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ISRN Algebra
Volume 2011 (2011), Article ID 659672, 12 pages
http://dx.doi.org/10.5402/2011/659672
Research Article

Examples of Stability of Tensor Products in Positive Characteristic

Universidad Autónoma, “Benito Juárez” de Oaxaca, Mexico

Received 5 June 2011; Accepted 3 July 2011

Academic Editor: V. Futorny

Copyright © 2011 A. S. Argáez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let X be projective smooth variety over an algebraically closed field k and let , be μ-semistable locally free sheaves on X. When the base field is , using transcendental methods, one can prove that the tensor product is always a μ-semistable sheaf. However, this theorem is no longer true over positive characteristic; for an analogous theorem one needs the hypothesis of strong μ-semistability; nevertheless, this hypothesis is not a necessary condition. The objective of this paper is to construct, without the strongly μ-semistability hypothesis, a family of locally free sheaves with μ-stable tensor product.

1. Introduction

When the base field is , the Kobayashi-Hitchin correspondence ensures that a vector bundle on a complex projective variety is polystable if and only if it admits a Hermitian-Einstein metric with respect to the Kähler metric induced by 𝒪𝑋(1) (see [1] for curves and [2] for complex compact varieties). In this way, one can prove that the tensor product of Hermitian-Einstein bundles is again Hermitian-Einstein, therefore polystable, and the same is true for symmetric and exterior products. However, in positive characteristic, this is false; in [3], Gieseker proved the existence of stable vector bundles on curves with nonsemistable symmetric products. When these bundles are of degree zero, the nonsemistability of the symmetric product imply the nonsemistability of the tensor product. One way to solve this problem is to introduce the concept of strong 𝜇-semistability. Let 𝐹𝑋𝑋 be a total Frobenius morphism of 𝑋; we say that is strongly 𝜇-semistable if for all 𝑚0 the pullback (𝐹𝑚) is 𝜇-semistable with respect to the induced polarization (𝐹𝑚)𝒪𝑋(1). Under these assumptions, the tensor product of strongly 𝜇-semistable bundles is again strongly 𝜇-semistable (see [4, Section 7] for curves and [5] for general case). However, in general, there are no conditions to ensure the 𝜇-semistability of a tensor product of 𝜇-semistable bundles (at least not known to the author).

The aim of this work is the construction of examples, in any characteristic, of families of 𝜇-stable bundles with 𝜇-stable tensor products, this without the assumption of strong 𝜇-stability. The key result is Proposition 3.1, which shows that, if 𝜋𝑌𝑋 is étale and Galois with g.c.d.(|𝐺|,char(𝑘))=1, then is 𝜇-semistable (𝜇-polystable) if and only if 𝜋 is 𝜇-semistable (𝜇-polystable, resp.). We remark that, in positive characteristic, this result is false for an arbitrary finite morphism. Under these hypotheses, if is a line bundle on 𝑌, in Corollary 3.2 we show that 𝜋 is 𝜇-(semi)polystable if is 𝜇-(semi)stable. If in addition g.c.d.(deg,𝑟𝑘())=1=(|𝐺|,𝑟𝑘()) and 𝑔 for all 𝑔𝐺, then in Corollary 3.5 it is proved that 𝜋 is 𝜇-stable. The other examples that are constructed come from the 𝐺-invariant decomposition of 𝜋𝒪𝑌. Now, the general outline of this construction is as follows. In Section 1, we observe that 𝜋𝒪𝑌 has a natural 𝐺-invariant decomposition: 𝜋𝒪𝑌=𝑉𝑉𝑘𝑉,(1.1) where is the set of irreducible representations of 𝐺 over 𝑘 and each 𝑉 is a locally free sheaf of rank dim𝑉. Also, we have the relations 𝑉1𝑉2𝑇𝑛𝑇𝑇,𝑠𝑉𝑊𝑛𝑊𝑊,(1.2) where 𝑉𝑊𝑇𝑇𝑛𝑇 and 𝑠𝑉=𝑊𝑊𝑛𝑊. In Section 2, we show that 𝑉 is 𝜇-𝐻-stable for any polarization on 𝑋 and also that 𝑉 is 𝜇-𝐻-polystable (𝜇-𝐻-semistable) whenever is 𝜇-𝐻-stable (𝜇-𝐻-semistable, resp.). Furthermore, let 𝑋(𝑟,𝑑) be the moduli space parametrizing 𝜇-𝐻-stable sheaves of rank 𝑟 and degree 𝑑, if g.c.d.(𝑑,𝑟)=1=g.c.d.(|𝐺|,𝑟), we proved in Theorem 3.12 that, for any irreducible representation 𝑉 of 𝐺 and 𝑋(𝑟,𝑑), 𝑉 is 𝜇-𝐻-stable and the natural morphism 𝑋(𝑟,𝑑)𝑋(𝑟dim𝑉,𝑑dim𝑉) given by 𝑉 is injective.

