Examples of Stability of Tensor Products in Positive Characteristic

Let X be projective smooth variety over an algebraically closed field k and let E, F be μ-semistable locally free sheaves on X. When the base field is C, using transcendental methods, one can prove that the tensor product E ⊗ F 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 μsemistability; 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.


Introduction
When the base field is C, 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 O X 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 F : X → X be a total Frobenius morphism of X; we say that E is strongly μ-semistable if for all m ≥ 0 the pullback F m * E is μ-semistable with respect to the induced polarization F m * O X 1 .Under these assumptions, the tensor product of strongly μ-semistable bundles is again strongly μsemistable 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 .

ISRN Algebra
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 μstability.The key result is Proposition 3.1, which shows that, if π : Y → X is étale and Galois with g.c.d.|G|, char k 1, then F is μ-semistable μ-polystable if and only if π * F 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 L is a line bundle on Y , in Corollary 3.2 we show that F ⊗ π * L is μ-semi polystable if F is μ-semi stable.If in addition g.c.d.deg F, rk F 1 |G|, rk F and g * L / L for all g ∈ G, then in Corollary 3.5 it is proved that F ⊗ π * L is μ-stable.The other examples that are constructed come from the Ginvariant decomposition of π * O Y .Now, the general outline of this construction is as follows.
In Section 1, we observe that π * O Y has a natural G-invariant decomposition: where I is the set of irreducible representations of G over k and each E V is a locally free sheaf of rank dim V .Also, we have the relations where V ⊗W T ∈I T ⊕n T and ∧ s V W∈I W ⊕n W .In Section 2, we show that E V is μ-H-stable for any polarization on X and also that F ⊗ E V is μ-H-polystable μ-H-semistable whenever F is μ-H-stable μ-H-semistable, resp. .Furthermore, let M 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 In Section 3, we proved that, if g.c.d.

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 O X 1 , and we denote by H any divisor in the linear system |O X 1 |.
We denote by k G the group algebra of G with coefficients in k.Also, if T 1 and T 2 are representations of G over k then we denote by T 1 , T 2 the dim Hom k T 1 , T 2 G dim Hom k G T 1 , T 2 .We will identify vector bundles on X with locally free O X -modules.

Étale Covers
Let π : Y → X be an étale Galois cover.Now, we recall that a locally free sheaf F on Y is a G-sheaf see 6, page 69 if G acts on F in a way compatible with the action on Y .Since π is étale, it is flat; hence, we have that π * F is a locally free O X -module with an action of G. From this, we have defined a natural morphism of k-algebras k G → End O X π * F .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 I {V 0 , . . ., V r } the set of irreducible representations of G over k, that is, the set of irreducible k G -modules, and by {e V } V ∈I the set of corresponding idempotents.Thus, we have that where V * is the dual representation of V and U is the trivial representation.Then, where I is the set of all irreducible representations of G over k and rank where where Proof.We recall that k G V ∈I V ⊗ k V * as k G × G -modules, where actions are given by , respectively.Thus, we have the following G-invariant isomorphisms:

2.5
Now, Theorem 1 in 6, page 111 asserts that π * defines an equivalence between the category of locally free O X -modules of finite rank and the category of locally free O Y -modules of finite rank with G-action.This proves that π In particular, we have that On the other hand, at the generic point of X, π * O Y is the function field K Y of Y which, by the normal basis theorem, is isomorphic to K X G as K X G -modules, where K X is the function field of X.Thus, we have a natural K X G -isomorphisms: and we conclude that E V is a locally free sheaf with rank dim V .
For the last part of 1 , we need to see that 2 This is immediate from 4 We know that tensor functors commutes with the pull back, so we have that 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.

Stable Sheaves
Let E be a locally free sheaf over X and It is well known that this polynomial can be written as Define the degree of E as and its slope by where rk E is the rank of E. Recall that a locally free sheaf is μ-H-stable μ-H-semistable if μ F < μ E for all subsheaf F μ F ≤ μ E , 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 E 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 π : Y → X be an étale Galois cover, with Galois group G. Let E be a locally free sheaf on X.
Proof.We have that the cover is étale, thus Proof.We have that the cover is étale, thus Theorem 3.6.Let X be a smooth projective variety over an algebraically closed field k and let π : Y → X be an étale Galois cover with Galois group G and Proof.We have that π * π * π * F g∈G π * F, so, by Lemma 5.
semistable for all V .Applying Proposition 3.1 we get the second assertion by an analogous argument.
Let M H r, d M 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 M H dim V, 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 S d .Then, there exists a nonempty open set Proof.As the characteristic is zero, the wedge product of μ-H-stables sheaves is μ-Hpolystable, in particular μ-H-semistable.So, we have defined a morphism: given by E → k E. As the variety By hypothesis, we assume the existence of an étale Galois cover with Galois group, the symmetric group on d letters S d ; let V be the standard representation of S d and E V the corresponding sheaf obtained in Theorem 3.6, and recalling that in characteristic zero the exterior algebra of the standard representation k V is irreducible for all k, we have, from part 4 of Proposition 2.1, that k E V E k V 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 E is μ-H-stable for the generic vector bundle E of degree zero and rank d.
Proof.In this case, the moduli spaces M H d, 0 and M H d k , 0 are irreducible, so we only need to prove the existence of a Galois cover with Galois group S d 1 and, by the Lefschetz Principle see 10 , it suffices to do it for k 1 be the fundamental group of X, with base point x 0 ; on the other hand, the symmetric group S d 1 is generated by one transposition and one cycle of length d 1, so we can define a surjective morphism π 1 X, x 0 → S d 1 , for example, the one given by α Proposition 3.11.Let X be projective smooth variety over an algebraically closed field k and let π : Y → X be an étale Galois cover with Galois group G, and set Let F be a μ-H-stable vector bundle.Then, the following statements are equivalent:

