Journal of Geometry and Physics 61 (2011) 773–780

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Journal of Geometry and Physics

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Direct image and parabolic structure on symmetric product of curvesIndranil Biswas a,∗, Fatima Laytimi ba School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, Indiab Mathématiques – Bât. M2, Université Lille 1, F-59655 Villeneuve d’Ascq Cedex, France

a r t i c l e i n f o

Article history:Received 26 June 2010Received in revised form 29 November2010Accepted 3 December 2010Available online 23 December 2010

MSC:14F0514H60

Keywords:Parabolic bundleSymmetric productDirect image

a b s t r a c t

Given a complex smooth projective curve X and a vector bundle E on it; there is acorresponding vector bundle F (E) on the symmetric product Sn(X) for any n. We showthat there is a natural parabolic structure on the vector bundle F (E). We prove that F (E)is parabolic semistable (respectively, parabolic polystable) if and only if E is semistable(respectively, polystable). If E is not the trivial line bundle on X , then we prove that F (E)is parabolic stable if and only if E is stable.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Let E be a vector bundle over an irreducible smooth projective curve defined over the field of complex numbers. Take asymmetric power Sn(X) of X for some positive integer n. Let q1 (respectively, q2) be the projection of Sn(X) × X on Sn(X)(respectively, X). Let∆ ⊂ Sn(X)× X be the universal divisor. The direct image

F (E) := q1∗(O∆ ⊗OSn(X)×X q∗

2E)

is a vector bundle over Sn(X). These vector bundles F (E) are extensively studied (see [1,2]).The symmetric product Sn(P1

C) is isomorphic to PnC. In this case the vector bundle F (E) −→ Pn

C is known to be stable ifE is a line bundle [3].

Parabolic vector bundles on curves were introduced in [4], and parabolic vector bundles on higher dimensional varietieswere introduced in [5]. They are vector bundles equipped with certain extra structures over a fixed divisor.

Our first aim here is to show that the vector bundle F (E) has a natural parabolic structure. This is done by identifyingF (E)with another vector bundle.

Let f be the natural projection of Xn to Sn(X), and let pi, 1 ≤ i ≤ n, be the projection of Xn to the ith factor. Note thatSn(X) is the quotient of Xn for the action of the permutation groupΣ(n) of {1, . . . , n}. This action ofΣ(n) lifts to the directsumE :=

ni=1 p

∗

i E.We prove the following (see Proposition 2.1):

Proposition 1.1. The vector bundle F (E) is canonically isomorphic to the invariant direct image

VE := (f∗E)Σ(n) ⊂ f∗E.∗ Corresponding author.

E-mail addresses: indranil@math.tifr.res.in (I. Biswas), Fatima.Laytimi@math.univ-lille1.fr (F. Laytimi).

0393-0440/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.geomphys.2010.12.005

774 I. Biswas, F. Laytimi / Journal of Geometry and Physics 61 (2011) 773–780

In Section 3 we construct a parabolic structure on the vector bundle VE on Sn(X). This construction is similar to theconstruction in [6] of a parabolic vector bundle from an orbifold vector bundle. It should be clarified that in [6], the parabolicdivisor is assumed to be a normal crossing divisor. This assumption is needed to construct an orbifold bundle fromaparabolicbundle. More precisely, it is needed to be able to use the ‘‘covering lemma’’ of Kawamata to construct a suitable Galoiscovering whose total space is smooth. Since in the situation here we are given the Galois covering f : Xn

−→ Sn(X), thisassumption on the parabolic divisor is not needed here.

Using Proposition 1.1, the parabolic structure on VE produces a parabolic structure on F (E). The resulting parabolicvector bundle is denoted by F (E)∗.

We prove the following (see Lemmas 4.2 and 4.3):

Lemma 1.2. The parabolic vector bundleF (E)∗ is parabolic semistable (respectively, parabolic polystable) if and only if the vectorbundle E is semistable (respectively, polystable).

