss the behaviour of our stability condition under a field extension . A similar claim holds for any projective surface. Let us denote by the base change of X, and set standard morphisms as
For given on X, we denote by the pull-backs on . The stability function is defined in the same way as :
The function is also defined in the same way:
Assume that the extension is finite.
For an object we have
For objects we have
Note that for an object we have
Here on the right hand side means the intersection product on the Chow ring . To explain , let us note that the degree-zero part of is a rank-1 free abelian module and the class of the fundamental cycle is its generator. Thus the degree-zero part of is in . We denote the coefficient by .
Then we have
Here at the third equality we used the projection formula. Thus we have (1). Then (2) follows from the definition of . ∎
Recall the notation for a category and a field extension given in Definition 1.2.4. Then we see that the stability condition on is well-defined.
The following statements hold.
The derived pull-back induces an exact functor
The derived push-forward induces an exact functor
What should be shown is that the image (resp. ) is indeed in (resp. in ). Case (1) is clear, since the twisted semi-stability is preserved under pull-back (which is a consequence of the uniqueness of Harder–Narasimhan filtration).
For case (2), let E be a -twisted semi-stable object of . It is enough to prove that is also β-twisted semi-stable.
We may assume that L is a normal extension of . Indeed, for a normal extension of containing L, , where is the projection.
Let F be a subobject of . Then is a subobject of . By Lemma A.1.3 below, is a -twisted semi-stable object with
Therefore is β-twisted semi-stable. ∎
The object is a successive extension of the object with .
We note that . Since L is a normal extension of , is a successive extension of R-modules , where are maximal ideals of R and . Since as -isomorphisms, is a successive extension of (). ∎
Assume that the extension is finite. If E is a -semi-stable object, then is -semi-stable,
Assume that is not -semi-stable. (Here we are implicitly using Lemma A.1.2 (1).) Take a destabilizing subobject F of , so that we have
Then by Lemma A.1.1 (2) we have .
On the other hand, since with , it follows that the object is -semi-stable. Then since and it is a subobject of by Lemma A.1.2 (2), we have . Therefore by contradiction the claim holds. ∎
This claim implies that our stability condition is a member of in [35, Definition 3.1].
Let , the algebraic closure of . If E is a -semi-stable object, then is -semi-stable.
Assume that is not -semi-stable. Let F be the destabilizing subobject of . Since F is a class of a complex consisting of coherent sheaves, we may assume that F is defined on a field such that and . Then by Proposition A.1.4, is semi-stable with respect to , where and are pull-backs of via . But it contradicts the choice of F. ∎
A.2 Non-emptiness of the moduli spaces
Over a field of characteristic 0, the moduli spaces of stable objects are non-empty if the expected dimension is non-negative. We will explain that a similar claim also holds over a field of positive characteristic under a technical condition. Let be a field of characteristic and let X be an abelian or a K3 surface defined over . Let α be a 2-cocycle of the étale sheaf . Let be the Grothendieck group of α-twisted sheaves on X, where α is a representative of . The main result of this subsection is the following proposition.
Let be a locally free α-twisted sheaf such that
where is the class of in and is the subgroup generated by torsion sheaves. Assume that . Let v be the Mukai vector of an α-twisted sheaf E. If v is primitive and , then for a general . Moreover, is deformation equivalent to , .
The rest of this subsection is devoted to the proof of this proposition. We may assume that is algebraically closed. We first treat the case where α is trivial. By [30, Corollary 3.2] and [11, Corollary 1.8], for a polarized abelian or a K3 surface , we have a lifting to characteristic 0.
Assume that is a family of abelian surfaces with polarizations. By [29, Section 6.2], a polarization is a T-homomorphism such that for any geometric point , for some ample line bundle L on . By [29, Proposition 6.10], we have a relatively ample line bundle which induces the polarization . We consider
where is the connected component containing H on X. Since π is a projective morphism, is a closed subscheme of T. For the category of Artinian rings, polarizations come from ample line bundles ([33, Lemma 2.3.2]). Thus [30, Corollary 3.2] implies that the infinitesimal lifting of H is unobstructed. Then π is dominant. Then for a suitable finite covering of T, we have a family of line bundles which is an extension of H.
