Moduli spaces of stable pairs and non-abelian zeta functions of curves via wall-crossing

Moduli spaces of stable pairs and non-abelian zeta functions of curves via wall-crossing

Tome 1 (2014), p. 117-146.

<http://jep.cedram.org/item?id=JEP_2014__1__117_0>

© Les auteurs, 2014.

Certains droits réservés.

Cet article est mis à disposition selon les termes de la licence CREATIVECOMMONS ATTRIBUTION–PAS DE MODIFICATION3.0 FRANCE. http://creativecommons.org/licenses/by-nd/3.0/fr/

L’accès aux articles de la revue « Journal de l’École polytechnique — Mathématiques » (http://jep.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://jep.cedram.org/legal/).

Publié avec le soutien

du Centre National de la Recherche Scientifique

cedram

Article mis en ligne dans le cadre du

Centre de diffusion des revues académiques de mathématiques

MODULI SPACES OF STABLE PAIRS AND NON-ABELIAN ZETA FUNCTIONS OF CURVES VIA WALL-CROSSING

by Sergey Mozgovoy & Markus Reineke

Abstract. — In this paper we study and relate the non-abelian zeta functions introduced by Weng and invariants of the moduli spaces of arbitrary rank stable pairs over curves. We prove a wall-crossing formula for the latter invariants and obtain an explicit formula for these invariants in terms of the motive of a curve. Previously, formulas for these invariants were known only for rank 2 due to Thaddeus and for rank 3 due to Muñoz. Using these results we obtain an explicit formula for the non-abelian zeta functions, we check the uniformity conjecture of Weng for the ranks 2 and 3, and we prove the counting miracle conjecture.

Résumé(Espaces de modules de paires stables et fonctions zêta non abéliennes des courbes via le « wall-crossing »)

Dans cet article nous étudions et mettons en relation les fonctions zêta non abéliennes introduites par Weng et les invariants des espaces de modules de paires stables de rang arbitraire sur les courbes. Nous prouvons une formule « wall-crossing » pour ces invariants et obtenons une formule explicite pour ceux-ci en terme du motif de la courbe. Auparavant, des formules pour ces invariants n’étaient connues qu’en rang2par Thaddeus et en rang3par Muñoz. En utilisant ces résultats nous obtenons une formule explicite pour les fonctions zêta non abéliennes, nous vérifions la conjecture d’uniformité de Weng pour les rangs2et3, et nous montrons sa conjecture de dénombrement miracle.

Contents

1. Introduction. . . 118

2. Preliminaries. . . 120

3. Semistable pairs and triples. . . 123

4. Wall-crossing. . . 125

5. Invariants. . . 128

6. Zagier-type formula. . . 131

7. Non-abelian zeta functions. . . 134

8. Appendix. Slice and inversion formulas. . . 137

References. . . 145

Mathematical subject classification (2010). — 14H60, 14D20.

Keywords. — Stable pairs, vector bundles, wall-crossing formulas, higher zeta functions.

1. Introduction

This paper has two motivations. The first one is the study of motivic invariants (like Poincaré polynomials, Hodge polynomials, or motives) of moduli spaces of pairs on a smooth projective curve. The moduli spaces of pairs were studied extensively in the last two decades [4, 7, 22, 12]. Their Poincaré resp. Hodge polynomials were computed by Thaddeus [22] in the rank two case and by Muñoz [18] in the rank three case. For rank four it was proved [17], and conjectured for general rank, that the motive of the moduli space can be expressed in terms of the motive of the curve. We will compute the motives of these moduli spaces for arbitrary rank in terms of an explicit Zagier-type formula, and in particular confirm the above conjecture.

Our second motivation is the work of Weng [25] on the (pure) non-abelian zeta functions of curves. Given a curveX over a finite fieldFq, letM(r, d)denote the set of isomorphism classes of semistable vector bundles onX having rankrand degreed.

Define the rankrpure non-abelian zeta function by ZX,r(t) =X

k>0

X

E∈M(r,kr)

q^h0(X,E)−1

|AutE| t^k.

The special uniformity conjecture of Weng [25, Conj. 9] suggests that the rankrpure zeta function coincides with the group zeta function associated to the special linear group SLr [25, §2]. This conjecture was announced to be a theorem in [23, Th. 5].

This result can be used to express the rankrpure zeta functions in terms of the usual zeta function of a curve. We will use a different approach based on moduli spaces of pairs to compute rank r zeta functions by an explicit Zagier-type formula. We will also check the uniformity conjecture for the rank 2and3zeta functions.

Let us now describe our results in more detail. Let X be a smooth projective complex curve of genusg. A pair(E, s)onX consists of a vector bundleEonX and a nonzero sections∈H⁰(X, E). There is a notion of stability of such pairs depending on a parameter τ ∈R (see §3.1). For any(r, d)∈Z>0×Z, let Mτ(r, d)denote the moduli stack of τ-semistable pairs with a vector bundle having rank r and degreed and let

f_τ(r, d) = (q−1)q^(1−g)(^r₂)[Mτ(r, d)]

be its motive up to some factor, where q denotes the Lefschetz motive. Define the generating function

f_τ=X

r,d

f_τ(r, d)x^r₁x^d₂x₃

in a certain completion of a skew polynomial ring (see §5.2). For example, forτ=∞ (this means that for each pair(E, s), we verify its semistability with respect toτ asτ goes to infinity; this is equivalent to the requirement thatcokersis finite), we have

f_∞=x1x3ZX(x2), whereZX(t)is the motivic zeta function ofX.

On the other hand, let u_>τ(r, d) be the twisted motive of the moduli stack of vector bundles having rankr, degree dand such that the quotients of their Harder- Narasimhan filtrations have slopes > τ. These invariants can be computed by the formula of Zagier [26], based on the works of Harder and Narasimhan [11], Desale and Ramanan [6], and Atiyah and Bott [1]. Define the generating function

u_>τ = X

d/r>τ

u_>τ(r, d)x^r₁x^d₂.