Lemma 1.2 raises the question whether the same statement on stability is true.If we take E to be the trivial line bundle on X , then the corresponding parabolic vector bundle is not parabolic stable if

n > 1; if n = 1, then Sn(X) = X , and the parabolic bundle is the trivial line bundle equipped with the trivial parabolicstructure.

If E is not the trivial line bundle, then the following is proved (see Theorem 4.4):

Theorem 1.3. Assume that E −→ X is not isomorphic to the trivial line bundle on X. The parabolic vector bundle F (E)∗ isparabolic stable if and only if the vector bundle E is stable.

2. Direct image and an isomorphism

Let X be an irreducible smooth projective curve defined over C. Fix a positive integer n. Let Σ(n) be the group ofpermutations of {1, . . . , n}. The Cartesian product Xn is considered as the space ofmaps from {1, . . . , n} to X . The symmetricgroupΣ(n) has a tautological action of Xn. The quotient Xn/Σ(n) is the symmetric product Sn(X). Let

f : Xn−→ Sn(X) (2.1)

be the quotient morphism.Let ∆′

⊂ (Xn) × X be the reduced effective divisor parametrizing all (α, x) such that x ∈ image(α), where α is a mapfrom {1, . . . , n} to X . Let

∆ := (f × IdX )(∆′) ⊂ Sn(X)× X (2.2)

be the image of∆′; it is known as the universal divisor (see [1, p. 336]). Let

q1 : Sn(X)× X −→ Sn(X) and q2 : Sn(X)× X −→ X

be the natural projections. Given a vector bundle E on X , we have the vector bundle

F (E) := q1∗(O∆ ⊗OSn(X)×X q∗

2E) −→ Sn(X), (2.3)

where O∆ is the structure sheaf of the subscheme∆ ⊂ Sn(X)× X . We note that rank(F (E)) = rank(E)n.The Chern character of F (E) is computed in [1]; we will recall it. Let

u : Sn(X) −→ Picn(X)

be the morphism that sends any unordered collection {x1, . . . , xn} to the divisor∑n

i=1 xi on X . Let

θ ∈ H2(Sn(X),Q)

be the pullback, by u, of the canonical theta class in H2(Picn(X),Q). Let

x ∈ H2(Sn(X),Q)

be the Künneth component of the Poincaré dual of ∆ lying in H2(Sn(X) × X,Q) (see [1, p. 338]). Then the Chern characterhas the following expression:

ch(F (E)) = degree(E)(1 − exp(−x))− r(g − 1)+ r(n + g − 1 + θ) exp(−x), (2.4)

where r = rank(E), and g = genus(X) [1, p. 340, Lemma 2.5] (see the last sentence in the proof of Lemma 2.5 of [1]). In [1,Lemma 2.5], this is proved under the assumption that E is a line bundle. For a vector bundle E, fix a filtration of subbundles ofE such that each successive quotient is a line bundle. This filtration produces a filtration of subbundles ofF (E). If the gradedobject for the filtered vector bundle E is

k Lk, then the graded object for the filtered vector bundle F (E) is

k F (Lk).

Hence (2.4) follows from the computation for line bundles.

I. Biswas, F. Laytimi / Journal of Geometry and Physics 61 (2011) 773–780 775

For each i ∈ [1, n], let

pi : Xn−→ X (2.5)

be the projection to the ith factor. For a vector bundle E over X , let

E :=

ni=1

p∗

i E −→ Xn (2.6)

be the direct sum. Consider the quotient morphism f in (2.1). There is a natural lift of the action of the Galois groupGal(f ) = Σ(n) on Xn to the vector bundleE defined in (2.6). Any element of Σ(n) permutes the fibers of E the sameway it permutes any collection of ordered n points of X .

Consider the direct image

f∗E −→ Sn(X),

where f is the morphism in (2.1). The above mentioned action ofΣ(n) onE produces an action ofΣ(n) on f∗E. This actionis a lift of the trivial action ofΣ(n) on Sn(X). Let

VE := (f∗E)Σ(n) ⊂ f∗E (2.7)

be the invariant part of the direct image. Since f is a finite proper morphism between smooth varieties, the direct image f∗Eis locally free. The subsheaf (f∗E)Σ(n) is locally free because it is a direct summand of the vector bundle f∗E.Proposition 2.1. The vector bundle VE in (2.7) is canonically isomorphic to F (E) constructed in (2.3).

Proof. Consider the Zariski open subset

U0 ⊂ Sn(X) (2.8)

that parametrizes distinct unordered n points of X; so U0 is the complement of the big diagonal. For any point

z = {z1, . . . , zn} ∈ U0,

both the fibers F (E)z and (VE)z are identified with the direct sumn

i=1 Ezi . So we have a natural identification of F (E)|U0with VE |U0 . We will show that this identification between F (E)|U0 and VE |U0 extends to a homomorphism F (E) −→ VE .

Take any pointz = {z1, . . . , zn} ∈ Xn.

Let z := f (z) ∈ Sn(X) be its image. This point z defines a zero dimensional subscheme of X of length n; this subscheme of Xwill also be denoted by z. Note that z is reduced if and only if the n points {z1, . . . , zn} inz are all distinct. The fiber of F (E)over the point z is as follows:

F (E)z = H0(z, E|z) (2.9)

[1, p. 339].Let

Γz ⊂ Σ(n)

be the isotropy subgroup of the pointz for the action of Gal(f ) = Σ(n) on Xn. Consider the action of Γz on the fiberEz ofEoverz obtained by restricting the action ofΣ(n) on the vector bundleE defined in (2.6) (recall thatΣ(n) acts onE). Let

(Ez)Γz ⊂Ezbe the space of invariants for the action of Γz onEz . We note that

(Ez)Γz = H0(z, E|zred),

where zred ⊂ z ⊂ X is the reduced subscheme. Therefore, using (2.9), the restriction homomorphism

H0(z, E|z) −→ H0(zred, E|zred)

produces a homomorphism

F (E)z −→ (VE)z . (2.10)

Using the homomorphism in (2.10), the identification between F (E)|U0 and (VE)|U0 extends to a homomorphism

φ : F (E) −→ VE (2.11)

between OSn(X)-coherent sheaves. The proposition will be proved by showing that φ is an isomorphism.

776 I. Biswas, F. Laytimi / Journal of Geometry and Physics 61 (2011) 773–780

Let U1 ⊂ Xn be a Zariski open subset such that U1 is left invariant by the action of Gal(f ) = Σ(n) on Xn. Let

s ∈ H0(U1,E|U1)Σ(n)

be an invariant section, whereE is constructed in (2.6). In view of the description of the fibers of F (E) given in (2.9), thissection s clearly defines a section of F (E) over U1. Therefore, we get a homomorphism

φ′: VE −→ F (E).

Both φ (constructed in (2.11)) and φ′ coincide over the open subset U0 (defined in (2.8)) with the natural identificationbetween F (E)|U0 and VE |U0 . Therefore, the restriction φ|U0 is the inverse of the restriction φ′

|U0 . From this it follows that φand φ′ are inverses of each other; see Lemma 2.2. This completes the proof of the proposition. �

Lemma 2.2. Let V be an algebraic vector bundle over a smooth projective variety M. Let

ϕ : V −→ V

be an endomorphism of V which is an isomorphism over a nonempty Zariski open subset U ⊂ M. Then ϕ is an isomorphismover M.

Proof. Let r be the rank of V . The endomorphism ϕ produces an endomorphism of the determinant bundle det V :=r V ,

det(ϕ) : det V −→ det V . (2.12)

Since det V is a line bundle, it follows that det(ϕ) is multiplication by an algebraic function on M; this function, which wewill denote by β , is nonzero over U because ϕ|U is an isomorphism. Since there are no nonconstant functions, the functionβ is nowhere vanishing. This implies that the homomorphism det(ϕ) in (2.12) is an isomorphism. Consequently, ϕ is anisomorphism. �

3. A parabolic structure on F (E)

For any ordered pair i, jwith 1 ≤ i < j ≤ n, let

Di,j := {{z1, . . . , zn} ∈ Xn| zi = zj} ⊂ Xn (3.1)

be the divisor. The subgroup of Σ(n) that fixes Di,j pointwise is generated by the element of order two that exchanges iand j keeping everything else unchanged. The action of the symmetric group Σ(n) on Xn preserves the divisor

∑i,j Di,j.

Therefore, the action of Σ(n) on Xn lifts to an action of Σ(n) on the line bundle OXn−∑

i,j Di,j. In other words, the line

bundle OXn−∑

i,j Di,jis equipped with aΣ(n)-linearization.

As before, E is a vector bundle over X . Define

E ′:=E ⊗ OXn

−

−i,j

Di,j

, (3.2)

whereE is constructed in (2.6) and Di,j is defined in (3.1). The actions ofΣ(n) onE and OXn−∑

i,j Di,jtogether define an

action ofΣ(n) on the vector bundleE ′ in (3.2).Consider the direct image

f∗E ′−→ Sn(X),

where f is the morphism in (2.1). The action ofΣ(n) onE ′ produces an action ofΣ(n) on f∗E ′. Let

V ′

E := (f∗E ′)Σ(n) ⊂ f∗E ′ (3.3)

be the invariant part of the direct image. This coherent sheaf V ′

E is locally free for the same reason VE defined in (2.7) islocally free.

We note that V ′

E is a subsheaf of VE becauseE ′ is a subsheaf ofE. The inclusion map

V ′

E ↩→ VE (3.4)

is an isomorphism over the open subset U0 defined in (2.8). We will construct a parabolic structure on VE .Let

D := Sn(X) \ U0 ⊂ Sn(X) (3.5)

be the hypersurface. Let OD be the structure sheaf of D defined in (3.5). Let

VE |D := VE ⊗OSn(X) OD and V ′

E |D := V ′

E ⊗OSn(X) OD

I. Biswas, F. Laytimi / Journal of Geometry and Physics 61 (2011) 773–780 777

be the restrictions, to D, of VE and V ′

E respectively. The homomorphism in (3.4) produces a homomorphism

h : V ′

E |D −→ VE |D

of coherent sheaves on D. Let

I := image(h) ⊂ VE |D (3.6)

be the image.The parabolic structure on VE has the following description:The parabolic divisor is D. The quasiparabolic filtration over D is the filtration

I ⊂ VE |D

in (3.6), and the parabolic weights are 1/2 and 0.

Definition 3.1. The parabolic vector bundle constructed above with VE as the underlying vector bundle will be denotedby VE∗.

In [5], Maruyama and Yokogawa defined parabolic vector bundles as filtrations of coherent sheaves, parametrized by R,on the ambient variety satisfying certain conditions (see [5, p. 80, Definition 1.2]). This filtration for the parabolic bundleVE∗ in Definition 3.1 is the following:

The coherent sheaf for 0 ∈ R is VE , for any t ∈ (0, 1/2] the coherent sheaf is V ′

E , and for any t ∈ (1/2, 1] the coherentsheaf isVE ⊗OSn(X) OSn(X)(−D) (since the sheaf for t+1 is determined by the sheaf for t , these conditions uniquely determinethe filtration parametrized by R).

4. Stability criterion for VE∗

Fix a polarization ζ ∈ H2(Sn(X),Q); thismeans that ζ is a positivemultiple of the first Chern class of an ample line bundleover Sn(X). A description of the cohomology algebra H⋆(Sn(X),Q) is given in [7, p. 321, (3.1)]. The degree of a coherent sheafV on Sn(X), which is denoted by degree(V ), is defined as follows:

degree(V ) := (c1(V ) ∪ ζ n−1) ∩ [Sn(X)] ∈ Q.

Let V∗ be a parabolic vector bundle over Sn(X) defined by a filtration {Vt}t∈R of coherent sheaves. Let D0 ⊂ be the divisorthat supports the parabolic structure of V∗. The parabolic degree of V∗, which is denoted by par-deg(V∗), is defined to be

par-deg(V∗) :=

∫ 1

0degree(Vt)dt + rank(V∗) · degree(OSn(X)(D0))

(see [5, p. 82, (1.9.2)]).

Remark 4.1. Let c0 = 1 ∈ H2(X,Q) = Q be the polarization on X . Let∑n

i=1 p∗

i c0 be the polarization on Xn, where pi isthe projection in (2.5). There is a unique polarization ζ0 on Sn(X) such that f ∗ζ0 =

∑ni=1 p

∗

i c0, where f is the projection in(2.1). The parabolic degree of a parabolic vector bundle V∗ on Sn(X) with respect to this polarization ζ0 will be denoted bypar-deg0(V∗). Consider the parabolic vector bundle VE∗ in Definition 3.1. It is straight forward to check that

par-deg0(VE∗) = degree(E)/(n − 1)!.

Let V be a vector bundle over Sn(X) equipped with a parabolic structure; this parabolic sheaf will be denoted by V∗. Theparabolic sheaf V∗ is called parabolic stable (respectively, parabolic semistable) if for any coherent subsheaf W ⊂ V with0 < rank(W ) < rank(V ),

par-deg(W∗)

rank(W )<

par-deg(V∗)

rank(V )

respectively,

par-deg(W∗)

rank(W )≤

par-deg(V∗)

rank(V )

,

where W∗ is the parabolic sheaf defined by the parabolic structure on the subsheaf W induced by the parabolic structureon V .

A parabolic semistable sheaf is called parabolic polystable if it is a direct sum of parabolic stable sheaves.

Lemma 4.2. The parabolic vector bundle VE∗ in Definition 3.1 is parabolic semistable if and only if the vector bundle E over X issemistable.

Proof. Since f in (2.1) is a finite morphism, the pullback f ∗ζ ∈ H2(Xn,Q) is a polarization on Xn. We will first show that Eis semistable if and only if the vector bundleE −→ Xn defined in (2.6) is semistable with respect to the polarization f ∗ζ .

778 I. Biswas, F. Laytimi / Journal of Geometry and Physics 61 (2011) 773–780

Take any i ∈ [1, n]. The vector bundle E is semistable if and only if the vector bundle p∗

i E, where pi is defined in (2.5),is semistable with respect to f ∗ζ ; to see this first note that statement (1) in [8, Theorem 1.2] holds for E, hence statement(4) in [8, Theorem 1.2] holds for E, therefore, statement (4) in [8, Theorem 1.2] holds for p∗

i E, hence p∗

i E is semistable bystatement (1) in [8, Theorem 1.2].

Take any i, i ∈ [1, n]. We will show that p∗

j E is semistable with respect to the polarization f ∗ζ if and only if p∗

i E issemistable with respect to f ∗ζ . We will also show that the degree if p∗

j E with respect to f ∗ζ coincides with that of p∗

i E withrespect to f ∗ζ . To prove these, let

τ : Xn−→ Xn

be the automorphism that interchanges the ith and jth coordinates keeping the rest unchanged; so

pi ◦ τ = pj, pj ◦ τ = pi, (4.1)

and pk ◦ τ = pk for all k = i, j. From (4.1),

τ ∗p∗

j E = p∗

i E. (4.2)

Note that the following diagram is commutative:

Xn τ−→ Xnf f

Sn(X)Id

−→ Sn(X).

