Toric sheaves stability and fibrations

2025-05-06 0 0 379.23KB 26 页 10玖币

侵权投诉

arXiv:2210.04587v2 [math.AG] 27 Jul 2023

TORIC SHEAVES, STABILITY AND FIBRATIONS

ACHIM NAPAME AND CARL TIPLER

Abstract. For an equivariant reﬂexive sheaf over a polarised toric variety,

we study slope stability of its reﬂexive pullback along a toric ﬁbration. Ex-

amples of such ﬁbrations include equivariant blow-ups and toric locally trivial

ﬁbrations. We show that stability (resp. unstability) is preserved under such

pullbacks for so-called adiabatic polarisations. In the strictly semistable situa-

tion, under locally freeness assumptions, we provide a necessary and suﬃcient

condition on the graded object to ensure stability of the pulled back sheaf.

As applications, we provide various stable perturbations of semistable tangent

sheaves, either by changing the polarisation, or by blowing-up a subvariety. Fi-

nally, our results apply uniformly in speciﬁc ﬂat families and induce injective

maps between the associated moduli spaces.

1. Introduction

The study of moduli spaces of torsion-free coherent sheaves on a given variety is a

fundamental problem in algebraic geometry. The construction of a quasi-projective

structure on the moduli space can be achieved by considering stable sheaves [14].

In this paper, we will be interested in slope stability as introduced by Mumford

and Takemoto [25]. Stable reﬂexive sheaves are of particular interest, given their

close relation to stable vector bundles [12]. Being tightly linked to the geometry

of the ambiant variety, it is natural to investigate how they behave with respect to

natural maps such as pullbacks. In this direction, a fundamental result of Mehta

and Ramanathan [18] asserts that the restriction of a slope (semi)stable torsion-free

sheaf to a generic complete intersection of high degree remains slope (semi)stable.

In this paper, we address the problem of pulling-back a (semi)stable reﬂexive sheaf

along a ﬁbration, in the equivariant context of toric geometry.

Consider a toric ﬁbration π:X′→Xbetween Q-factorial projective toric

varieties deﬁned over the complex numbers (see Section 2for precise deﬁnitions).

We will denote by Tthe torus of X. Assume that Lis an ample divisor on Xand L′

is a relatively ample divisor on X′. Then, for ε∈Qsmall enough, Lε:= π∗L+εL′

deﬁnes an ample Q-divisor on X′. Following the terminology used in diﬀerential

geometry, we will call the associated polarisation adiabatic. For a given torsion-free

sheaf Eon X, we will denote by µL(E) := degL(E)/rk(E) its slope with respect to

L, and call Eastable (resp. semistable) sheaf with respect to Lif for all coherent

subsheaves F⊆Ewith strictly smaller rank, we have µL(F)< µL(E) (resp.

µL(F)≤µL(E)). If for one subsheaf F⊆Ewe have µL(F)> µL(E), Eis called

unstable. Then, our ﬁrst result is the following:

Theorem 1.1. Let Ebe a T-equivariant stable reﬂexive sheaf on (X, L). Then

there is ε0>0such that for all ε∈(0, ε0)∩Q, the reﬂexive pullback (π∗E)∨∨ is

stable on (X′, Lε).

This result relies essentially on the fact that in the torus equivariant context, it is

enough to test slope inequalities for equivariant and reﬂexive saturated subsheaves

[16,13]. In general, the slope of a pulled back sheaf on X′with respect to Lεadmits

an expansion in ε, with the coeﬃcient of the smallest exponent given by the slope

of the initial sheaf on Xwith respect to L. Hence, the following is straightforward:

2 A. NAPAME AND C. TIPLER

Proposition 1.2. Let Ebe a T-equivariant unstable reﬂexive sheaf on (X, L).

Then there is ε0>0such that for all ε∈(0, ε0)∩Q, the reﬂexive pullback (π∗E)∨∨

is unstable on (X′, Lε).