In Section 3, we proved that, if g.c.d.(𝑑,|𝐺|)=1, then, for all 𝑟>0, 𝐺 acts without fixed points on 𝑌(𝑟,𝑑); in particular the morphisms 𝑌(𝑟,𝑑)𝑌(𝑟,𝑑)/𝐺 are étale Galois covers, thus we have nontrivial examples for Section 2. Finally, when 𝑋 is an abelian variety and 𝜋𝑌𝑋 is an isogeny of degree 𝑟, Proposition 4.5 shows that, if g.c.d(𝑟,𝑑)=1, then each irreducible component of Pic𝑑(𝑌)/𝐺 is an irreducible component of 𝑋(𝑟,𝑑), where 𝐺=Ker(𝜋). Section 5 is devoted to prove Proposition 3.1.

1.1. Notations and Conventions

Throughout the paper, 𝑘 denotes an algebraically closed field and 𝐺 a finite group satisfying that g.c.d.(|𝐺|,char(𝑘))=1. Also, 𝑋 denotes a smooth projective variety over 𝑘 with a fixed polarization, that is, with a fixed ample line bundle 𝒪𝑋(1), and we denote by 𝐻 any divisor in the linear system |𝒪𝑋(1)|.

We denote by 𝑘[𝐺] the group algebra of 𝐺 with coefficients in 𝑘. Also, if 𝑇1 and 𝑇2 are representations of 𝐺 over 𝑘 then we denote by 𝑇1,𝑇2 the dimHom𝑘(𝑇1,𝑇2)𝐺=dimHom𝑘[𝐺](𝑇1,𝑇2). We will identify vector bundles on 𝑋 with locally free 𝒪𝑋-modules.

2. Étale Covers

Let 𝜋𝑌𝑋 be an étale Galois cover. Now, we recall that a locally free sheaf on 𝑌 is a 𝐺-sheaf (see [6, page 69]) if 𝐺 acts on in a way compatible with the action on 𝑌. Since 𝜋 is étale, it is flat; hence, we have that 𝜋 is a locally free 𝒪𝑋-module with an action of 𝐺. From this, we have defined a natural morphism of 𝑘-algebras 𝑘[𝐺]End𝒪𝑋(𝜋). Now, as we suppose by hypothesis that g.c.d.(char(𝑘),|𝐺|)=1, Maschke's Theorem guarantees that 𝑘[𝐺] is a semisimple 𝑘-algebra of finite dimension |𝐺| over 𝑘. We denote by ={𝑉0,,𝑉𝑟} the set of irreducible representations of 𝐺 over 𝑘, that is, the set of irreducible 𝑘[𝐺]-modules, and by {𝑒𝑉}𝑉 the set of corresponding idempotents. Thus, we have that 𝜋=𝑉𝑒𝑉𝜋.(2.1)

On the other hand, by Theorem 1(B) in [6, page 111], there is a locally free sheaf on 𝑋 such that 𝜋, so we have natural 𝐺-invariant isomorphisms 𝜋𝜋𝜋𝜋𝒪𝑌𝑉𝑒𝑉(𝜋𝒪𝑌); therefore, to understand the 𝐺-structure of 𝜋 it suffices to do it for 𝜋𝒪𝑌. Thus, we have the next.

Proposition 2.1. Let 𝑋 be a smooth projective variety over an algebraically closed field 𝑘 and 𝜋𝑌𝑋 and étale Galois cover with Galois group 𝐺. Let 𝑉 and define 𝑉 as the 𝒪𝑋-module 𝜋(𝑉𝑘𝒪𝑌)𝐺=𝑒𝑈(𝜋(𝑉𝑘𝒪𝑌)), where 𝑉 is the dual representation of 𝑉 and 𝑈 is the trivial representation. Then, (1)𝜋𝒪𝑌=𝑉𝑉𝑘𝑉,(2.2) where is the set of all irreducible representations of 𝐺 over 𝑘 and 𝑟𝑎𝑛𝑘𝑉=dim𝑘𝑉. Also, one has that 𝑉=𝑒𝑈(𝑉𝑒𝑉(𝜋𝒪𝑌)) and 𝑉𝑉,(2)𝑉𝑖𝑉𝑗𝑟𝑙=0𝑛𝑙𝑉𝑙,(2.3) where 𝑉𝑖𝑉𝑗=𝑟𝑙=0𝑉𝑛𝑙𝑙,(3)each 𝑉 is simple (i.e., End𝒪𝑋(𝑉)𝑘) and 𝑉𝑖̸𝑉𝑗 if 𝑖𝑗,(4)for any irreducible representation 𝑉 of 𝐺 one has that 𝑘𝑉=𝑛𝑖𝑉𝑖,(2.4) where 𝑘𝑉(𝑉)=𝑛𝑖𝑖.

