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The wall-crossing behavior for Bridgeland’s stability conditions on abelian and K3 surfaces

  • Hiroki Minamide EMAIL logo , Shintarou Yanagida and Kōta Yoshioka

Abstract

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 σ(β,ω)=(𝔄(β,ω),Z(β,ω)) under a field extension L/𝔨. A similar claim holds for any projective surface. Let us denote by XL the base change of X, and set standard morphisms as

For given β,H,ω on X, we denote by β,H,ω the pull-backs on XL. The stability function Z(β,ω):𝐃(XL) is defined in the same way as Z(β,ω):

Z(β,ω)(EL):=eβ+-1ω,v(EL),EL𝐃(X).

The function Σ(β,ω)(EL,EL) is also defined in the same way:

Σ(β,ω)(EL,EL):=det(ReZ(β,ω)(EL)ReZ(β,ω)(EL)ImZ(β,ω)(EL)ImZ(β,ω)(EL)),EL,EL𝐃(XL).

Lemma A.1.1.

Assume that the extension L/k is finite.

  1. For an object EL𝐃(XL) we have

    [L:𝔨]Z(β,ω)(EL)=Z(β,ω)(p*EL).
  2. For objects EL,EL𝐃(XL) we have

    Σ(β,ω)(EL,EL)0Σ(β,ω)(p*EL,p*EL)0.

Proof.

Note that for an object EL𝐃(XL) we have

Z(β,ω)(EL)=[π*(v(EL)eβ+-1ω)]0.

Here on the right hand side means the intersection product on the Chow ring A*(XL). To explain []0, let us note that the degree-zero part A0(XL) of A*(XL) is a rank-1 free abelian module and the class [XL] of the fundamental cycle is its generator. Thus the degree-zero part of αA*(XL) is in [XL]. We denote the coefficient by [α]0.

Then we have

[L:𝔨]Z(β,ω)(EL)=p*[π*(v(EL)eβ+-1ω)]0=[π*p*(v(EL)eβ+-1ω)]0=[π*(v(p*EL)eβ+-1ω)]0=Z(β,ω)(p*EL).

Here at the third equality we used the projection formula. Thus we have (1). Then (2) follows from the definition of Σ(β,ω). ∎

Recall the notation 𝔨L for a category and a field extension L/𝔨 given in Definition 1.2.4. Then we see that the stability condition σ(β,ω),L:=(𝔄(β,ω)𝔨L,Z(β,ω)) on 𝐃(XL) is well-defined.

Lemma A.1.2.

The following statements hold.

  1. The derived pull-back p*:𝐃(X)𝐃(XL) induces an exact functor

    p*:𝔄(β,ω)𝔄(β,ω)𝔨L.
  2. The derived push-forward p*:𝐃(XL)𝐃(X) induces an exact functor

    p*:𝔄(β,ω)𝔨L𝔄(β,ω).

Proof.

What should be shown is that the image p*(𝔄(β,ω)) (resp. p*(𝔄(β,ω)𝔨L)) is indeed in 𝔄(β,ω)𝔨L (resp. in 𝔄(β,ω)). 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 (β)-twisted semi-stable object of L. It is enough to prove that p*(E) is also β-twisted semi-stable.

We may assume that L is a normal extension of 𝔨. Indeed, for a normal extension L of 𝔨 containing L, q*(q*(E))=E[L:L], where q:XLXL is the projection.

Let F be a subobject of p*(E). Then p*(F) is a subobject of p*p*(E). By Lemma A.1.3 below, p*p*(E) is a β-twisted semi-stable object with

χ(p*p*(E)(-β+nH))=χ(p*(E)(-β+nH)).

Hence

χ(F(-β+nH))rkF=χ(p*(F)(-β+nH))rkFχ(p*(E)(-β+nH))rkp*(E).

Therefore p*(E) is β-twisted semi-stable. ∎

Lemma A.1.3.

The object p*p*(E) is a successive extension of the object f*(E) with fAut(L/k).

Proof.

We note that p*p*(E)=EL(L𝔨L). Since L is a normal extension of 𝔨, R:=L𝔨L is a successive extension of R-modules R/𝔪i, where 𝔪i are maximal ideals of R and R/𝔪iL. Since EL(R/𝔪i)E as 𝔨-isomorphisms, p*p*(E) is a successive extension of f*(E) (fAut(L/𝔨)). ∎

Proposition A.1.4.