Our main result deals with the more delicate strictly semistable situation. Let E

be a strictly semistable torsion-free sheaf on (X, L). It then admits a Jordan-H¨older

ﬁltration

0 = E1⊆E2⊆...⊆Eℓ=E

by slope semistable coherent subsheaves with stable quotients of same slope as E

[14]. The reﬂexive pullbacks of the Ei’s form natural candidates to test for stability

of the reﬂexive pullback of Eon (X′, Lε). In fact, we will see shortly that if Eand

Gr(E) := Lℓ−1

i=1 Ei+1/Eiare locally free, it is actually enough to compare slopes

with these sheaves. In order to state our result, we will introduce some notations.

Let Ebe the set of equivariant and saturated reﬂexive subsheaves F⊆Earising

in a Jordan-H¨older ﬁltration of E. For two coherent sheaves F1and F2on X′, we

will write µ0(F1)< µ0(F2) (resp. µ0(F1)≤µ0(F2) or µ0(F1) = µ0(F2)) when

the coeﬃcient of the smallest exponent in the expansion in εof µLε(F2)−µLε(F1)

is strictly positive (resp. greater or equal to zero or equal to zero). Recall that

a locally free semistable sheaf is called suﬃciently smooth if its graded object is

locally free.

Theorem 1.3. Let Ebe a T-equivariant locally free and suﬃciently smooth strictly

semistable sheaf on (X, L). Then there is ε0>0such that for all ε∈(0, ε0)∩Q,

the reﬂexive pullback E′:= (π∗E)∨∨ on (X′, Lε)is:

(i) stable iﬀ for all F∈E,µ0((π∗F)∨∨)< µ0(E′),

(ii) strictly semistable iﬀ for all F∈E,µ0((π∗F)∨∨)≤µ0(E′)with at least

one equality,

(iii) unstable iﬀ there is one F∈Ewith µ0((π∗F)∨∨)> µ0(E′).

This theorem should be compared to [23, Theorem 1.4], where a similar result

is obtained, in a non necessarily toric setting, for pullbacks of strictly semistable

vector bundles on holomorphic submersions. The approach to the problem in [23]

is fairly diﬀerent, with diﬀerential geometric techniques, and requires some addi-

tional technical assumptions on Gr(E). Working in the toric setting, by mean of

combinatorial and algebraic methods, we are able to extend the results from [23]

to more general ﬁbrations, allowing singularities on Xand X′, and ﬁbers with

multiple irreducible components.

Remark 1.4.We will see in Section 3.4 that if πis a locally trivial ﬁbration, our

assumption on Eand Gr(E) to be locally free in Theorem 1.3 is not necessary. In

that situation, we can remove all the technical hypothesis that were required in [23,

Theorem 1.4].

Remark 1.5.In [3], a closely related problem is considered. Let (X′, L′) be a normal

toric variety and X=X′//G a GIT quotient under the action of a generic subtorus

Gof the torus of X′. Denote by ι:Xs→X′the inclusion of the stable locus under

this action, and π:Xs→Xthe quotient map. Then a combinatorial condition

on (X′, L′, G) ensures that there is an ample class αon Xsuch that for any torus

equivariant reﬂexive sheaf Ethat is stable on (X, α), the sheaf (ι∗(π∗E))∨∨ is

stable on (X′, L′) [3]. Toric locally trivial ﬁbrations can be seen as GIT quotients.

However, the results presented in this paper are diﬀerent in nature from the ones

in [3]. In the present situation, we ﬁx a single sheaf and study the stability of

its pullback for adiabatic polarisations, whereas in [3], the set of all pulled back

reﬂexive equivariant sheaves is considered at once, for non necessarily adiabatic

polarisations.