Therefore,

τ ∗f ∗ζ = f ∗ζ . (4.3)

From (4.3) it follows immediately that for any vector bundle V on Xn, the degrees of V and τ ∗V with respect to f ∗ζ coincide.In particular, degrees of p∗

j E and p∗

i E with respect to f ∗ζ coincide (see (4.2)). Hence using (4.2) we conclude that p∗

j E issemistable with respect to f ∗ζ if and only if p∗

i E is semistable with respect to f ∗ζ .Since the degrees of p∗

i E and p∗

j E with respect to f ∗ζ coincide, and p∗

j E is semistable with respect to f ∗ζ if and only if p∗

i Eis semistable with respect to f ∗ζ , it follows that the vector bundle p∗

i E is semistable with respect to f ∗ζ if and only if thedirect sumE (defined in (2.6)) is semistable with respect to f ∗ζ .

Lemmas 3.16 and 2.7 of [6] together imply thatE is semistable with respect to f ∗ζ if and only if the parabolic vectorbundle VE∗ is parabolic semistable. �

Lemma 4.3. The parabolic vector bundle VE∗ is parabolic polystable if and only if the vector bundle E is polystable.

Proof. First assume that the vector bundle E is polystable. Since E is polystable, the projective bundle P(E) is given by aprojective unitary representation of the fundamental group

π1(X, x0) −→ PU(r),

where r = rank(E), and x0 is a point of X [9]. Therefore, the endomorphism bundle End(E) = E ⊗ E∗ is given by a unitaryrepresentation

ρ : π1(X, x0) −→ U(r2).

Letx0 be the point (x0, x0, . . . , x0) of Xn. For any i ∈ [1, n], let

ρi := ρ ◦ pi∗

be the composition, where pi∗ : π1(Xn,x0) −→ π1(X, x0) is the homomorphism induced by the projection pi in (2.5). Theendomorphism bundle End(E) of the vector bundle in (2.6) is given by the unitary representation

ni=1

ρi : π1(Xn,x0) −→ U(nr2).

Therefore, the vector bundle End(E) is polystable. HenceE is polystable [10, Corollary 3.8]. This implies that the parabolicbundle VE∗ is parabolic polystable (see [11, Theorem 4.3]).

To prove the converse, assume that VE∗ is parabolic polystable. Therefore, the vector bundleE on Xn is polystable [11,Theorem 4.3]. SinceE is polystable, the direct summand p∗

1E ofE is also polystable. Note that

ck(End(p∗

1E)) = p∗

1ck(End(E)) = 0

for all k > 0. Therefore, the polystable vector bundle End(p∗

1E) is given by a unitary representation of π1(Xn,x0) [12,13].

I. Biswas, F. Laytimi / Journal of Geometry and Physics 61 (2011) 773–780 779

The vector bundle End(E) is isomorphic to the pullback of End(p∗

1E) by the embedding X −→ Xn defined by x −→

(x, x0, x0, . . . , x0). From this it follows that End(E) is given by a unitary representation of π1(X, x0) because End(p∗

1E) isgiven by a unitary representation of π1(Xn,x0). Hence End(E) is polystable. From this it follows that the vector bundle E ispolystable [10, Corollary 3.8]. �

Consider the trivial line bundle OX on X . In this case the vector bundle defined in (2.6) is the trivial vector bundle O⊕nXn ;

but the action of the Galois groupΣ(n) is nontrivial. We have

par-deg(VOX∗) = degree(O⊕nXn )/n! = 0. (4.4)

There is a natural inclusion of OSn(X) in VOX∗ which can be constructed as follows. Send any algebraic function ψ definedover a Zariski open subset U ⊂ Sn(X) to the section (ψ ◦ f , ψ ◦ f , . . . , ψ ◦ f ) of O⊕n

Xn over the open subset f −1(U) ⊂ Xn,where f is the quotient map in (2.1). This section of O⊕n