Assume that the extension L/k is finite. If E is a σ(β,ω)-semi-stable object, then p*E is σ(β,ω),L-semi-stable,

Proof.

Assume that p*(E) is not σ(β,ω),L-semi-stable. (Here we are implicitly using Lemma A.1.2 (1).) Take a destabilizing subobject F of p*(E), so that we have

Σ(β,ω)(F,p*(E))<0.

Then by Lemma A.1.1 (2) we have Σ(β,ω)(p*F,p*p*E)<0.

On the other hand, since p*p*EEd with d:=[L:𝔨], it follows that the object p*p*E is σ(β,ω)-semi-stable. Then since p*F𝔄(β,ω) and it is a subobject of p*p*E by Lemma A.1.2 (2), we have Σ(β,ω)(p*F,p*p*E)0. Therefore by contradiction the claim holds. ∎

Remark A.1.5.

This claim implies that our stability condition σ(β,ω),L is a member of Stab(XL)p in [35, Definition 3.1].

Corollary A.1.6.

Let L=k¯, the algebraic closure of k. If E is a σ(β,ω)-semi-stable object, then p*E is σ(β,ω),L-semi-stable.

Proof.

Assume that p*E is not σ(β,ω),L-semi-stable. Let F be the destabilizing subobject of p*E. Since F is a class of a complex consisting of coherent sheaves, we may assume that F is defined on a field L such that 𝔨LL=𝔨¯ and [L:𝔨]<. Then by Proposition A.1.4, pL/𝔨*E is semi-stable with respect to σ(β′′,ω′′),L, where pL/𝔨:XLX and β′′,ω′′ are pull-backs of β,ω via 𝔨L. 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 𝔨 be a field of characteristic p>0 and let X be an abelian or a K3 surface defined over 𝔨. Let α be a 2-cocycle of the étale sheaf 𝒪X×. Let Kα(X) be the Grothendieck group of α-twisted sheaves on X, where α is a representative of [α]Hét2(X,𝒪X×). The main result of this subsection is the following proposition.

Proposition A.2.1.

Let G0 be a locally free α-twisted sheaf such that

Kα(X)=G¯0+Kα(X)1,

where G¯0 is the class of G0 in Kα(X) and Kα(X)1 is the subgroup generated by torsion sheaves. Assume that (p,rkG0)=1. Let v be the Mukai vector of an α-twisted sheaf E. If v is primitive and v2-2ε, then MHG(v) for a general (G,H). Moreover, MHG(v) is deformation equivalent to MH(1,0,-), :=v2/2.

The rest of this subsection is devoted to the proof of this proposition. We may assume that 𝔨 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 (X,H), we have a lifting (𝒳,) to characteristic 0.

Remark A.2.2.

Assume that 𝒳T is a family of abelian surfaces with polarizations. By [29, Section 6.2], a polarization is a T-homomorphism λ:𝒳𝒳^ such that for any geometric point tT, λt=ϕL for some ample line bundle L on 𝒳t. By [29, Proposition 6.10], we have a relatively ample line bundle which induces the polarization 2λ. We consider

π:Pic𝒳/TξT,

where Pic𝒳/Tξ is the connected component containing H on X. Since π is a projective morphism, imπ 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 TT 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

v=(r,ξ,a)NS(X)

be a primitive Mukai vector with v>0 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 G=𝒪X. For the proof of MH(v), we may assume that 𝔨 is algebraically closed. Replacing v by venH, n0, we may assume that ξ is ample. Since H is sufficiently close to ξ, we may assume that Hξ. Thus we may assume that v=(r,dH,a). In this case, we have a relative moduli scheme M(𝒳,)/S(v)S which is smooth and projective. Therefore the claim holds. In order to cover general cases, we prepare the following lemmas.

Lemma A.2.3.

Assume that (p,rkG0)=1. Let (R,tR) be a discrete valuation ring with R/tR=k and (X,L) a flat family of polarized surfaces over T=Spec(R). There is an extension (R,t) of (R,t) and a flat family of projective bundle PXR such that the Brauer class of PRR/tR is [α]Hét2(X,OX×).

Proof.

We set 0:=𝔨. We take a smooth curve D|n0|, n0. Let G1 be a torsion free α-twisted sheaf on X such that rkG1>1 and rkG1rkG0modp. Since (p,rkG1)=1, G1G1𝐃(X) is decomposed as

G1G1𝒪X(G1G1)/𝒪X.