Our first main result is the following wall-crossing formula (see Theorem 5.4):

Theorem1.1. — For any τ∈R, we have

f_τ= (u⁻¹_>τ◦f_∞◦u_>τ)|_µ6τ,

where the truncation |µ6τ means that we keep only the coefficients x^r₁x^d₂x3 with d/r6τ.

This result implies that the motive of M_τ(r, d)can be expressed in terms of the motive of X and its symmetric products. The latter statement follows also from [9, Prop. 6.2] (as well as [8, Th. 1]) for a large class of stability parameters. It was also conjectured in [17]. Using generalizations of Zagier’s formula for the motive of the moduli stack of semistable bundles (to be discussed in an appendix which also contains a new proof of Zagier’s original formula), this yields the following explicit formula for the motive[Mτ(r, d)]of the moduli space ofτ-semistable pairs (see Theorem 6.2):

Theorem1.2. — Forr>2 and genericτ, we have [M_τ(r, d)] =q^(g−1)(^r₂) X

r₁+···+r_k=r−1

b_r1. . . b_rk Qk−1

i=1 (1−q^ri+ri+1)

coeff_td−d(r−1)τe

Z_X(t)· q^F0 1−q^r1+1t−

k−1

X

p=1

q^Fp(1−q^rp+rp+1)t^δp

(1−q^rp+1+1t)(1−q^−rpt)− q^Fk 1−q^−rkt

, wherebrequals (up to a twist) the motive of the moduli stack of rankrbundles, andFp

andδp are certain explicit exponents.

The wall-crossing formula can also be used to compute the higher zeta functions.

We can write the motivic version of the higher zeta-functions as follows ZX,r(t) = (q−1)X

k>0

[Mk(r, kr)]t^k=q^(g−1)(^r2)X

k>0

fk(r, kr)t^k.

This means that in order to find ZX,r(t) we have to compute fτ(r, d) for τ = d/r.

Applying the above theorem we obtain, for anyτ∈R: X

d r=τ

fτ(r, d)x^r₁x^d₂x3= (u⁻¹_>τ◦f_∞◦u_>τ)|µ=τ. The following result describes the higher zeta functions explicitly.

Theorem1.3. — Let ZbX,r(t) =t^1−gZX,r(t)andZbX(t) =t^1−gZX(t). Then Zb_X,r(t) =q^(g−1)(^r₂) X

r₁+···+r_k=r−1

b_r1· · ·b_rk Qk−1

i=1(1−q^ri+ri+1)

Zb_X(t) 1−q^r1+1t−

k−1

X

i=1

(1−q^ri+ri+1)q^r<itZb_X(q^r6it)

(1−q^r<it)(1−q^r6i+1+1t) −q^r<ktZb_X(q^r−1t) 1−q^r<kt

. This theorem implies a generalization of the counting miracle conjecture of Weng [24, Conj. 15]

q^(1−g)rZX,r(0) = [M(r−1,0)],

where M(r,0)denotes the moduli stack of semistable vector bundles having rank r and degree zero.

Our approach can be also used in order to find the higher zeta functions of curves over finite fields. In this case motives should be substituted by the so-called c-sequences introduced in [15]. All of the above formulas remain the same.

The reader should not be deceived by the apparent simplicity of our approach.

A lot of obnoxious geometry happens behind the innocent algebraic scene. While for ranks2and3it is possible, with some effort, to control destabilizing loci when crossing the walls, the situation becomes much more complicated for higher ranks. Our basic idea goes back to the work of Thaddeus. In order to find the motivic invariant of the moduli stackMτ(r, d)ofτ-semistable pairs, we first find this invariant forτ 0and then decrease τ, thoroughly analyzing the behavior of our invariants when crossing the walls, i.e. when τ goes through the critical values, where some semistable pairs become non-semistable. In this way we can find [M_τ(r, d)] for any τ > d/r. But in contrast to [16, 22], our approach does not use the geometry of the moduli spaces directly. Instead, we use ideas from motivic wall-crossing [13] and derive the behaviour of the motivic invariants from identities in a Hall algebra of a category of triples. For τ < d/rthe moduli space is empty. One might ask why we do not cross just one wall at τ =d/r and find the invariant [Mτ(r, d)] forτ = (d/r) +εwith 0 < ε1; the answer is that in order to prove the wall-crossing formula we need enough vanishing of secondExtin the category of triples, which only holds forτ > d/r.

Acknowledgments. — The authors would like to thank Oscar García-Prada, Tamás Hausel, Jochen Heinloth, and Lin Weng for helpful remarks about the results of the paper.

2. Preliminaries

All results of this section will be formulated for an algebraic curveX over an alge- braically closed fieldkof characteristic zero, and for the motives of moduli stacks over it. Motives will be considered as elements in the Grothendieck ringK₀(St_k)of stacks (of finite type overkand with affine stabilizers), which is related to the Grothendieck ring ofk-varieties via localization or dimensional completion (see e.g. [9]). We denote the Lefschetz motive byqand always work in the coefficient ringR=K0(St_k)[q^±1/2].

We can also substitute motives by virtual Poincaré polynomials or E-polynomials.

Also we can formulate all the results for a curve defined over a finite field, in which case we have to substitute motives by the so-called c-sequences introduced in [15].

2.1. Zeta function. — Given an algebraic varietyX we define its motivic zeta func- tion

(2.1) Z_X(t) =X

n>0

[SⁿX]tⁿ= Exp(t[X]).