Xn over f −1(U) is left invariant by the action ofΣ(n). Hence we get ahomomorphism of vector bundles

OSn(X) ↩→ (f∗O⊕nXn )

Σ(n)= VOX

(see (2.7)). The parabolic structure on the above subbundle OSn(X) induced by the parabolic structure on VOX∗ is the trivialone. Hence the parabolic degree of the subbundle OSn(X) equipped with the induced parabolic structure is zero. Therefore,the parabolic vector bundle VOX∗ is not parabolic stable if n ≥ 2. Note that VOX∗ is parabolic polystable by Lemma 4.3.

Theorem 4.4. Assume that E −→ X is not isomorphic to the trivial line bundle on X. The parabolic vector bundleVE∗ is parabolicstable if and only if the vector bundle E is stable.

Proof. Let W∗ be a parabolic vector bundle with W as the underlying vector bundle. Then a parabolic endomorphism of W∗

is a global endomorphism ofW preserving the parabolic filtration.Since a polystable vector bundle is a direct sum of stable bundles, a polystable vector bundle V is stable if and only if all

the global sections of End(V ) are scalar multiplications. Similarly, parabolic polystable vector bundleW∗ is parabolic stableif and only if all the global parabolic endomorphisms ofW∗ are scalar multiplications.

We have an injective homomorphism

δ : H0(X, End(E)) −→ H0(Xn, End(E))that sends any h ∈ H0(X, End(E)) to (p∗

1h, . . . , p∗nh). We note that

image(δ) ⊂ H0(Xn, End(E))Σ(n)(the space of invariants for the action ofΣ(n) on H0(Xn, End(E)) given by the action ofΣ(n) onE). Let

δ0 : H0(X, End(E)) −→ H0(Xn, End(E))Σ(n) (4.5)

be the homomorphism given by φ. Any Σ(n)-equivariant endomorphism of E produces a parabolic endomorphismof the corresponding parabolic vector bundle VE∗. This identifies H0(Xn, End(E))Σ(n) with the space of all parabolicendomorphisms of VE∗ (see [11, p. 344]).

Since δ0 in (4.5) is injective, if E is polystable but not stable, thenVE∗ is not parabolic stable because there are other globalparabolic endomorphisms of VE∗ apart from the scalar multiplications.

To prove the converse, assume that E is stable. We have an injective homomorphism

η : Cn−→ H0(Xn, End(E))

defined by

(a1, . . . , an) −→

ni=1

ai · Idp∗i E

∈ H0(Xn, End(E)),where pi is the projection in (2.5). Since E is nontrivial and stable, there is no nonzero global homomorphism from p∗

i E top∗

j E if j = i. Also, all global endomorphisms of E are scalar multiplications because E is stable. Using these it follows that theabove homomorphism η is surjective.

The homomorphism η evidently commutes with the actions ofΣ(n) on H0(Xn, End(E)) and Cn. Since any element of Cn

preserved by the action ofΣ(n) on Cn lies in the line C · (1, . . . , 1), we conclude thatH0(Xn, End(E))Σ(n) coincides with thespace of scalar multiplications. Hence all the global parabolic endomorphisms of VE∗ are scalar multiplications. Therefore,VE∗ is parabolic stable. �

Remark 4.5. We noted that VOX∗ is parabolic polystable, but it is not parabolic stable. We also noted in (4.4) that theparabolic degree of VOX∗ is zero. The parabolic stable subbundles of VOX∗ of parabolic degree zero are identified withthe irreducible submodules of the Σ(n)-module Cn. This follows from the fact that the Σ(n)-linearized vector bundle OX(defined in (2.6)) is identified with the trivial vector bundle Xn

× Cn−→ Xn.

780 I. Biswas, F. Laytimi / Journal of Geometry and Physics 61 (2011) 773–780

Acknowledgement

The first author wishes to thank the Université Lille 1, where the work was carried out, for its hospitality.

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