We first prove the simplest case in order to explain our strategy. Let
be a primitive Mukai vector with and consider the untwisted case. We assume that H is general with respect to v. In this case, the twisted semi-stability is independent of G. Hence we can set . For the proof of , we may assume that is algebraically closed. Replacing v by , , we may assume that ξ is ample. Since is sufficiently close to , we may assume that . Thus we may assume that . In this case, we have a relative moduli scheme which is smooth and projective. Therefore the claim holds. In order to cover general cases, we prepare the following lemmas.
Assume that . Let be a discrete valuation ring with and a flat family of polarized surfaces over . There is an extension of and a flat family of projective bundle such that the Brauer class of is .
We set . We take a smooth curve , . Let be a torsion free α-twisted sheaf on X such that and . Since , is decomposed as
Hence we have a decomposition
where is the trace free part of . There is a torsion free α-twisted sheaf such that fits an exact sequence
and . Let be the deformation space of a fixed determinant . Hence is smooth and we have a surjection
Thus is submersive. Since deforms to a μ-stable vector bundle, we find that deforms to a torsion free α-twisted sheaf such that is μ-stable. Then is a μ-stable α-twisted sheaf. Replacing by , we assume that is μ-stable with respect to and . Then there is a discrete valuation ring dominating such that is lifted to a projective bundle . ∎
For the morphism , is a line bundle on . Since , we have a non-trivial extension
We take an étale trivialization of :
where . Then the relative tautological line bundle on forms a twisted line bundle on . gives a twisted sheaf on with . Assume that with satisfies . Then we have a relative moduli scheme
which is smooth and projective. Moreover, if there is also a family of divisors on such that is nef and big and is a local projective generator of a category of perverse coherent sheaves associated to the contraction , then we have a relative moduli space
which is also smooth and projective.
In order to reduce to the case where G is general and , we will use a Fourier–Mukai transform. For this purpose, we prepare a lemma for the non-emptiness of the moduli spaces.
Let u be a primitive and isotropic Mukai vector such that and assume that there is a local projective generator of with . Then there is a u-twisted stable object E with .
We use a deformation argument as in [50, Section 2.5]. Let be a discrete valuation ring with and . Let be a family of polarized surfaces and a family of divisors such that . We apply Lemma A.2.3 to on X with . Replacing T by a finite extension , we have a relative moduli space
which is smooth and projective. Hence by replacing be a finite cover, we have a u-twisted semi-stable objects with Mukai vectors u over the generic point η of . By the properness of f, can be extended to a family of u-twisted semi-stable objects with Mukai vectors u over . Therefore there is a u-twisted semi-stable object E with . By the proof of [50, Corollary 2.5.5] (i.e., by using [32, Proposition 2.11, Proposition 2.17]), we deduce the claim. ∎
Let G be a local projective generator of in Proposition A.2.1. We have a local projective generator such that in ( denotes the class of G in ) and both G and belong to the same chamber for v, where N is sufficiently large ([50, Corollary 2.4.4]). We have that is isotropic. Hence there is a primitive and isotropic Mukai vector u such that
For an isotropic Mukai vector u, assume that there is a local projective generator of with . Then is isomorphic to the moduli space of stable (twisted) sheaves on an abelian surface or a K3 surface , where the polarization is a general one.
Let be a local projective generator of such that
is sufficiently small and general in . Applying Lemma A.2.4 to ,
is a smooth projective surface. Let be a universal family on the product as an object of the category , where is a 2-cocycle of representing a Brauer class . By [52, Theorem 2.7.1], there is a category of perverse coherent sheaves with a local projective generator of and an ample divisor on such that is -twisted stable object of and by the correspondence , where and is sufficiently small.