IfX is a curve of genusg then

ZX(t) = PX(t) (1−t)(1−qt),

wherePX(t)is a polynomial of degree2g. The valuePX(1) equals the motive of the Jacobian [JacX]. The functionZX(t)satisfies the functional equation

ZX(1/qt) = (qt²)^1−gZX(t).

Therefore the function

(2.2) ZbX(t) =t^1−gZX(t)

satisfiesZbX(1/qt) =ZbX(t).

2.2. Stacks of bundles. — LetX be a curve of genusgand letα= (r, d)∈Z>0×Z. LetBunr,ddenote the stack of vector bundles overX having rankrand degreed. Its motive is independent ofd[2, Section 6]:

[Bunr,d] = [JacX]

q−1 q^{(r2−1)(g−1)}

r

Y

i=2

ZX(q⁻ⁱ) =PX(1) q−1

r−1

Y

i=1

ZX(qⁱ).

Define

(2.3) br=q^(1−g)(^r2)[Bunr,d] = PX(1) q−1

r−1

Y

i=1

ZbX(qⁱ).

Let

(1) M(α) =M(r, d)be the moduli stack, (2) M(α) =M(r, d)be the moduli space,

(3) M(α) =M(r, d)be the set of isomorphism classes of semistable vector bundlesE overX withch(E) =α. Define

(2.4) βα=q^(1−g)(^r₂)[M(α)].

2.3. Chern characters. — There is group homomorphism ch :K0(CohX)−→Z²

given bych(E) = (rkE,degE). For anyE, F ∈CohX, define χ(E, F) = dim HomX(E, F)−dim Ext¹_X(E, F).

LetchE=α= (r, d)andchF=β = (r⁰, d⁰). Then by the Riemann-Roch theorem χ(E, F) =rd⁰−r⁰d+ (1−g)rr⁰.

Define

χ(α, β) =rd⁰−r⁰d+ (1−g)rr⁰, (2.5)

χ(α) =χ((1,0), α) =d+ (1−g)r, (2.6)

hα, βi=χ(α, β)−χ(β, α) = 2(rd⁰−r⁰d).

(2.7)

2.4. Integration map. — Define the quantum affine plane A0 to be the completion of the algebraR[x1, x^±1₂ ]with multiplication

x^α◦x^β = (−q^1/2)^hα,βix^α+β, where we allow only elementsf =P

α∈N×Zfαx^αwith infn

− d r+ 1

fr,d6= 0o

>−∞.

LetH(A0)be the Hall algebra of the categoryA0= CohX [15] (we use the opposite multiplication where the product[E]◦[F]counts extensions fromExt¹(F, E)). There is an algebra homomorphismI:H(A₀)→A0[20], called an integration map, defined by

E7→(−q^1/2)^χ(E,E) x^chE [AutE].

For example, if1α∈H(A0)(resp.1^sst_α ∈H(A0)) is an element counting all (resp. all semistable) vector bundles having Chern characterα= (r, d), then

I(1α) = (−q^1/2)^(1−g)r2[Bunr,d]x^α= (−q^1/2)^(1−g)rbrx^α, I(1^sst_α ) = (−q^1/2)^(1−g)r2[M(α)]x^α= (−q^1/2)^(1−g)rβ_αx^α.

Using the Harder-Narasimhan filtrations and applying the integration map, we obtain

(2.8) br= X

(α₁,...,α_k)∈P_α^d

q^{12Pi<jhαi,αji}βα₁· · ·βα_k,

whereP_α^d is the set of slope decreasing partitions ofα.

2.5. Zagier formula. — It was proved by Zagier [26] that if there are families of elements(br)r>1,(βα)α∈Z>0×Zsatisfying (2.8), then

(2.9) βα= X

r₁,...,r_k>0 r₁+···+rk=r

^k−1 Y

i=1

q^{(ri+ri+1){(r1+···+ri)d/r}}

1−q^ri+ri+1

br₁· · ·br_k.

This gives an effective way to compute the motives of the moduli stacks M(α) of semistable vector bundles.

3. Semistable pairs and triples

3.1. Semistable pairs. — Throughout the paper, letX be a smooth projective curve over a fieldk, letτ∈Rand(r, d)∈Z>0×Z.

Definition3.1. — A pair (E, s) over X consists of a vector bundle E over X and a nonzero sections∈H⁰(X, E). Pairs overX form ak-linear category: a morphism f : (E, s)→(E⁰, s⁰)between two pairs is an elementf = (f0, f1)∈k×HomX(E, E⁰) such thatf1s=s⁰f0.

Definition3.2. — A pair(E, s)overX is called τ-semistable (resp. stable) if (1) For any subbundleF ⊂E we haveµ(F)6τ (resp.µ(F)< τ).

(2) For any subbundle F ⊂ E with s ∈ H⁰(X, F) we have µ(E/F) > τ (resp.

µ(E/F)> τ).

Definition 3.3. — Given (r, d) ∈ Z>0×Z, we say that τ ∈ R is (r, d)-generic if τ6=d/randτ /∈ _r¹0Zfor any16r⁰< r. In this case anyτ-semistable pair(E, s)with chE= (r, d)isτ-stable.

We denote

(1) byMτ(r, d)the moduli stack (see [9]), (2) byMτ(r, d)the moduli space ([12]), (3) byMτ(r, d)the set of isomorphism classes

ofτ-semistable pairs(E, s)with ch(E) = (r, d), i.e. of rankrand degreed.

Remark3.4. — If k is algebraically closed of characteristic zero, we can define the motives[Mτ(r, d)],[Mτ(r, d)]as elements ofK0(St_k), the Grothendieck ring of stacks overk. Ifτ is(r, d)-generic then

[Mτ(r, d)] = [Mτ(r, d)]

q−1

as for any τ-stable pair (E, s) we have End(E, s) = k. The analogue of [Mτ(r, d)]

over a finite fieldF^q is

X

(E,s)∈Mτ(r,d)

1

|Aut(E, s)|.