Proof. We recall that 𝑘[𝐺]𝑉𝑉𝑘𝑉 as 𝑘[𝐺×𝐺]-modules, where actions are given by 𝑔1×𝑔2(𝜆𝑔𝜆𝑔)=𝑔𝑔1𝑔𝑔21 and 𝑔1×𝑔2(𝑔𝑣𝛿)=1(𝑣)𝑔2(𝛿), respectively. Thus, we have the following 𝐺-invariant isomorphisms: 𝜋𝑉𝑉𝑘𝑉=𝑉𝑉𝑘𝜋𝑉𝑉𝑉𝑘𝑉𝑘𝒪𝑌[𝐺]𝑘𝑘𝒪𝑌𝑔𝐺𝑔𝒪𝑌𝜋𝜋𝒪𝑌.(2.5) Now, Theorem 1 in [6, page 111] asserts that 𝜋 defines an equivalence between the category of locally free 𝒪𝑋-modules of finite rank and the category of locally free 𝒪𝑌-modules of finite rank with 𝐺-action. This proves that 𝜋𝒪𝑌𝑉𝑉𝑘𝑉. In particular, we have that 𝑉=𝑒𝑈(𝑉𝑒𝑉(𝜋𝒪𝑌)), where 𝑈 is the trivial 𝑘[𝐺]-module.
On the other hand, at the generic point 𝜖 of 𝑋, (𝜋𝒪𝑌)𝜖 is the function field 𝐾𝑌 of 𝑌 which, by the normal basis theorem, is isomorphic to 𝐾𝑋[𝐺] as 𝐾𝑋[𝐺]-modules, where 𝐾𝑋 is the function field of 𝑋. Thus, we have a natural 𝐾𝑋[𝐺]-isomorphisms: 𝜋𝒪𝑌𝜖=𝑉𝑉𝑘𝑉𝜖𝐾𝑋[𝐺]=𝑉𝑉𝑘𝐾𝑋dim𝑉(2.6) and we conclude that 𝑉 is a locally free sheaf with rank (dim𝑉).
For the last part of (1), we need to see that (𝑉𝑖)𝑉𝑖, but this is a consequence that the trace morphism Tr𝜋𝒪𝑌(𝜋𝒪𝑌) is a 𝐺-invariant isomorphism.
(2) This is immediate from 𝜋𝑉𝑘𝑊𝑘𝒪𝑌𝐺=𝜋𝑇𝑇𝑘𝒪𝑌𝑇,𝑉𝑊𝐺=𝑇𝜋𝑇𝑘𝒪𝑌𝑇,𝑉𝑊𝐺=𝑉𝑇𝑇,𝑉𝑊.(2.7)
(3) We observe that Hom𝒪𝑋𝑉,𝑊=Hom𝒪𝑌𝑉𝑘𝒪𝑌,𝑊𝑘𝒪𝑌𝐺=Hom𝑘[𝐺](𝑉,𝑊),(2.8) and from Schur's Lemma, dim𝑘Hom𝑘[𝐺](𝑉,𝑊) is 0 if 𝑉𝑊 and 1 if 𝑉𝑊 (see [7, page 181]).
(4) We know that tensor functors commutes with the pull back, so we have that 𝜋𝑘𝑉𝑘𝜋𝑉𝑘𝑉𝒪𝑌𝑘𝑉𝒪𝑌𝑇𝑇𝑇,𝑘𝑉𝒪𝑌𝑇𝑇𝒪𝑌𝑇,𝑘𝑉𝑇𝜋𝑇,𝑘𝑇𝑉𝜋𝑇𝑇,𝑘𝑇𝑉,(2.9) and, applying Theorem 1(B) in [6, page 111], we get the desired isomorphism.

Remark 2.2. Note that (4) is valid for each of the Schur functors.

3. Stable Sheaves

Let be a locally free sheaf over 𝑋 and 𝑚𝜒(𝒪(𝑚))(3.1) its Hilbert polynomial, where 𝜒()=(1)𝑖𝑖(𝑋,) is the Euler characteristic of . It is well known that this polynomial can be written as 𝑃(,𝑚)=𝑑𝑖=0𝛼𝑖𝑚()𝑖𝑖!.(3.2) Define the degree of as deg()=𝛼𝑑1()𝑟𝑘()𝛼𝑑1𝒪𝑋(3.3) and its slope by 𝜇()=deg()𝑟𝑘(),(3.4) where 𝑟𝑘() is the rank of . Recall that a locally free sheaf is 𝜇-𝐻-stable (𝜇-𝐻-semistable) if 𝜇()<𝜇() for all subsheaf (𝜇()𝜇(), resp.); also a 𝜇-𝐻-semistable locally free sheaf is said 𝜇-𝐻-polystable if it is a direct sum of 𝜇-𝐻-stable sheaves. Any 𝜇-𝐻-stable sheaf is simple, that is, End()=𝑘, in particular, a 𝜇-𝐻-polystable sheaf is 𝜇-𝐻-stable if and only if it is a simple sheaf.

The following proposition is proved in [8, pages 62-63] under the assumption of characteristic zero in the base field, but the arguments that surround the proof are valid for any characteristic when we consider only étale covers; however, per clarity we will repeat the proof in Section 5.

Proposition 3.1. Let 𝑋 be a complete variety over 𝑘 and 𝜋𝑌𝑋 be an étale Galois cover, with Galois group 𝐺. Let be a locally free sheaf on 𝑋. Then, is 𝜇-𝐻-polystable if and only if 𝜋 is 𝜇-𝜋𝐻-polystable.

Corollary 3.2. Let be a line bundle on 𝑌; then; for all 𝜇-𝐻-(semi)stable bundle on 𝑋, 𝜋 is 𝜇-𝐻-(semi)polystable bundle.

Proof. We have that the cover is étale, thus 𝜋(𝜋)𝜋(𝜋𝜋)𝑔𝐺𝑔(𝜋)𝑔𝐺𝜋𝑔. Then, by Proposition 3.1, 𝜋 is 𝜇-𝐻-(semi)polystable because 𝜋𝑔 is 𝜇-𝜋𝐻-(semi)polystable.

Corollary 3.3. Let 𝒢 be a locally free 𝜇-𝜋𝐻-stable 𝒪𝑌-module. Then, 𝜋𝒢 is 𝜇-𝐻-polystable, and is 𝜇-𝐻-stable if and only if 𝒢𝑔𝒢 for all 𝑔Id.

Proof. We have that the cover is étale, thus 𝜋𝜋𝒢𝑔𝐺𝑔𝒢, so 𝜋𝒢 is 𝜇-𝐻-polystable because 𝑔𝒢𝒢 is 𝜇-𝜋𝐻-stable. On the other hand, 𝜋dimHom𝒢,𝜋𝒢=0𝑌,𝑔𝐺𝑔𝒢𝒢=𝑔𝐺0𝑌,𝑔𝒢𝒢,(3.5) so 𝜋𝒢 is simple if and only if 𝑔𝒢𝒢 for all 𝑔1𝑑𝑌.

Now, we recall the following lemma of the theory of semistable bundles.

Lemma 3.4. Let be a 𝜇-𝐻-semistable sheaf; if g.c.d(deg(),𝑟𝑘())=1, then is 𝜇-𝐻-stable.

Proof. If is not 𝜇-𝐻-stable, then there exists a subsheaf such that 𝜇(𝐹)=𝜇() and 𝑟𝑘()<𝑟𝑘(); thus, we have deg()𝑟𝑘()=deg()𝑟𝑘() which contradicts the assumption g.c.d.(deg(),𝑟𝑘())=1.

Corollary 3.5. Let 𝑑,𝑟 be integers such that, 𝑟>0, g.c.d.(𝑑,𝑟)=1=g.c.d.(|𝐺|,𝑟), and suppose that is a line bundle on 𝑌 such that 𝜋 is 𝜇-𝐻-stable; then, for all 𝜇-𝐻-stable bundle on 𝑋 of degree 𝑑 and rank𝑟, one has that 𝜋 is 𝜇-𝐻-stable bundle.

Proof. From Proposition 3.1, we have that 𝜋 is 𝜇-𝜋𝐻-polystable with degree 𝑑|𝐺| and rank 𝑟, and from the fact that g.c.d.(𝑑|𝐺|,𝑟)=1 and Lemma 3.4, we have that it is 𝜇-𝜋𝐻-stable. Now, by Corollary 3.3, 𝜋 is 𝜇-𝐻-stable if and only if 𝑔 for all 𝑔𝐺; thus, 𝜋𝑔𝜋 for all 𝑔𝐺, then 𝜋(𝜋)𝜋 is 𝜇-𝐻-stable.

Theorem 3.6. Let 𝑋 be a smooth projective variety over an algebraically closed field 𝑘 and let 𝜋𝑌𝑋 be an étale Galois cover with Galois group 𝐺 and 𝜋𝒪𝑌=𝑉𝑉𝑘𝑉(3.6) the isotypical decomposition of 𝜋𝒪𝑌. Then, each 𝑉 is 𝜇-𝐻-stable with respect to any ample line bundle 𝒪𝑋(1).

Proof. As 𝜋𝑌𝑋 is étale, we have that 𝜋𝜋𝒪𝑌𝒪𝑌|𝐺|, and from Proposition 3.1, it follows that 𝜋𝒪𝑌 is 𝜇-𝐻-polystable; now, we just have to see that each 𝑉 is simple, but this is consequence of Proposition 2.1, Part (3).

Corollary 3.7. Let be a 𝜇-𝐻-semistable vector bundle on 𝑋, then 𝑉 is 𝜇-𝐻-semistable for all 𝑉. Moreover, if is 𝜇-𝐻-stable, then 𝑉 is 𝜇-𝐻-polystable.

Proof. We have that 𝜋𝜋𝜋𝑔𝐺𝜋, so, by Lemma 5.3of Section 5,𝜋𝜋 is 𝜇-𝐻-semistable; now, 𝜋𝜋𝜋𝒪𝑌𝑉𝑉𝑉, then 𝑉 is 𝜇-𝐻-semistable for all 𝑉. Applying Proposition 3.1 we get the second assertion by an analogous argument.

Let 𝐻(𝑟,𝑑) (𝐻(𝑟,𝑑)) be the moduli space parametrizing locally free 𝜇-𝐻-stable sheaves (𝜇-𝐻-polystables, resp.), of rank 𝑘 and degree zero over 𝑘 over 𝑋.