TORIC SHEAVES, STABILITY AND FIBRATIONS 3

Remark 1.6.In fact, we will see in Section 3.5 that our results hold for speciﬁc ﬂat

families of equivariant reﬂexive sheaves. More precisely, if (Es)s∈Sis a family of

stable equivariant reﬂexive sheaves over (X, L) with either

(I) Eis locally free on X×S, or

(II) the characteristic function (χ(Es))s∈Sis constant,

then the ε0in Theorem 1.1 can be taken uniformly for (Es)s∈S. Similarly, if we as-

sume all Esto be suﬃciently smooth, the ε0in Theorem 1.3 can be taken uniformly

in s∈Sprovided one of conditions (I) or (II) above is satisﬁed. As a corollary,

we will see that the reﬂexive pullback induces injective maps between the relevant

moduli spaces of stable equivariant sheaves. As those moduli spaces arise as ﬁxed

point loci under the torus action on moduli spaces of reﬂexive sheaves on toric

varieties [16], we hope to extract more information from those injective maps, at

least for some simple ﬁbrations.

We then specify our results to various types of toric ﬁbrations. The ﬁrst one

that we address in Section 3.4 is when X′=Xand πis the identity. The only

modiﬁcation comes then from the change in polarisation from Lto L+εL′. As

noticed in [23], in that case, our result already provides interesting information on

the behaviour of a semistable reﬂexive sheaf when the polarisation varies. On a

global level, moduli spaces of stable sheaves are subject to modiﬁcations related

to wall-crossing phenomena in the ample cone (see [14, Chapter 4, Section C] and

reference therein for results on variations of moduli spaces of stable bundles on

surfaces). Restricting to a single semistable reﬂexive sheaf E, Theorem 1.3 gives a

simple and eﬀective criterion on perturbations of the polarisation that send Eto

the stable locus. As an illustration, we describe in Section 3.4 stable perturbations

of the tangent sheaf of a normal Del Pezzo surface, which is strictly semistable with

respect to the anticanonical polarisation.

Another case of interest is when π:X′→Xis an equivariant blow-up along a

torus invariant subvariety Z⊆Xand Lε=π∗L−εE where Eis the exceptional

divisor of π. Assuming Xto be smooth, we push further the study of pulling back

semistable sheaves under that setting in Section 4. In particular, if Sis a set of

invariant points under the torus action of X, we obtain (see Section 4.3):

Theorem 1.7. Let (X, L)be a smooth polarised toric variety and Sa set of in-

variant points under the torus action of X. Let π:X′→Xbe the blow-up along S

and let Lε=π∗L−εE for Ethe exceptional divisor of π. Let Ebe a T-equivariant

reﬂexive sheaf that is strictly semistable on (X, L). Then there is ε0>0such that

for all ε∈(0, ε0)∩Q, the reﬂexive pullback E′:= (π∗E)∨∨ on (X′, Lε)is

(i) strictly semistable iﬀ for any subsheaf F∈E,(π∗F)∨∨ is saturated in E′,

(ii) unstable otherwise.

Corollary 1.8. With the notations of Theorem 1.7, if Eis suﬃciently smooth,

then E′satisﬁes (i)and thus is strictly semistable on (X′, Lε)for ε≪1.

Remark 1.9.Corollary 1.8, together with Theorem 1.1, show that blowing-up points

strictly preserves (semi)stability of a suﬃciently smooth vector bundle for adiabatic

polarisations. However, in general, the reﬂexive pullback of a saturated subsheaf

might not be saturated, see Example 4.4. Hence, pulling back along a single point

blow-up might “destabilize” a semistable reﬂexive sheaf.

Hence, for adiabatic polarisations, blowing-up a point will never push a strictly

semistable toric sheaf to the stable locus. This is no longer true if we blow-up

higher dimensional varieties. In Section 4.4, we prove :

Theorem 1.10. Let (X, L)be a smooth polarised toric variety. Let π:X′→X

be the blow-up along a T-invariant irreducible curve Z⊆Xand let Lε=π∗L−εE

4 A. NAPAME AND C. TIPLER

for Ethe exceptional divisor of π. Let Ebe a T-equivariant reﬂexive sheaf that is

strictly semistable on (X, L). Then there is ε0>0such that for all ε∈(0, ε0)∩Q,

the pullback E′:= (π∗E)∨∨ on (X′, Lε)is

(i) stable iﬀ for all F∈E,(π∗F)∨∨ is saturated in E′and

c1(E)·Z

rkE<c1(F)·Z

rkF;

(ii) semistable iﬀ for all F∈E,(π∗F)∨∨ is saturated in E′and

c1(E)·Z

rkE≤c1(F)·Z

rkF;

(iii) unstable otherwise.