It follows from [21] that the stackMτ(r, d)is of finite type. Similarly one can show thatMτ(r, d)is finite if the field is finite.

Remark3.5. — Let(r, d)∈Z>0×Zand let τ∈R. We will see in Lemma 5.2 that if Mτ(r, d)6=∅thend>0andd/r6τ < d/(r−1).

Lemma3.6. — Let(r, d)∈Z>0×Zand letτ =d/r. Then a pair(E, s)withch(E) = (r, d)isτ-semistable if and only ifE is semistable.

Proof. — Assume that E is semistable. Then for any subbundle F ⊂ E we have µ(F)6τ =µ(E). ThereforeE is semistable.

Assume that E is semistable. Then for any subbundle F ⊂ E we have µ(F) 6 µ(E) =τ andµ(E/F)>µ(E) =τ. Therefore(E, s)isτ-semistable.

Corollary3.7. — Assume that k=Fq andτ =d/r. Then X

(E,s)∈Mτ(r,d)

1

|Aut(E, s)| = 1 q−1

X

E∈M(r,d)

q^h0(X,E)−1

|AutE| .

Proof. — LetE∈M(r, d). There is a natural action of the groupG_E=k^∗×AutE on the setME=H⁰(X, E)r{0}. The orbits of this action can be identified with the isomorphism classes of pairs (E, s). The stabilizer of s∈ME can be identified with Aut(E, s). Therefore

X

[s]∈ME/G_E

1

|Aut(E, s)| = X

s∈ME

1

|GE| = |M_E|

|GE| = q^h0(X,E)−1

(q−1)|AutE|. 3.2. The category of triples

Definition3.8. — LetQbe the quiver with two vertices0,1and one arrows: 0→1.

We considerQas a category and define the categoryT_Xof triples onXas the category of functors fromQtoCohX. This is an abelian category. An objectE ∈ TX can be represented as a triple(E1, E0, sE)whereE0, E1∈CohX andsE∈Hom_OX(E0, E1).

Theorem3.9( [10, Th. 4.1]). — Let E, F ∈ TX be two triples on the curve X. Then there is a long exact sequence

0−→Hom(E, F)−→ L

i=0,1

Hom_OX(Ei, Fi)−→Hom_OX(E0, F1)−→Ext¹(E, F)

−→ L

i=0,1

Ext¹_OX(Ei, Fi)−→Ext¹_OX(E0, F1)−→Ext²(E, F)−→0.

The following results about the vanishing of Ext² in the category of triples are crucial for this paper. They will allow us to apply a Hall algebra formalism for the computation of motivic invariants.

Proposition 3.10. — Let E, F be two triples and assume that sE : E0 → E1 is a monomorphism. ThenExt²(E, F) = 0.

Proof. — According to the previous theorem it is enough to show that Ext¹_OX(E1, F1)−→Ext¹_OX(E0, F1)

is surjective. By Serre duality this is equivalent to the injectivity of Hom_OX(F₁, E₀⊗ω_X)−→Hom_OX(F₁, E₁⊗ω_X)

which holds assE :E0→E1 is a monomorphism.

Corollary 3.11. — Let E, F be two triples and assume that one of the following conditions is satisfied

(1) E0= 0.

(2) E0=OX andsE6= 0.

ThenExt²(E, F) = 0.

Definition3.12. — For anyσ∈Rand for any tripleE= (E₁, E₀, s_E)we define the σ-slope ofE by

µσ(E) = degE1+ degE0+σrkE0

rkE1+ rkE0

∈R∪ {∞}.

A triple E is called semistable (resp. stable) with respect to µσ if for any proper nonzero subobjectF ⊂E we haveµ_σ(F)6µ_σ(E)(resp.µ_σ(F)< µ_σ(E)).

4. Wall-crossing 4.1. Framed categories

Definition4.1. — A framed category is a pair(A, v), whereAis an abelian category and v : K₀(A)→ Z is a group homomorphism such thatv(E) >0 for any E ∈ A.

For anyk>0 we denote by Ak the category of objects E ∈ A withv(E) =k. The objects of the abelian category A0 are called unframed objects. The objects of the categoryA₁ are called framed objects.

We assume that for any objectE∈ Athere exists a maximal unframed subobject E₁⊂E. Similarly, we assume that there exists a maximal unframed quotientE→E₂. Definition 4.2 (see e.g. [3]). — A torsion pair in an abelian category C is a pair (T,F)of strict (i.e. closed under isomorphisms) full subcategories ofC satisfying

(1) C(T, F) = 0 for anyT ∈ T,F ∈ F.

(2) For anyX∈ C there exists a short exact sequence inC 0−→T −→X −→F −→0 such thatT ∈ T andF ∈ F.

The category T is called a torsion class and the categoryF is called a torsion-free class.

Definition 4.3. — Let (T,F) be a torsion pair in the category A0 of unframed objects. A framed objectE∈ A₁ is called

(1) (T,F)-stable ifE₁∈ F andE₂∈ T.

(2) +∞-stable if it is(0,A0)-stable, i.e. if E2= 0.

(3) −∞-stable if it is(A0,0)-stable, i.e. if E1= 0.

Proposition4.4(Canonical filtration). — Any framed object E ∈ A1 has a unique filtration

E⁰⊂E⁰⁰⊂E

such that E⁰∈ T,E⁰⁰/E⁰∈ A1 is(T,F)-stable, andE/E⁰⁰∈ F.

Proof. — LetE be a framed object. We define E⁰ ⊂E₁⊂E to be the torsion part ofE1and we defineE⁰⁰= ker(E→E2→E₂^f), whereE2→E^f₂ is the free part ofE2.

Uniqueness is left to the reader.

4.2. Framed category of triples. — LetX be a curve. LethOXibe the subcategory ofCohX generated from OX by extensions. One can easily see that it is an abelian subcategory of CohX. We define the categoryA to be the category of triples E = (E1, E0, sE)such thatE0 ∈ hOXi. We define the framingv :K0(A)→Zbyv(E) = rkE0. Then the categoryA0of unframed objects can be identified with the category CohX. Framed objects in A have the form (E₁,OX, s_E), where E₁ ∈ CohX and s_E∈Hom_OX(O_X, E₁)'H⁰(X, E₁).

Remark 4.5. — If E = (E₁, E₀, s_E) ∈ A then the maximal unframed subobject of E is (E1,0,0) which we denote byE1. The maximal unframed quotient of E is (cokersE,0,0)which we denote byE2. This is in accordance with the conventions in

§4.2.

Definition4.6. — Let τ ∈R. A framed object E∈ A1 is calledτ-semistable (resp.

stable) if

(1) For any monomorphismF →Ewith unframedFwe haveµ(F)6τ(resp.< τ).

(2) For any epimorphismE→F with unframedF we haveµ(F)>τ (resp.> τ).

A framed objectE∈ A1is calledτ+-stable (resp.τ−-stable) ifEis(τ+ε)-semistable (resp.(τ−ε)-semistable) for0< ε1. This means that

(1) For any monomorphismF→E with unframedF we haveµ(F)6τ (resp.<).

(2) For any epimorphismE→F with unframedF we haveµ(F)> τ (resp.>).

It is clear from Definitions 3.2 and 4.6 that a pair (E, s) is τ-semistable if and only if the triple (E,OX, s) is τ-semistable. Therefore the stack Mτ(r, d) can be identified with the moduli stack of framedτ-semistable triplesE= (E1,OX, sE)with chE₁ = (r, d) and s_E 6= 0. In the next lemma we will see that the last condition sE6= 0 is automatically satisfied for almost allτ.

Lemma4.7. — Let E = (E₁,O_X, s_E) be a framed τ-semistable object with E₁ 6= 0.

Thenµ(E1)6τ. Ifµ(E1)< τ thensE 6= 0.

Proof. — Since E1 is an unframed subobject of E, we have µ(E1) 6 τ. If sE = 0 then E1 is an unframed quotient of E. Therefore µ(E1) >τ, contradicting our as-

sumption.

Lemma4.8. — Let E ∈ A1 be a framed object with E1 6= 0. Let chE1 = (r, d) and τ∈R. ThenE isτ-semistable if and only if it is semistable with respect toµσ, where σ= (r+ 1)τ−d.

Proof. — We note first thatµ_σ(E) = (d+σ)/(r+ 1) =τ. ThereforeE is semistable with respect toµσ if and only if for any unframedF ⊂E we have

µ(F) =µ_σ(F)6µ_σ(E) =τ and for any unframed quotientE→F we have

µ(F) =µ_σ(F)>µ_σ(E) =τ.

This is equivalent toτ-stability ofE.

Define the categoryA_>τ to be the category of sheavesE∈ A0= CohX such that the quotients of their Harder-Narasimhan filtration have slope>τ. Similarly we define the categories A₆τ,A>τ,A<τ. The pairs of categories (A>τ,A₆τ) and (A_>τ,A<τ) are torsion pairs inA0.

Lemma4.9. — Let E∈ A1 be a framed object. Then (1) E isτ-semistable ⇔E1∈ A₆τ,E2∈ A_>τ. (2) E isτ-stable⇔E₁∈ A<τ,E₂∈ A>τ.

(3) E isτ₊-stable⇔ E₁∈ A_6τ,E₂∈ A_>τ⇔E is (A_>τ,A_6τ)-stable.

(4) E isτ₋-stable⇔ E1∈ A<τ,E2∈ A_>τ⇔E is(A_>τ,A<τ)-stable.

The unique filtration of a framed objectE ∈ A1 with respect to the torsion pair (A>τ,A₆τ)(see Prop. 4.4) will be called the canonical filtration with respect toτ+. The unique filtration ofE with respect to the torsion pair(A_>τ,A_<τ)will be called the canonical filtration with respect toτ₋.

Lemma4.10. — LetE∈ Abe a framedτ-semistable object withsE6= 0. Letch(E1) = (r, d)andσ= (r+ 1)τ−d(i.e.µσ(E) =τ). Then

(1) The canonical filtration ofEwith respect toτ+ has the form0 =E⁰⊂E⁰⁰⊂E.

If E⁰⁰ 6=E then E/E⁰⁰ is semistable and µ(E/E⁰⁰) = µ_σ(E⁰⁰) = τ. If s_E 6= 0 then s_E⁰⁰ 6= 0.

(2) The canonical filtration of E with respect to τ₋ has the form E⁰ ⊂E⁰⁰ =E.

If E⁰ 6= 0 then E⁰ is semistable and µ(E⁰) = µσ(E/E⁰) = τ. If µ(E1) < τ then s_E/E0 6= 0.

Proof

(1) Consider the canonical filtration E⁰ ⊂ E⁰⁰ ⊂ E with respect to τ₊. Then E⁰ ∈ A>τ, while E1 ∈ A₆τ. This implies E⁰ = 0. We have E/E⁰⁰ ∈ A₆τ, while if E/E⁰⁰ 6= 0 then µ(E/E⁰⁰)> τ. Therefore µ(E/E⁰⁰) = τ =µσ(E⁰⁰). If sE 6= 0 then automaticallysE⁰⁰6= 0.

(2) Consider the canonical filtration E⁰ ⊂ E⁰⁰ ⊂ E with respect to τ₋. Then E/E⁰⁰ ∈ A<τ, but µσ(E) = τ, and if E/E⁰⁰ 6= 0 then µ(E/E⁰⁰) > τ. This implies E/E⁰⁰ = 0 and E⁰⁰ = E. We have E⁰ ∈ A_>τ, while if E⁰ 6= 0 then µ(E⁰) 6 τ. Thereforeµ(E⁰) =τ =µ_σ(E/E⁰). Assume thatµ(E₁)< τ. The framed objectE/E⁰ is σ₋-stable and therefore is indecomposable. To prove that s_E/E⁰ 6= 0 we have to show thatE₁⁰ 6=E1. IfE₁⁰ = 0then we are done. If E₁⁰ 6= 0thenµ(E⁰₁) =τ > µ(E1),

soE₁⁰ 6=E₁.

Remark4.11. — In the first case of the previous lemma we haveExt²(E/E⁰⁰, E⁰⁰) = 0 as E/E⁰⁰is unframed (see Corollary 3.11). In the second case of the previous lemma we haveExt²(E/E⁰, E⁰) = 0ifµ(E1)< τ (assE/E⁰ 6= 0and we can apply Corollary 3.11).

Lemma4.12. — Let E∈ A1 be a framed object, A⊂E andB=E/A.

(1) IfA∈ A1 isτ₊-stable, B is semistable and µ(B) =τ, thenE isτ-semistable.

(2) IfB∈ A1 isτ₋-stable, Ais semistable and µ(A) =τ, thenE isτ-semistable.

Proof. — We prove just the first statement. Our assumption that A is τ+-stable implies thatA isτ-semistable. Letσ∈Rbe such that µσ(B) = τ. Then both A, B are semistable with slope τ with respect toµ_σ. Therefore their extension E is also semistable with slopeτ with respect toµσ. This implies thatE isτ-semistable.

5. Invariants

5.1. The class of a triple. — There is a group homomorphism cl : K0(A) → Z³ defined, for anyE= (E₁, E₀, s_E)∈ A, by

cl(E) = (rkE₁,degE₁,rkE₀).

For anyE= (E₁, E₀, s_E)∈ AandF= (F₁, F₀, s_F)∈ A, define

(5.1) χ(E, F) =

2

X

k=0

(−1)^kdim Ext^k_A(E, F).

Then

(5.2) χ(E, F) =χ(E0, F0) +χ(E1, F1)−χ(E0, F1).

Therefore, assumingclE=α= (α, v),clF=β = (β, w), we obtain χ(E, F) = (1−g)vw+χ(α, β)−vχ(β) Define

χ(α, β) = (1−g)vw+χ(α, β)−vχ(β), α, β

=χ(α, β)−χ(β, α) =hα, βi −vχ(β) +wχ(α).

5.2. Integration map. — Define the quantum affine planeAto be the completion of the algebraR[x1, x^±1₂ , x3](as in Section 2.4) with multiplication

x^α◦x^β = (−q^1/2)^hα,βix^α+β, where we allow only elementsf =P

α∈N×Z×Nfαx^αwith infn d

r+ 1

f_r,d,v6= 0o

>−∞.

Let H(A) be the Hall algebra of the category A. Define an integration map I:H(A)→A

E= (E1, E0, sE)7→(−q^1/2)^χ(E,E) x^clE [AutE].

Remark 5.1. — This integration map restricts to an algebra homomorphism I : H(A0) → A0 considered in Section 2.4. Note, however, that I : H(A) → A is not an algebra homomorphism. But if Ext²(F, E) = 0 then (see e.g. the proof of [20, Lem. 3.3])

I([E]◦[F]) =I([E])◦I([F]).

5.3. The wall-crossing formula. — Letα= (r, d)∈Z>0×Zand letτ∈R. Recall that M(α) denotes the moduli stack of semistable vector bundles over X having rankrand degreed. Let

(5.3) u(α) = (−q^1/2)^χ(α,α)+d[M(α)] = (−q^1/2)^χ(α)βα

be the motivic invariant “counting” (unframed) semistable bundles E over X with chE=α. Similarly, we define an invariant

(5.4) f_τ(α) = (q−1)(−q^1/2)χ(α,α)−χ(α)+d[M_τ(α)] = (q−1)q^(1−g)(^r₂)[Mτ(α)],

“counting” framedτ-semistable triplesE∈ Awiths_E6= 0andchE₁=α. Our main goal is to compute these invariants.

Letu^h(α)∈H(A0) andf^h_τ(α)∈ H(A)be elements in the Hall algebras counting semistable vector bundles and framed τ-semistable triples as above. Then

u(α)x^α= (−q^1/2)^dI(u^h(α)).

IfEis a triple withclE= (α,1)thenχ(E, E) = (1−g) +χ(α, α)−χ(α). This implies fτ(α)x^(α,1)= (q−1)(−q^1/2)^g−1+dI(f^h_τ(α)).

Define

(5.5) u_τ = 1 + X

µ(α)=τ

u(α)x^α∈A0, f_τ =X

α

f_τ(α)x^(α,1). We will see in the next lemma that f_τ ∈A. Finally, define

(5.6) u_>τ =

y

Y

τ⁰>τ

uτ⁰, where the product is taken in the decreasing slope order.

Lemma5.2. — If fτ(r, d)6= 0 thend>0andd/r6τ < d/(r−1).

Proof. — We know from [5, Th. 6.1] that if there exists aτ-stable triple inMτ(r, d) then d/r 6τ < d/(r−1). Assume now that there exists a τ-semistable triple E ∈ M_τ(r, d). Then, according to Lemma 4.10, there exists a τ₊-stable framed object E⁰⁰⊂Ewithµ(E/E⁰⁰) =τ. LetclE⁰⁰= (r⁰⁰, d⁰⁰,1)andch(E/E⁰⁰) = (r⁰, d⁰). Then

d⁰⁰

r⁰⁰ 6τ < d⁰⁰

r⁰⁰−1, d⁰ r⁰ =τ.

This implies

d⁰+d⁰⁰

r⁰+r⁰⁰ 6τ < d⁰+d⁰⁰ r⁰+r⁰⁰−1.

Finally, the inequalityd/r6d/(r−1)impliesd>0.

Remark 5.3. — Let E be a framed ∞-semistable object with chE1 = (r, d). This means thatcokersEis a finite sheaf. Thereforer= 1,d>0andcokersEhas lengthd.

The endomorphism ring ofE equalsk. The moduli spaceM∞(1, d)can be identified with a Hilbert schemeHilb^dX 'S^dX. Therefore

(5.7) f_∞=x1x3

X

d>0

[S^dX]x^d₂=x1x3ZX(x2).

For any series of the formf =P

αf_αx^(α,1)∈A, define its truncation f|µ<τ = X

µ(α)<τ

fαx^(α,1). Theorem5.4. — For any τ∈R we have

fτ = u⁻¹_>τ◦f_∞◦u_>τ

_µ

6τ. In particular

f_τ−=

u⁻¹_>τ◦f_∞◦u_>τ

_µ<τ, f_τ+= u⁻¹_>τ◦f_∞◦u_{>τ µ}

6τ. Proof. — Let

u^h_τ = 1 + X

µ(α)=τ

u^h(α), f^h_τ =X

α

f^h_τ(α)

be the elements of the completed Hall algebras. Note that f^h_τ(α) = 0 if µ(α) > τ. Therefore

f^h_τ =f^h_{τ µ}

6τ, f^h_τ+ =f^h_{τ+ µ}

6τ, f^h_τ− =f^h_{τ− µ<τ}. It follows from Lemma 4.10 and Lemma 4.12 that

f^h_τ =f^h_τ+◦u^h_τ, f^h_τ

_µ<τ =u^h_τ◦f^h_τ−.

Applying the integration map and using Remarks 4.11 and 5.1, we obtain f_τ=f_τ+◦u_τ, f_τ|_µ<τ =u_τ◦f_τ−.

This implies

fτ− = u⁻¹_τ ◦fτ+◦uτ

_µ<τ.

Applying the same formula for allτ⁰>τ we obtain f_τ− =

u⁻¹_>τ◦f_∞◦u_{>τ µ<τ}.

The other statements of the theorem are derived from this formula.

6. Zagier-type formula

The wall-crossing formula Theorem 5.4 for the motives of the moduli of stable pairs can now be made explicit, since explicit formulas for all three series are available.

Namely, we havef∞=x1x3P

d>0[S^dX]x^d₂ by formula (5.7); writing u_>τ = 1 + X

µ(α)>τ

a^>_α^τ(−q^1/2)^χ(α)x^(α,0), u⁻¹_>τ = 1 + X

µ(α)>τ

c^>τ_α (−q^1/2)^χ(α)x^(α,0),

we have by Remark 8.11 (replacing each sequence (r₁, . . . , r_k) by (r_k, . . . , r₁)) and Remark 8.14:

a^>_(r,d)^τ = X

r1+···+r_k=r

br1. . . br_kq^−(r−r1)d

k−1

Y

i=1

q^{(ri+ri+1)dr>i+1τe} 1−q^ri+ri+1 , c^>τ_(r,d)=− X

r1+···+rk=r

b_r1. . . b_rkq^(r−rk)d

k−1

Y

i=1

q^{−(ri+ri+1)br6iτc} 1−q^ri+ri+1 , wherer₆i=r1+· · ·+ri andr_>i+1=ri+1+· · ·+rk.

For everyr>2, comparison of coefficients ofx^(r,d,1) in Theorem 5.4 yields fτ(r, d) =X

e>0

[S^eX]a^>_{(r−1,d−e)}^τ q^d−re+X

e>0

[S^eX]c^>τ_{(r−1,d−e)}q(1−g+e)(r−1)

+X

e>0

X

r−1=r⁰+r⁰⁰

X

d−e=d⁰+d⁰⁰

[S^eX]c^>τ_(r0,d⁰)b^>_(r^τ00,d⁰⁰)q^{(1−g+e)r0−r00e+(r0+1)d00−r00d0}.

We insert the above formulas fora^>_α^τ andc^>τ_α into this expression. First, this bounds the summation over e to e 6 d−(r−1)τ, resp. to e < d−(r−1)τ, in the first resp. second sum. Second, the resulting summation over decompositionsr=r⁰+r⁰⁰, together with decompositions r⁰ = r⁰₁+· · ·+r⁰_k0 and r⁰⁰ = r⁰⁰₁ +· · ·+r_k⁰⁰00, can be replaced by the summation over decompositionsr−1 =r₁+· · ·+r_k, together with the choice of an index p= 1, . . . , k−1 which splits the latter decomposition into a part (r1, . . . , rp) = (r⁰₁, . . . , r⁰_k0)and a part (rp+1, . . . , rk) = (r⁰⁰₁, . . . , r⁰⁰_k00). This gives the following formula forfτ(r, d)(withr∗ indicating summation over decompositions

ofr−1as before):

bd−(r−1)τc

X

e=0

[S^eX]X

r∗

b_r1. . . b_rkq^{−(r−1−r1})(d−e)+d−re k−1

Y

i=1

q^{(ri+ri+1)dr>i+1τe} 1−q^ri+ri+1

−

dd−(r−1)τe−1

X

e=0

[S^eX]X

r∗

br₁. . . br_kq^(r−1−rk)(d−e)+(1−g+e)(r−1) k−1

Y

i=1

q^{−(ri+ri+1)br6iτc} 1−q^ri+ri+1

−X

e>0

X

r∗

br₁. . . br_k

X

d−e=d⁰+d⁰⁰ k−1

X

p=1

[S^eX]q^C

p−1

Y

i=1

q^{−(ri+ri+1)br6iτc} 1−q^ri+ri+1

k−1

Y

i=p+1

q^{(ri+ri+1)dr>i+1τe} 1−q^ri+ri+1 , where

C=r_6p−1d⁰−r_>p+2d⁰⁰+ (1−g+e)r_6p−r_>p+1e+ (r_6p+ 1)d⁰⁰−r_>p+1d⁰. We consider the summation overd⁰andd⁰⁰. We can replaced⁰=d−e−d⁰⁰; thend⁰⁰ is bound by