3.8
Thus, π * F is a simple vector bundle if and only if Hom F, E V ⊗ F 0 for all V different from the trivial representation V 0 . 1 . Also, we have, from Proposition 3.1, that π * F is μ-π * H-polystable; then π * F is simple, and then, μ-π * H-stable if and only if dim Hom π * F, π * F ⊕|G| |G|.

3.9
On the other hand, we have the following relations:

3.10
Now, |G| V ∈I dim V 2 and, from previous relationships, we have that equality 3.9 is possible if and only if dimHom F ⊗ E V , F ⊗ E W is zero if V / W and F ⊗ E 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 F ∈ M X r, d one has that F ⊗ E V is μ-Hstable.In addition, the natural induced morphism Proof.From Proposition 3.1, we have that π * F is μ-π * H-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 μπ * H-stable.Thus, applying Proposition 3.11, we have that F ⊗ E V is μ-H-stable for any representation V .
Let us now prove the injectivity of the morphism From the μ-H-stability of F i ⊗ E V , we have that these vector bundles are simple, so that 1 dim Hom and we can deduce the existence of a unique representation T ∈ I of dimension 1 such that F 1 F 2 ⊗ E T .From here we have that det F 1 det F 2 ⊗ E T ⊗r , where r rk F 2 and, thus, E T ⊗r O X and applying part 4 of Proposition 2.1, we obtain that E T ⊗r O X and so T ⊗r 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.

Space of Stable Bundles
Again, let π : Y → X be an étale Galois cover with Galois group G, thus the action of G on Y determines an action on the moduli space g∈G H 1 Y, g * L ⊗ L −1 ; now, by the vanishing theorem for abelian varieties in 6, page 76 , we have that h

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 O X -module.
Lemma 5.1.For a locally free sheaf E,

5.1
In particular, if E and F are locally free sheaves, then one has Proof.By Hirzebruch-Riemann-Roch formula, we have where ch O X m k mH k /k! for some ample divisor H; thus,

5.4
In particular, for E O X , we have and the first part of the lemma follows.Proof.If E is not μ-H-semistable, then there is a submodule F with μ F > μ E , so, by Lemma 5.2, μ π * F > μ π * E and π * E is not μ-H-semistable.
For the converse, suppose that π * E is not μ-π * H-semistable, then, by the Harder-Narasimhan filtration theorem, there exists a unique submodule G such that it is μ-π * Hsemistable and μ G ≥ μ H for all submodule of π * E; by the uniqueness, it is invariant under the action of G, so, by Theorem 1 in 6, page 111 , there exists an O X -submodule F such that G π * F and, by previous lemma, μ F > E.
Proof of Proposition 3.1.Suppose that π * E is μ-H-polystable, so by the previous lemma E is μ-H-semistable and by the Jordan-Holder filtration theorem there exists a destabilizing → F ⊗ V ⊗ E V for each irreducible representation V ; in particular for the trivial representation V 0 we have E V 0 O X , then F is a direct summand of E. Now, suppose that E is μ-H-stable; again, by the lemma above, π * E is μ-π * Hsemistable, so let G be a destabilizing submodulo for it.Thus, taking the sum g∈G g * G, we obtain a G-invariant μ-π * H-polystable submodule of π * E which must be the pullback of a μ-H-polystable subsheaf F of E with the same slope, so F E and then π * E g∈G g * G.
d, |G| 1, then, for all r > 0, G acts without fixed points on M Y r, d ; in particular the morphisms M Y r, d → M Y r, d /G are étale Galois covers, thus we have nontrivial examples for Section 2. Finally, when X is an abelian variety and π : Y → X is an isogeny of degree r, Proposition 4.5 shows that, if g.c.d r, d 1, then each irreducible component of Pic d Y /G is an irreducible component of M X r, d , where G Ker π .Section 5 is devoted to prove Proposition 3.1.