Hence we have a decomposition

Exti(G1,G1(KX+D))Hi(𝒪X(KX+D))Exti(G1,G1(KX+D))0,

where Exti(G1,G1(KX+D))0 is the trace free part of Exti(G1,G1(KX+D)). There is a torsion free α-twisted sheaf such that G2 fits an exact sequence

0G2G1i𝔨xi0

and Hom(G2,G2(KX+D))0=0. Let Def0(E) be the deformation space of a fixed determinant detG1=detG2. Hence Def0(G2) is smooth and we have a surjection

Ext𝒪X1(G2,G2)0Ext𝒪D1(G2|D,G2|D)0.

Thus Def0(G2)Def0(G2|D) is submersive. Since G2|D deforms to a μ-stable vector bundle, we find that G2 deforms to a torsion free α-twisted sheaf G3 such that G3|D is μ-stable. Then G3 is a μ-stable α-twisted sheaf. Replacing G1 by G3, we assume that G1 is μ-stable with respect to 0 and H2((G1G1)/𝒪X)=0. Then there is a discrete valuation ring (R,t) dominating (R,t) such that (G1)X is lifted to a projective bundle P𝒳R. ∎

For the morphism g:P𝒳RSpec(R), L:=Extg1(TP/𝒳,𝒪𝒳) is a line bundle on Spec(R). Since Homg(TP/𝒳,𝒪𝒳)=0, we have a non-trivial extension

0g*(L)𝒫TP/𝒳0.

We take an étale trivialization U𝒳 of P𝒳:

P×𝒳Um×U,

where m=rkG1-1. Then the relative tautological line bundle on ×U forms a twisted line bundle 𝒪P(1) on 𝒳. 𝒫𝒪P(-1) gives a twisted sheaf 𝒢 on 𝒳 with 𝒢RR/tRG1. Assume that EKα(X) with v(E)=v satisfies v(EG1)0ϱX. Then we have a relative moduli scheme

M¯(v)(𝒳,)𝒢Spec(R),

which is smooth and projective. Moreover, if there is also a family of divisors on 𝒳 such that H:=𝔨 is nef and big and G1 is a local projective generator of a category of perverse coherent sheaves associated to the contraction |mH|:XY, then we have a relative moduli space

M¯(v)(𝒳,)𝒢Spec(R),

which is also smooth and projective.

In order to reduce to the case where G is general and =Cohα(X), we will use a Fourier–Mukai transform. For this purpose, we prepare a lemma for the non-emptiness of the moduli spaces.

Lemma A.2.4.

Let u be a primitive and isotropic Mukai vector such that (p,rku)=1 and assume that there is a local projective generator Gu of C with v(Gu)Z>0u. Then there is a u-twisted stable object E with v(E)=u.

Proof.

We use a deformation argument as in [50, Section 2.5]. Let (R,tR) be a discrete valuation ring with R/tR=𝔨 and char(R)=0. Let (𝒳,,)Spec(R) be a family of polarized surfaces (𝒳,) and a family of divisors such that (𝒳𝔨,𝔨)=(X,H). We apply Lemma A.2.3 to G1 on X with v(G1)=umodϱX. Replacing T by a finite extension TT, we have a relative moduli space

f:M¯(u)(𝒳,)𝒢Spec(R),

which is smooth and projective. Hence by replacing T be a finite cover, we have a u-twisted semi-stable objects η with Mukai vectors u over the generic point η of T. By the properness of f, can be extended to a family of u-twisted semi-stable objects with Mukai vectors u over T. Therefore there is a u-twisted semi-stable object E with v(E)=u. 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 G2 such that G¯2=pNG¯+G¯0 in Kα(X) (G¯ denotes the class of G in Kα(X)) and both G and G2 belong to the same chamber for v, where N is sufficiently large ([50, Corollary 2.4.4]). We have that (rkG2)G¯2+(v(G2),v(G2)/2)𝔨x¯Kα(X) is isotropic. Hence there is a primitive and isotropic Mukai vector u such that

(rkG2)v(G2)+v(G2),v(G2)2ϱXu.

Moreover, MHG(v)=MHu(v).

Lemma A.2.5.

For an isotropic Mukai vector u, assume that there is a local projective generator Gu of C with v(Gu)Zu. Then MHu(v) is isomorphic to the moduli space of stable (twisted) sheaves on an abelian surface or a K3 surface X, where the polarization is a general one.