Corollary 3.8. Let 𝑋 be a smooth projective variety over an algebraically closed field 𝑘. Suppose that 𝑋 admits an étale Galois cover with Galois group 𝐺. Then, for all irreducible representation 𝑉 of 𝐺 over 𝑘 and any polarization 𝐻, one has that 𝐻(dim𝑉,0) is non empty.

Now, on characteristic zero, we have that exterior products of the standard representation of symmetric groups are irreducible (see [9, page 31]); thus, we have the next.

Corollary 3.9. Let 𝑋 be a smooth projective variety over 𝑘, char(𝑘)=0, and let 𝐻 be any polarization. Suppose that the variety admits a Galois cover with Galois group, the symmetric group on d letters 𝔖𝑑. Then, there exists a nonempty open set 𝑈𝐻(𝑑1,0) such that 𝑘 is 𝜇-𝐻-stable for all 𝑈.

Proof. As the characteristic is zero, the wedge product of 𝜇-𝐻-stables sheaves is 𝜇-𝐻-polystable, in particular 𝜇-𝐻-semistable. So, we have defined a morphism: 𝐻(𝑑1,0)𝐻𝑘𝑑1,0(3.7) given by 𝑘. As the variety 𝐻(𝑘𝑑1,0) is an open set of 𝐻(𝑘𝑑1,0), we only need to see that there is an 𝐻(𝑑1,0) such that 𝑘 is 𝜇-𝐻-stable.
By hypothesis, we assume the existence of an étale Galois cover with Galois group, the symmetric group on 𝑑 letters 𝔖𝑑; let 𝑉 be the standard representation of 𝔖𝑑 and 𝑉 the corresponding sheaf obtained in Theorem 3.6, and recalling that in characteristic zero the exterior algebra of the standard representation 𝑘𝑉 is irreducible for all 𝑘, we have, from part (4) of Proposition 2.1, that 𝑘𝑉=𝑘𝑉 is 𝜇-𝐻-stable.

In particular we have the following.

Corollary 3.10. Let 𝑋 be a smooth projective curve of genus 𝑔>1 over a field of characteristic zero. Then, 𝑘 is 𝜇-𝐻-stable for the generic vector bundle of degree zero and rank𝑑.

Proof. In this case, the moduli spaces 𝐻(𝑑,0) and 𝐻(𝑑𝑘,0) are irreducible, so we only need to prove the existence of a Galois cover with Galois group 𝔖𝑑+1 and, by the Lefschetz Principle (see [10]), it suffices to do it for 𝑘=. Let 𝜋1(𝑋,𝑥0)=𝛼1,𝛽1,,𝛼𝑔,𝛽𝑔|[𝛼𝑖𝛽𝑖𝛼𝑖1𝛽𝑖1]=1 be the fundamental group of 𝑋, with base point 𝑥0; on the other hand, the symmetric group 𝔖𝑑+1 is generated by one transposition and one cycle of length 𝑑+1, so we can define a surjective morphism 𝜋1(𝑋,𝑥0)𝔖𝑑+1, for example, the one given by 𝛼1(1,2), 𝛼2(1,,𝑑+1), 𝛼𝑖1 if 𝑖>2 and 𝛽1(1,2), 𝛽2(1,,𝑑+1)1=(𝑑+1,𝑑,𝑑1,,2,1), 𝛽𝑖1 if 𝑖>2.

Proposition 3.11. Let 𝑋 be projective smooth variety over an algebraically closed field 𝑘 and let 𝜋𝑌𝑋 be an étale Galois cover with Galois group 𝐺, and set 𝜋𝒪𝑌=𝑉𝐼𝑉𝑘𝑉. Let be a 𝜇-𝐻-stable vector bundle. Then, the following statements are equivalent: (1)𝜋 is 𝜇-𝜋𝐻-stable;(2)Hom(,𝑉)=0forall𝑉𝑉0;(3)𝑉 is 𝜇-𝐻-stable forall𝑉 and 𝑉𝑊 for 𝑉𝑊.