Proposition 4 . 5 .
Let X be a polarized abelian variety and π : Y → X 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 Q π 1,d is a smooth irreducible component of M X r, d .Proof.By previous Corollary 4.4, Q π 1,d is a smooth subvariety of M X r|G|, d and by Proposition 4.1 g *

Proposition 2.1. Let
O Y ; therefore, to understand the G-structure of π * F it suffices to do it for π * O Y .Thus, we have the next.X be a smooth projective variety over an algebraically closed field k and π : Y → X and étale Galois cover with Galois group G. Let V ∈ I and define E * is simple if and only if g * G / G for all g / 1d Y .Now, we recall the following lemma of the theory of semistable bundles.Proof.If E is not μ-H-stable, then there exists a subsheaf F such that μ F μ E and rk F < rk E ; thus, we have deg F rk E deg E rk F which contradicts the assumption g.c.d.deg E , rk E 1. |G|, r , and suppose that L is a line bundle on Y such that π * L is μ-H-stable; then, for all μ-H-stable bundle F on X of degree d and rank r, one has that F ⊗ π * L is μ-H-stable bundle.Proof.From Proposition 3.1, we have that π * F is μ-π * H-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-π * H-stable.Now, by Corollary 3.3, π * L is μ-H-stable if and only if g Lemma 3.4.Let E be a μ-H-semistable sheaf; if g.c.d deg E , rk E 1, then E is μ-H-stable.Corollary 3.5.Let d, r be integers such that, r > 0, g.c.d.d, r 1 g.c.d.
O Y is μ-H-polystable; now, we just have to see that each E V is simple, but this is consequence of Proposition 2.1, Part 3 .
stable with respect to any ample line bundle O X 1 .Proof.As π : Y → X is étale, we have that π * π * O Y O ⊕|G| Y , and from Proposition 3.1, it follows that π * Then, for all integer r > 0, the natural action of G on M Y r, d is without fixed points; in particular, for all G ∈ M Y r, d , one has that π * G is μ-H-stable.Proof.Let Picd O Y 1 Y be the subvariety of the Picard variety Pic Y formed by line bundles of degree d with respect to O Y 1 π * O X 1 .Thus, we have defined the determinant morphism det : M Y r, d → Pic d O Y 1 Y and this satisfies that det g * E g * det E , so it suffices to prove the proposition for Pic d O Y 1 Y .Let L ∈ Pic d O Y 1 Y ; from Lemma 5.2, we have that deg π * L d and rk π * L |G|, so Lemma 3.4 implies the μ-H-stability of π * L; finally, from Corollary 3.3, g * L / L for all g / Id, then we conclude that there are no fixed points.The last part of the statement is consequence of Corollary 3.3.the quotient variety M Y r, d /G and δ : Q π r,d → Q π 1,d the induced morphism by det : M Y r, d → Pic d Y M Y 1, d .The fiber of this map is described in the following.Let L ∈ Pic d Y .Then, if g.c.d |G|, d 1, then the restriction of the quotient morphism defines an isomorphism det −1 L → δ −1 L .Proof.In order to see this, it suffices to show that g det −1 L ∩ det −1 L ∅ which is consequence of the proof of the above proposition.Let r be a positive integer and suppose that g.c.d |G|, d 1.Then, one has a natural injective morphism π * : Q π r,d → M X r|G|, d ⊂ M X r|G|, d .Consider Y and X curves.If g.c.d.r, d 1, then we have that M Y r, d M Y r, d and that the moduli space is smooth see 8, Section 4.5 .So, we have the following.In general, there is a natural map Pic d O Y 1 Y → M X |G|, d whose image is naturally isomorphic to the quotient variety Q π 1,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 .
stables sheaves on Y , and from Corollary 3.3, we have a natural morphism M Y r, d → M X r|G|, d given by G → π * G, which factorizes by the quotient M 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.
In general, for a coherent sheaf G, we have that degG α d−1 G − rk G α d−1 O Y α d−1 π * G − rk G |G|α d−1 O X deg π * G .Finally, let E be a locally free sheaf on X, then deg π * E deg π * π * E deg E ⊗ π * O Y rk π * O Y deg E rk E deg π * O Y |G| deg E .Lemma 5.3.Let E be a locally free O X -module.Then, E is μ-H-semistable if and only if π * E is μ-H-semistable.
ISRN Algebrasubmodule F such that it is μ-H-stable and μ F μ E , so by previous lemmas π * F is μ-π * H-semistable and μ π * F μ π * E and so it must be a direct summand of π F be the inclusion and projection morphisms, so taking the direct imagewe have G-invariant morphisms F ⊗ π * O Y O Y with p * • i * Id F⊗π * O Y , but, now, π * O Y V ∈I V ⊗ E V ; hence, we have * E. Let π * F i → π * E p → π * *