Proof.

Let Gu be a local projective generator of such that

c1(Gu)rkGu-c1(Gu)rkGu

is sufficiently small and general in NS(X). Applying Lemma A.2.4 to u,

X:=MHGu(u)

is a smooth projective surface. Let be a universal family on the product X×X as an object of the category 𝐃(α-1,α)(X×X), where α is a 2-cocycle of 𝒪X× representing a Brauer class [α]Hét2(X,𝒪X×). By [52, Theorem 2.7.1], there is a category of perverse coherent sheaves 𝐃α(X) with a local projective generator G of and an ample divisor H on X such that |{x}×X is G-twisted stable object of and XMHG(u) by the correspondence x|{x}×X, where u=v(|{x}×X) and v(G)/rkG-u/rkuNS(X) is sufficiently small.

Indeed, [52, Assumption 2.1.1] holds for D by Remark 1.4.2 and [52, Assumption 2.1.1] is preserved under the dual (i.e., 𝔨x[2]D is -γ-stable). Under the following correspondence of the notations

(Y,H^,Per(X/Y)D,w0,(w0-β^))(M¯(u)Hu,H,,u,v(G)),

we apply [52, Theorem 2.7.1], where the left hand side is the notations in [52].

For the Fourier–Mukai transform Φ^:=ΦXX we will apply [52, Proposition 2.8.2] under the correspondence of notations:

(w,H^,H,v(G2),v(G1))(v,H,H,u,u),

where H=H^. Then Φ^ induces an isomorphism

MHu(venH)MH′′(v)(n0),

where v=Φ^(venH), H′′ is a general ample divisor on X which is sufficiently close to H and MH′′(v) consists of α1-twisted stable sheaves. ∎

Proof of Proposition A.2.1.

By Lemmas A.2.4A.2.5, we may assume that H is general with respect to v and =Cohα(X). Replacing v by venH, n0,

vrkv-v(G1)rkG1=(0,ξ,a),

where G1 is the α-twisted sheaf in the proof of Proposition A.2.3 and ξ is sufficiently close to H. Replacing H by ξ, we may assume that vv(G1)+H+ϱX. Then we have a relative moduli scheme M(𝒳,)𝒢(v)Spec(R), 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 𝒜-modules, where 𝒜 is a sheaf of 𝒪Y-algebras on a projective scheme Y via Morita equivalence. Then by Simpson’s result [34, Theorem 4.7] on the moduli of semi-stable 𝒜-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 [21].

Let S be a scheme of finite type over a universally Japanese ring. Let (Y,𝒪Y(1))S be a flat family of polarized schemes over S. Let 𝒜 be a sheaf of 𝒪Y-algebras such that 𝒜 is a coherent 𝒪Y-module, which is flat over S. For an 𝒜s-module E on Xs, we write the Hilbert polynomial of E as

χ(Ys,E(m))=iai(E)(n+ii),ai(E).

If χ(E(m)) 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.

Lemma A.3.1.

We take a surjective morphism

𝒪Y(-m)N𝒜.

Let E be a μ-semi-stable A-module of dimension d. Then

ad-1(F)ad(F)ad-1(E)ad(E)+m

for any subsheaf F of E.

Proof.

We may assume that F is μ-semi-stable. For the multiplication morphism

ϕ:F𝒜E,

imϕ is an 𝒜-submodule of E. Hence

ad-1(imϕ)ad(imϕ)ad-1(E)ad(E).

By our assumption, we have a surjective morphism F(-m)Nimϕ. Hence

ad-1(imϕ)ad(imϕ)ad-1(F(-m))ad(F)=ad-1(F))ad(F)-m.

Therefore the claim holds. ∎

By this lemma, the set of μ-semi-stable 𝒜s-modules E on Ys (sS) with the Hilbert polynomial P is bounded by Langer’s boundedness theorem. Hence we can parametrize semi-stable 𝒜s-modules by the Quot scheme

𝔔:=Quot𝒜(-n)N/Y/S𝒜,P

whose points correspond to quotient 𝒜s-modules 𝒜s(-n)NE with χ(E(x))=P(x), where N=P(n). For a purely d-dimensional 𝒜s-module E on Ys,

μmax(E):=max{ad-1(F)ad(F):F is an 𝒜s-submodule of E}.

Combining this with Langer’s important result [21, Corollary 3.4], we have the following:

Lemma A.3.2.

For any purely d-dimensional A-module E on Ys, sS,

h0(Ys,E)ad(E)[1d!(μmax(E)+c)d]+,

where c depends only on (Y,OY(1)), A, d and ad(E).

Let (R,𝔪) be a discrete valuation ring R with maximal ideal 𝔪. Let K be the fractional field and k the residue field. Let be an R-flat family of 𝒜R-modules such that RK is pure.

Lemma A.3.3.

There is an R-flat family of coherent AR-modules F and a homomorphism ψ:EF such that FRk is pure, ψK is an isomorphism and ψk is an isomorphism at generic points of Supp(FRk).

Proof.

By using [34, Lemma 1.17], we first construct as a usual family of sheaves. Then by the very construction of it, becomes an 𝒜R-module. ∎

Let 𝔔ss be the open subscheme of 𝔔 consisting of semi-stable 𝒜s-modules. By using Lemma A.3.2 and Lemma A.3.3, we can construct the moduli of semi-stable 𝒜s-modules as a GIT-quotient of 𝔔 by the action of PGL(N), where n is sufficiently large. Thus we have a good quotient 𝔮:𝔔ss𝔔ss/PGL(N) and 𝔔ss/PGL(N) is the coarse moduli space of semi-stable 𝒜s-modules.

Theorem A.3.4.

We have a coarse moduli scheme of semi-stable A-modules

M¯(Y,𝒪Y(1))/S𝒜,PS,

which is projective over S.

Let M(Y,𝒪Y(1))/S𝒜,P be the open subscheme of stable 𝒜s-modules. By this construction,

𝔮:𝔔sM(Y,𝒪Y(1))/S𝒜,P

is a principal PGL(N)-bundle; 𝔔s is the open subscheme parametrizing stable 𝒜s-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 XS is smooth. Let π:XY be a family of S-morphisms and G a locally free sheaf on X such that Gs is a local projective generator of a family of abelian categories s𝐃(Xs) as in [50, Section 1.3]. Since [50, Corollary 1.3.10] holds for any base S, we have the following.

Corollary A.3.5.

The following statements hold.

  1. We have a relative moduli stack of Gs-twisted semi-stable objects with the Hilbert polynomial P as a quotient stack [Qss/GL(N)], where Qss is an open subscheme of QuotGN/X/S,P parametrizing Gs-twisted semi-stable objects.

  2. We have a relative moduli scheme

    M¯(X,𝒪X(1))/SG,PS

    of Gs-twisted semi-stable objects with the Hilbert polynomial P as a GIT-quotient:

    M¯(X,𝒪X(1))/SG,PQss/PGL(N).

Let 𝒬 be the universal quotient on X×SQss.

Proposition A.3.6.

Assume that there is a bounded family of locally free (twisted) sheaves V on X such that

χ((V)s,𝒬s)=1,sS.

Then there is a universal family on X×SM(X,OX(1))/SG,P.

Proof.

We set

𝒜:=π*(G𝒪XG).

Then π*(G𝒬) is a GL(N)-equivariant 𝒜𝒪Qss-module. Hence

𝒬π-1(π*(G𝒬))𝐋π-1(𝒜)G

is GL(N)-equivariant. Then

F:=𝐑pQss(V𝒬)

is a bounded complex of GL(N)-equivariant locally free sheaves on Qss, where

pQss:X×SQssQss

is the projection. Then 𝒬pQss*(det(F)) is a PGL(N)-linearized object on X×SQss. Let Qs be the open subscheme parametrizing stable objects. Since QsQs/PGL(N) is a principal bundle, it follows that

(𝒬pQss*(det(F)))|X×SQs

is the pull-back of a family of perverse coherent sheaves on X×SM(X,𝒪X(1))/SG,P, which gives a universal family. ∎

Corollary A.3.7.

Assume that (X,H) is a polarized K3 or an abelian surface over a scheme S. For v=(r,ξ,a)ZNS(X/S)Z, if gcd(r,(ξ,D),a)=1, DNS(X/S), then there is a universal family on X×SMHβ(v).

Acknowledgements

We would like to thank referees very much for their valuable comments and suggestions on improvements of our paper.

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Received: 2012-11-26
Revised: 2014-9-3
Published Online: 2015-6-10
Published in Print: 2018-2-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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