For the Fourier–Mukai transform we will apply [52, Proposition 2.8.2] under the correspondence of notations:
where . Then induces an isomorphism
where , is a general ample divisor on which is sufficiently close to and consists of -twisted stable sheaves. ∎
Proof of Proposition A.2.1.
where is the α-twisted sheaf in the proof of Proposition A.2.3 and is sufficiently close to . Replacing H by ξ, we may assume that . Then we have a relative moduli scheme , which is smooth and projective. Therefore the claim holds. ∎
A.3 Moduli of perverse coherent sheaves
In [50, Proposition 1.4.3], one of the authors constructed the moduli scheme of semi-stable perverse coherent sheaves under the characteristic 0 assumption. The construction is reduced to the construction of semi-stable -modules, where is a sheaf of -algebras on a projective scheme Y via Morita equivalence. Then by Simpson’s result [34, Theorem 4.7] on the moduli of semi-stable -modules, the moduli scheme exists. In this subsection, we will remark that Simpson’s construction also works for any characteristic case by Langer’s results .
Let S be a scheme of finite type over a universally Japanese ring. Let be a flat family of polarized schemes over S. Let be a sheaf of -algebras such that is a coherent -module, which is flat over S. For an -module E on , we write the Hilbert polynomial of E as
If is a polynomial of degree d, then E is of dimension d. By using the Hilbert polynomial of E, we have the notion of semi-stability and also the μ-semi-stability.
We take a surjective morphism
Let E be a μ-semi-stable -module of dimension d. Then
for any subsheaf F of E.
We may assume that F is μ-semi-stable. For the multiplication morphism
is an -submodule of E. Hence
By our assumption, we have a surjective morphism . Hence
Therefore the claim holds. ∎
By this lemma, the set of μ-semi-stable -modules E on () with the Hilbert polynomial P is bounded by Langer’s boundedness theorem. Hence we can parametrize semi-stable -modules by the Quot scheme
whose points correspond to quotient -modules with , where . For a purely d-dimensional -module E on ,
Combining this with Langer’s important result [21, Corollary 3.4], we have the following:
For any purely d-dimensional -module E on , ,
where c depends only on , , d and .
Let be a discrete valuation ring R with maximal ideal . Let K be the fractional field and k the residue field. Let be an R-flat family of -modules such that is pure.
There is an R-flat family of coherent -modules and a homomorphism such that is pure, is an isomorphism and is an isomorphism at generic points of .
By using [34, Lemma 1.17], we first construct as a usual family of sheaves. Then by the very construction of it, becomes an -module. ∎
Let be the open subscheme of consisting of semi-stable -modules. By using Lemma A.3.2 and Lemma A.3.3, we can construct the moduli of semi-stable -modules as a GIT-quotient of by the action of , where n is sufficiently large. Thus we have a good quotient and is the coarse moduli space of semi-stable -modules.
We have a coarse moduli scheme of semi-stable -modules
which is projective over S.
Let be the open subscheme of stable -modules. By this construction,
is a principal -bundle; is the open subscheme parametrizing stable -modules.
Let X and Y be flat families of projective varieties over a scheme S of finite type over a universally Japanese ring and assume that is smooth. Let be a family of S-morphisms and G a locally free sheaf on X such that is a local projective generator of a family of abelian categories as in [50, Section 1.3]. Since [50, Corollary 1.3.10] holds for any base S, we have the following.
The following statements hold.
We have a relative moduli stack of -twisted semi-stable objects with the Hilbert polynomial P as a quotient stack , where is an open subscheme of parametrizing -twisted semi-stable objects.
We have a relative moduli scheme
of -twisted semi-stable objects with the Hilbert polynomial P as a GIT-quotient:
Let be the universal quotient on .
Assume that there is a bounded family of locally free (twisted) sheaves on X such that
Then there is a universal family on .
Then is a -equivariant -module. Hence
is -equivariant. Then
is a bounded complex of -equivariant locally free sheaves on , where
is the projection. Then is a -linearized object on . Let be the open subscheme parametrizing stable objects. Since is a principal bundle, it follows that
is the pull-back of a family of perverse coherent sheaves on , which gives a universal family. ∎
Assume that is a polarized K3 or an abelian surface over a scheme S. For , if , , then there is a universal family on .
We would like to thank referees very much for their valuable comments and suggestions on improvements of our paper.
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