Proof. (3.9)(2) Since is 𝜇-𝐻-stable, from Proposition 3.1, we have that 𝜋 is 𝜇-𝜋𝐻-polystable; on the other hand, 𝜋Hom,𝜋=𝐻0𝑌,𝜋𝜋=𝐻0𝑋,𝜋𝜋𝜋=𝐻0𝑋,𝜋𝜋=Hom,𝜋𝜋=Hom,𝜋𝒪𝑌=Hom,𝑉𝐼𝑉𝑘𝑉=𝑉𝐼𝑉𝑘Hom,𝑉.(3.8) Thus, 𝜋 is a simple vector bundle if and only if Hom(,𝑉)=0 for all 𝑉 different from the trivial representation 𝑉0.
 (3.9)(3) Notice that 𝜋𝜋𝜋𝑔𝜋𝜋|𝐺| is 𝜇-𝜋𝐻-polystable; thus, 𝜋𝜋𝜋𝒪𝑌 and 𝑉 are 𝜇-𝐻-polystable. Also, we have, from Proposition 3.1, that 𝜋 is 𝜇-𝜋𝐻-polystable; then 𝜋 is simple, and then, 𝜇-𝜋𝐻-stable if and only if 𝜋dimHom𝜋,|𝐺|=||𝐺||.(3.9) On the other hand, we have the following relations: 𝜋Hom𝜋,|𝐺|𝜋=Hom,𝑔𝐺𝑔𝜋𝜋=Hom,𝜋𝜋𝜋=𝐻0𝑌,𝜋𝜋𝜋𝜋=𝐻0𝑋,𝜋𝜋𝜋𝜋𝜋=Hom𝜋,𝜋𝜋=Hom𝒪𝑌,𝒪𝑌=Hom𝑉𝐼𝑉𝑘𝑉,𝑉𝐼𝑉𝑘𝑉=𝑉,𝑊𝐼Hom𝑉𝑘𝑉,𝑊𝑘𝑊𝑉,𝑊𝐼Hom𝑉,𝑊(dim𝑉)×(dim𝑊)𝑉𝐼Hom𝑉,𝑉(dim𝑉)2𝑉𝑊Hom𝑉,𝑊(dim𝑉)×(dim𝑊).(3.10)
Now,|𝐺|=𝑉𝐼(dim𝑉)2 and, from previous relationships, we have that equality (3.9) is possible if and only if dimHom(𝑉,𝑊) is zero if 𝑉𝑊 and are simple for all 𝑉. This tested the statement.

Theorem 3.12. Let d,r be integers such that r>0 and g.c.d.(d,r)=1=g.c.d.(|G|,r). Then, for every irreducible representation V of G and X(r,d) one has that V is 𝜇-H-stable. In addition, the natural induced morphism X(r,d)X(rdimV,ddimV)X(rdimV,ddimV) is injective.

Proof. From Proposition 3.1, we have that 𝜋 is 𝜇-𝜋𝐻-polystable with degree 𝑑|𝐺| and rank 𝑟, and from the fact that g.c.d.(𝑑|𝐺|,𝑟)=1 and Lemma 3.4, we have that it is 𝜇-𝜋𝐻-stable. Thus, applying Proposition 3.11, we have that 𝑉 is 𝜇-𝐻-stable for any representation 𝑉.
Let us now prove the injectivity of the morphism 𝑋(𝑟,𝑑)𝑋(𝑟dim𝑉,𝑑dim𝑉).
Let 1,2𝑋(𝑟,𝑑) and suppose that 1𝑉2𝑉; notice that this isomorphism implies that det(1)=det(2). From the 𝜇-𝐻-stability of 𝑖𝑉, we have that these vector bundles are simple, so that 1=dimHom(1𝑉,2𝑉)=dimHom(1,2𝑉𝑉). On the other hand, from Proposition 2.1, we have that 𝑉=𝑉 and 𝑉𝑉=𝑇,𝑉𝑉T, so 1=dimHom(1𝑉,2𝑉)=𝑇dimHom(1,2𝑇)𝑇,𝑉𝑉 and we can deduce the existence of a unique representation 𝑇 of dimension 1 such that 12𝑇. From here we have that det(1)det(2)(𝑇)𝑟, where 𝑟=𝑟𝑘(2) and, thus, (𝑇)𝑟𝒪𝑋 and applying part (4) of Proposition 2.1, we obtain that 𝑇𝑟=𝒪𝑋 and so 𝑇𝑟 is the trivial representation. However, from representation theory, the order of the cyclic group generated by the isomorphism class of 𝑇 should be divided by |𝐺|, but by hypothesis g.c.d.(𝑟,|𝐺|)=1, then 𝑟=1 and 𝑇 is the trivial representation.

4. Examples of Varieties with Group Action: Actions in the Moduli Space of Stable Bundles

Again, let 𝜋𝑌𝑋 be an étale Galois cover with Galois group 𝐺, thus the action of 𝐺 on 𝑌 determines an action on the moduli space 𝑌(𝑟,𝑑) (𝑌(𝑟,𝑑)) of 𝜇-𝐻-(semi)stables sheaves on 𝑌, and from Corollary 3.3, we have a natural morphism 𝑌(𝑟,𝑑)𝑋(𝑟|𝐺|,𝑑) given by 𝒢𝜋𝒢, which factorizes by the quotient 𝑌(𝑟,𝑑)/𝐺. The aim of this section is the study of such morphism.

Proposition 4.1. Suppose that g.c.d(|𝐺|,𝑑)=1. Then, for all integer 𝑟>0, the natural action of 𝐺 on 𝑌(𝑟,𝑑) is without fixed points; in particular, for all 𝒢𝑌(𝑟,𝑑), one has that 𝜋𝒢 is 𝜇-𝐻-stable.

Proof. Let Pic𝑑𝒪𝑌(1)(𝑌) be the subvariety of the Picard variety Pic(𝑌) formed by line bundles of degree 𝑑 with respect to 𝒪𝑌(1)=𝜋𝒪𝑋(1). Thus, we have defined the determinant morphism det𝑌(𝑟,𝑑)Pic𝑑𝒪𝑌(1)(𝑌) and this satisfies that det(𝑔())=𝑔(det()), so it suffices to prove the proposition for Pic𝑑𝒪𝑌(1)(Y).
Let Pic𝑑𝒪𝑌(1)(𝑌); from Lemma 5.2, we have that deg(𝜋)=𝑑 and 𝑟𝑘(𝜋)=|𝐺|, so Lemma 3.4 implies the 𝜇-𝐻-stability of 𝜋; finally, from Corollary 3.3, 𝑔 for all 𝑔Id, then we conclude that there are no fixed points. The last part of the statement is consequence of Corollary 3.3.

Denote by 𝒬𝜋𝑟,𝑑 the quotient variety 𝑌(𝑟,𝑑)/𝐺 and 𝛿𝒬𝜋𝑟,𝑑𝒬𝜋1,𝑑 the induced morphism by det𝑌(𝑟,𝑑)Pic𝑑(𝑌)=𝑌(1,𝑑). The fiber of this map is described in the following.

Corollary 4.2. Let Pic𝑑(𝑌). Then, if g.c.d(|𝐺|,𝑑)=1, then the restriction of the quotient morphism defines an isomorphism det1()𝛿1([]).

Proof. In order to see this, it suffices to show that 𝑔(det1())det1()= which is consequence of the proof of the above proposition.

Corollary 4.3. Let 𝑟 be a positive integer and suppose that g.c.d(|𝐺|,𝑑)=1. Then, one has a natural injective morphism 𝜋𝒬𝜋𝑟,𝑑𝑋(𝑟|𝐺|,𝑑)𝑋(𝑟|𝐺|,𝑑).

Consider 𝑌 and 𝑋 curves. If g.c.d.(𝑟,𝑑)=1, then we have that 𝑌(𝑟,𝑑)=𝑌(𝑟,𝑑) and that the moduli space is smooth (see [8, Section 4.5]). So, we have the following.

Corollary 4.4. 𝒬𝜋𝑟,𝑑 is a smooth subvariety of 𝑋(𝑟|𝐺|,𝑑).

In general, there is a natural map Pic𝑑𝒪𝑌(1)(𝑌)𝑋(|𝐺|,𝑑) whose image is naturally isomorphic to the quotient variety 𝒬𝜋1,𝑑. A special case is for abelian varieties; we note that, by a theorem of Serre-Lang, every étale cover of an abelian variety 𝑋 has the structure of an abelian variety (see [6, page 167]).

Proposition 4.5. Let 𝑋 be a polarized abelian variety and 𝜋𝑌𝑋 an étale cyclic cover of degree 𝑟. Let 𝑑 be an integer such that g.c.d(𝑟,𝑑)=1. Then, each irreducible component of 𝒬𝜋1,𝑑 is a smooth irreducible component of 𝑋(𝑟,𝑑).

Proof. By previous Corollary 4.4, 𝒬𝜋1,𝑑 is a smooth subvariety of 𝑋(𝑟|𝐺|,𝑑) and by Proposition 4.1  𝑔 if 𝑔Id. Now, let Pic𝑑(𝑌); thus, 𝑇𝜋𝑋(𝑟,𝑑) is given by 𝐻1(𝑋,𝜋𝜋1)𝐻1(𝑋,𝜋(𝜋𝜋𝐿1))𝐻1(𝑌,𝜋𝜋1)𝐻1(𝑌,𝑔𝐺𝑔1)𝑔𝐺𝐻1(𝑌,𝑔1); now, by the vanishing theorem for abelian varieties in [6, page 76], we have that 1(𝐿)=0 for all Pic0(𝑌), 𝒪𝑌. Then, 𝑇𝜋𝑋(𝑟,𝑑)𝐻1(𝑌,𝒪𝑌)𝑇Pic(𝑌)𝑇𝜋𝒬𝜋1,𝑑.

5. Proof of Proposition 3.1

We will need three lemmas; on them, we will be under the assumptions of Proposition 3.1.

Let us recall an equivalent definition of the degree of an 𝒪𝑋-module.

Lemma 5.1. For a locally free sheaf , deg()=𝑐1()𝐻𝑑1.(5.1) In particular, if and are locally free sheaves, then one has deg()=𝑟𝑘()deg()+𝑟𝑘()deg().(5.2)

Proof. By Hirzebruch-Riemann-Roch formula, we have 𝑃()(𝑚)=𝜒𝒪𝑋(𝑚)=degch𝒪𝑋(𝑚)td(𝒯𝑋)𝑛=𝑘𝑖+𝑗=𝑛𝑘degch()𝑖𝒯td𝑋𝑗𝐻𝑑𝑘𝑚𝑘,𝑘!(5.3) where ch(𝒪𝑋(𝑚))=𝑘((𝑚𝐻)𝑘)/𝑘! for some ample divisor 𝐻; thus, 𝛼𝑑1()=degch()1td0𝒯𝑋𝐻𝑑1+degch0()td1𝒯𝑋𝐻𝑑1𝑐=deg1()𝐻𝑑1+𝑟𝑘()2𝑐deg1𝒯𝑋𝐻𝑑1.(5.4) In particular, for =𝒪𝑋, we have 𝛼𝑑1𝒪𝑋=12𝑐deg1𝒯𝑋𝐻𝑑1(5.5) and the first part of the lemma follows. The second part is consequence of basic properties of Chern classes.