r_>p+1τ6d⁰⁰< d−e−r_6pτ,

and thuse < d−(r−1)τ. Analyzing the occurrences ofd⁰ andd⁰⁰in theq-exponentC above, this shows that the only part of the last sum depending ond⁰⁰ is

dd−e−r_6pτe−1

X

d⁰⁰=dr_>p+1τe

q^{(rp+rp+1+1)d00}= q^{(rp+rp+1+1)dr>p+1τe}−q^{(rp+rp+1+1)dd−e−r6pτe} 1−q^rp+rp+1+1 . We can change the order of summation, summing over decompositions ofr−1and overefirst (adding one extra term for the cased−(r−1)τ ∈N), which gives:

Lemma6.1. — Forr>2, the motivic invariantf_τ(r, d) is given by

(6.1) X

r1+···+rk=r−1

br₁. . . br_k

Qk−1

i=1 (1−q^ri+ri+1)

[S^{d−(r−1)τ}X]q^A0δ_{d−(r−1)τ∈N}

+

dd−(r−1)τe−1

X

e=0

[S^eX]·

q^A−q^B−

k−1

X

p=1

q^Cp(q^Dp−q^Ep) 1−q^rp+rp+1 1−q^rp+rp+1+1

, where

A₀= (r−1)((r₁+ 1)τ−d) +Pk−1

i=1(r_i+r_i+1)dr_>i+1τe, A=−(r−1−r1)(d−e) +d−re+Pk−1

i=1(ri+ri+1)dr_>i+1τe, B = (r−1−rk)(d−e) + (1−g+e)(r−1)−Pk−1

i=1(ri+ri+1)br₆iτc, Cp = (1−g)r₆p+ (r_6p−1−r_>p+1)d+rpe−Pp−1

i=1(ri+ri+1)br₆iτc +Pk−1

i=p+1(r_i+r_i+1)dr_>i+1τe, Dp = (rp+rp+1+ 1)dr_>p+1τe,

Ep = (rp+rp+1+ 1)dd−e−r₆pτe.

Consider the case of(r, d)-genericτ, i.e.τ 6=d/r andτ /∈ _r¹0Zfor all16r⁰< r.

Theorem6.2. — Forr>2,d∈Z, and(r, d)-generic τ∈R, we have [M_τ(r, d)] =q^(g−1)(^r₂) X

r1+···+rk=r−1

br₁. . . br_k

Qk−1

i=1 (1−q^ri+ri+1)coeff_td−d(r−1)τe

Z_X(t)· q^F0 1−q^r1+1t−

k−1

X

p=1

q^Fp(1−q^rp+rp+1)t^δp

(1−q^rp+1+1t)(1−q^−rpt)− q^Fk 1−q^−rkt

, where

Fp= (1−g)r₆p+ (r₆p−r_>p+1)d−rpdr₆pτe+ (rp+1+ 1)dr_>p+1τe

−

p−1

X

i=1

(r_i+r_i+1)br_6iτc+

k−1

X

i=p+1

(r_i+r_i+1)dr_>i+1τe andδp equals1 if{r₆pτ}+{r_>p+1τ}<1 and zero otherwise.

Proof. — In the formula of the previous lemma, we perform summation overeusing the simple identity

N

X

e=0

[S^eX]q^ae=coeff_tN

ZX(t) q^aN 1−q^−at

for N > 0 and a ∈ Z. This simplifies the term Pd−d(r−1)τe

e=0 [S^eX]q^A in the above formula to

coeff_td−d(r−1)τe

ZX(t)q^{−(r−1)d+(r1}+1)d(r−1)τe+Pk−1

i=1(ri+ri+1)dr_>i+1τe

1−q^r1+1t

;

note that theq-exponent equalsF0ifr0is interpreted as zero. Similarly we treat the termPd−d(r−1)τe

e=0 [S^eX]q^B, interpretingrk+1 as zero.

After some calculation, the termPd−d(r−1)τe

e=0 [S^eX](q^Cp+Dp−q^Cp+Ep)is rewritten as thet^{d−d(r−1)τe}-coefficient of

ZX(t)q^{(1−g)r6p+(r6p−r>p+1)d−Pp−1i=1(ri+ri+1)br6iτc+Pk−1i=p+1(ri+ri+1)dr>i+1τe}

·q^{(rp+rp+1+1)dr>p+1τe−rpd(r−1)τe}

1−q^−rpt −q^{−(rp+rp+1+1)br6pτc+(rp+1}+1)d(r−1)τe

1−q^rp+1+1t

. We use the following simple identity which holds for all a, b, c, d ∈ Z and generic τ∈R:

q(a+b)ddτe−ad(c+d)τe

1−q^−at −q−(a+b)bcτc+bd(c+d)τe

1−q^bt = qbddτe−adcτe(1−q^a+b)t^δ(c,d,τ) (1−q^−at)(1−q^bt) , where δ(c, d, τ)equals one if {cτ}+{dτ} <1 and zero otherwise. We apply this to a=r_p,b=r_p+1+ 1,c=r_6p andd=r_>p+1 to simplify the previous expression to

coeff_td−d(r−1)τe

ZX(t) q^Fpt^{δ(r6p,r>p+1,τ)} (1−q^rp+1+1t)(1−q^−rpt)

.