Remark 1.11.In Theorem 1.10, if Eis suﬃciently smooth, then for all F∈E,

(π∗F)∨∨ is saturated in E′(cf. Lemma 3.9). In that case, to study stability of E′

on (X′, Lε) for ε≪1, it is enough to compare the intersection numbers c1(E)·Z

rkEand

c1(F)·Z

rkFfor Fin the ﬁnite set E. We provide in Section 4.5 an explicit semistable

example, namely the tangent sheaf of a Picard rank 2 toric variety, that becomes

stable when pulled back to the blow-up along a curve.

Theorem 1.1 and Theorem 1.7 recover and generalize some of the results in [2]

and [7] on pullbacks of stable bundles along blow-ups of points. While we restrict

ourselves to toric varieties, our results cover the cases of stable reﬂexive sheaves

and semistable suﬃciently smooth vector bundles. In comparison, in [2] the base

manifold is a surface, while in [7], the method is via a gluing construction for

Hermite-Einstein metrics, providing more precise information on the behaviour of

the metrics when ε→0, but with a restriction to stable bundles. The closer result

in [10, Proposition 5.1] is more general than our Corollary 1.8 as it deals with

pullbacks of semistable torsion-free sheaves over normal projective varieties, but

Theorem 1.7 seems to provide more information when Eis not suﬃciently smooth.

On the other hand, Theorem 1.10 seems to be, to the knowledge of the authors,

the ﬁrst result in the direction of pushing a semistable bundle to the stable locus

by blowing-up higher dimensional sub-varieties.

The results in Theorem 1.10 rely on a more general formula for slopes of pullback

sheaves under blow-ups. In general, if Z⊆Xis an ℓ-dimensional smooth subvariety

of a smooth projective variety Xwith 1 ≤ℓ≤dim(X)−2, and Eis a reﬂexive

sheaf on X, then, setting π:X′→Xthe blow-up along Z, we have:

(1) µLε((π∗E)∨∨) = µL(E)−n−1

ℓ−1µL|Z(E|Z)εn−ℓ+O(εn−ℓ+1).

This formula is quite striking as from Mehta-Ramanathan’s restriction theorem, if

Zis generic and an intersection of divisors coming from linear systems H0(X, Lki)

with large ki’s, then E|Zwill be semistable provided Eis. Hence, in that situation,

formula (1) shows that subsheaves F⊆Ewith µL(F) = µL(E) tend to destabilise

(π∗E)∨∨. Setting ourselves in a typically non-generic situation, we can avoid this

no go result. We obtain:

Theorem 1.12. Let (X, L)be a smooth polarised toric variety. Let π:X′→X

be the blow-up along a T-invariant irreducible subvariety Z⊆Xof codimension

at least 2and let Lε=π∗L−εE for Ethe exceptional divisor of π. Let Ebe a

T-equivariant reﬂexive sheaf that is strictly semistable on (X, L). Assume that for

all F∈E,(π∗F)∨∨ is saturated in E′:= (π∗E)∨∨ and that

µL|Z(E|Z)< µL|Z(F|Z).

Then there is ε0>0such that for all ε∈(0, ε0)∩Q, the pullback E′is stable on

(X′, Lε).