Lemma 5.2. Let be a locally free 𝒪𝑋-module. Then, 𝑟𝑘(𝜋)=𝑟𝑘() and deg(𝜋)=|𝐺|deg(), in particular 𝜇(𝜋)=|𝐺|𝜇(). Let 𝒢 be a locally free 𝒪𝑌-module. Then, 𝑟𝑘(𝜋𝒢)=|𝐺|𝑟𝑘(𝒢) and deg(𝜋𝒢)=deg(𝒢), in particular |𝐺|𝜇(𝜋𝒢)=𝜇(𝒢).

Proof. Statements about the rank follow from the general theory of finite covers. Now, as an étale cover is affine, we have 𝑖(𝑌,𝒢)=𝑖(𝑋,𝜋𝒢) for any coherent sheaf on 𝑌; thus, 𝑃(𝑌,𝒢)=𝑃(𝑋,𝜋𝒢) and 𝛼𝑑1(𝒢)=𝛼𝑑1(𝜋𝒢). On the other hand, by Lemma 5.1 we have that deg(𝜋𝒪𝑌)=𝑐1(𝜋𝒪𝑌)𝐻𝑑1=𝑐1(𝜋𝜋𝒪𝑋)𝐻𝑑1=𝜋(𝑐1(𝜋𝒪𝑋))𝐻𝑑1=|𝐺|𝑐1(𝒪𝑋)𝐻𝑑1=0, thus 𝛼𝑑1(𝜋𝒪𝑌)=|𝐺|𝛼𝑑1(𝒪𝑋). In general, for a coherent sheaf 𝒢, we have that deg(𝒢)=𝛼𝑑1(𝒢)𝑟𝑘(𝒢)𝛼𝑑1(𝒪𝑌)=𝛼𝑑1(𝜋𝒢)𝑟𝑘(𝒢)|𝐺|𝛼𝑑1(𝒪𝑋)=deg(𝜋𝒢).
Finally, let be a locally free sheaf on 𝑋, then deg(𝜋)=deg(𝜋𝜋)=deg(𝜋𝒪𝑌)=𝑟𝑘(𝜋𝒪𝑌)deg()+𝑟𝑘()deg(𝜋𝒪𝑌)=|𝐺|deg().

Lemma 5.3. Let be a locally free 𝒪𝑋-module. Then, is 𝜇-𝐻-semistable if and only if 𝜋 is 𝜇-𝐻-semistable.

Proof. If is not 𝜇-𝐻-semistable, then there is a submodule with 𝜇()>𝜇(), so, by Lemma 5.2, 𝜇(𝜋)>𝜇(𝜋) and 𝜋 is not 𝜇-𝐻-semistable.
For the converse, suppose that 𝜋 is not 𝜇-𝜋𝐻-semistable, then, by the Harder-Narasimhan filtration theorem, there exists a unique submodule 𝒢 such that it is 𝜇-𝜋𝐻-semistable and 𝜇(𝒢)𝜇() for all submodule of 𝜋; by the uniqueness, it is invariant under the action of 𝐺, so, by Theorem 1 in [6, page 111], there exists an 𝒪𝑋-submodule such that 𝒢𝜋 and, by previous lemma, 𝜇()>𝐸.

Proof of Proposition 3.1. Suppose that 𝜋 is 𝜇-𝐻-polystable, so by the previous lemma is 𝜇-𝐻-semistable and by the Jordan-Holder filtration theorem there exists a destabilizing submodule such that it is 𝜇-𝐻-stable and 𝜇()=𝜇(), so by previous lemmas 𝜋 is 𝜇-𝜋𝐻-semistable and 𝜇(𝜋)=𝜇(𝜋) and so it must be a direct summand of 𝜋. Let 𝜋𝑖𝜋𝑝𝜋 be the inclusion and projection morphisms, so taking the direct image we have 𝐺-invariant morphisms 𝜋𝒪𝑌𝑖𝜋𝒪𝑌𝑝𝜋𝒪𝑌 with 𝑝𝑖=Id𝜋𝒪𝑌, but, now, 𝜋𝒪𝑌=𝑉𝑉𝑉; hence, we have 𝑉𝑉𝑖𝑉𝑉𝑝𝑉𝑉 for each irreducible representation 𝑉; in particular for the trivial representation 𝑉0 we have 𝑉0=𝒪𝑋, then is a direct summand of .
Now, suppose that is 𝜇-𝐻-stable; again, by the lemma above, 𝜋 is 𝜇-𝜋𝐻-semistable, so let 𝒢 be a destabilizing submodulo for it. Thus, taking the sum 𝑔𝐺𝑔𝒢, we obtain a 𝐺-invariant 𝜇-𝜋𝐻-polystable submodule of 𝜋 which must be the pullback of a 𝜇-𝐻-polystable subsheaf of with the same slope, so = and then 𝜋=𝑔𝐺𝑔𝒢.

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