TORIC SHEAVES, STABILITY AND FIBRATIONS 5

Finally, Theorem 1.1 has another consequence on resolution of singularities. An

application of Hironaka’s resolution of indeterminacy locus shows that for a given

equivariant reﬂexive sheaf Eon X, there is a ﬁnite sequence of blow-ups along

smooth irreducible torus invariant centers πi:Xi→Xi−1for 1 ≤i≤pwith

X0=Xsuch that, if we set Ei= (π∗

iEi−1)∨∨, the sheaf E′:= Epis locally free

on X′:= Xp. Each map πiis a toric ﬁbration, and thus we can iterate Theorem

1.1. Starting with a stable sheaf E, we then obtain a stable locally free sheaf E′

on (X′, L′) that is isomorphic to Eaway from the exceptional locus. In [1] and

[24], a similar result is obtained, without the toric hypothesis, but with diﬀerential

geometric methods, and for a diﬀerent polarisation on X′. The polarisation in

[1,24] is of the form L+εH, where His an ample divisor on X′. In contrast,

the polarisation L′, at the level of K¨ahler forms, only aﬀects the geometry of X

on a small neighborhood of the exceptional divisor. We believe that this can be

useful regarding the resolution of admissible Hermite-Einstein metrics as introduced

by Bando and Siu in [1]. We will come back to these explicit resolutions in the

forthcoming [19].

Remark 1.13.We should note that while we restrict ourselves to toric varieties and

equivariant sheaves, we believe that all results in this paper should be true without

the torus equivariant assumption, on normal varieties. Nevertheless, working with

equivariant structures provides several crucial simpliﬁcations in the arguments, and

enables to produce explicit examples that might be diﬃcult to ﬁnd in general. For

the sake of generality, we aim to relax our equivariant hypothesis in future work.

Organisation and conventions. All varieties considered in this paper are deﬁned

over the complex numbers and assumed to be normal. In Section 2we recall

the necessary background on toric varieties, their morphisms and their equivariant

sheaves. We also recall the basics of slope stability. In Section 3, we prove Theorem

1.1, Proposition 1.2 and Theorem 1.3. We then give the ﬁrst applications in the

case of locally trivial ﬁbrations. Section 4is a more in depth study of the blow-

up case, in which proofs of Theorems 1.7,1.10 and 1.12, as well as Corollary 1.8,

together with applications, are given.

Acknowledgments. The authors would like to thank Lars Martin Sektnan for

stimulating discussions on this problem, Ruadha´ı Dervan for several enlightening

suggestions and Michel Brion for his careful reading of the manuscript and his

advice. We also thank the anonymous referee for their advices that improved the

exposition and the content of this paper. The authors are partially supported by

the grants MARGE ANR-21-CE40-0011 and BRIDGES ANR–FAPESP ANR-21-

CE40-0017.

2. Background

In this ﬁrst section we gather the necessary background about toric varieties [4]

and equivariant reﬂexive sheaves [15,21].

2.1. Toric varieties and divisors. Let Nbe a rank nlattice and Mbe its dual

with pairing h · ,· i :M×N→Z. The lattice Nis the lattice of one-parameter

subgroups of N⊗ZC∗. For K=Ror C, we deﬁne NK=N⊗ZKand MK=M⊗ZK.

Afan Σ in NRis a set of rational strongly convex polyhedral cones in NRsuch that:

•Each face of a cone in Σ is also a cone in Σ;

•The intersection of two cones in Σ is a face of each.

We will denote τσthe inclusion of a face τin σ∈Σ. A cone σin NRis smooth

if its minimal generators form part of a Z-basis of N. We say that σis simplicial

if its minimal generators are linearly independent over R. A fan Σ is smooth (resp.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

arXiv:2210.04587v2[math.AG]27Jul2023TORICSHEAVES,STABILITYANDFIBRATIONSACHIMNAPAMEANDCARLTIPLERAbstract.Foranequivariantreﬂexivesheafoverapolarisedtoricvariety,westudyslopestabilityofitsreﬂexivepullbackalongatoricﬁbration.Ex-amplesofsuchﬁbrationsincludeequivariantblow-upsandtoriclocallytrivialﬁbrati...

展开>> 收起<<

Toric sheaves stability and fibrations.pdf

共26页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Toric sheaves stability and fibrations

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: