Principal bundles over a curve in positive characteristic, II

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Bull. Sci. math. 129 (2005) 267–273www.elsevier.com/locate/bulsc

Principal bundles over a curve in positivecharacteristic, II

Indranil Biswas∗, A.J. Parameswaran

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005

Received 27 October 2004; accepted 27 October 2004

Available online 16 December 2004

Abstract

This is a postscript to our earlier paper [Bull. Sci. Math. 128 (2004) 761–773]. In [Bull.Math. 128 (2004) 761–773] a criterion was given for a principal bundle over a curve to be strsemistable. This led to a conjectural criterion to characterize semistable principal bundles. Hconstruct examples to show that this generalization of the criterion fails. 2004 Elsevier SAS. All rights reserved.

MSC:14L15; 14H60

Keywords:Principal bundle; Separable morphism; Numerically effectiveness

1. Separably numerically effective line bundle

We will first recall the main result of [3].Let k be an algebraically closed field of positive characteristic andX a connected

smooth projective curve defined overk. Let

FX :X → X (1)

be the Frobenius morphism. For any integerr � 1 ther-fold iteration of the self-mapFX

will be denoted byF rX , andF 0 will denote the identity map ofX.

* Corresponding author.E-mail addresses:[email protected] (I. Biswas), [email protected] (A.J. Parameswaran).

0007-4497/$ – see front matter 2004 Elsevier SAS. All rights reserved.doi:10.1016/j.bulsci.2004.10.002

268 I. Biswas, A.J. Parameswaran / Bull. Sci. math. 129 (2005) 267–273

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Let G be a connected simple linear algebraic group defined over the fieldk. Given aG-bundleEG over X, for any parabolic subgroupP ⊂ G and any characterλ of P , theassociated line bundle(EG × kλ)/P overEG/P will be denoted byEG(λ).

A G-bundleEG over X is calledsemistableif for every triple of the form(P,λ,σ ),where

(i) P ⊂ G is a proper (reduced) parabolic subgroup,(ii) λ is an anti-dominant character ofP , and

(iii) σ :X → EG/P is a section of the natural projectionEG/P → X,

the pulled back line bundleσ ∗EG(λ) overX is of nonnegative degree.A G-bundleEG overX is calledstrongly semistableif the iterated pull-back(F r

X)∗EG

is semistable for eachr � 0, whereFX is the Frobenius map in (1).Now fix once and for all a proper parabolic subgroupP � G, and also fix a nontrivia

anti-dominant characterλ of P . The main theorem of [3] says that aG-bundleEG overXis strongly semistable if any only if the line bundleEG(λ) overEG/P , for the given fixedpair (P,λ), is numerically effective (see [3, p. 766, Theorem 3.1]).

The condition in [3, Theorem 3.1] that the line bundleEG(λ) overEG/P is numericallyeffective can be weaken as follows.

We recall that a line bundleξ overEG/P is numerically effective if and only it for everpair (C,f ), whereC is a smooth projective curve andf :C → EG/P is a map, we have

degree(f ∗ξ) � 0.

Let

φ :EG → X (2)

be the natural projection.

Definition 1.1. A line bundleξ overEG/P will be calledseparably numerically effectivif for every pair(C,f ), where

(i) C is a smooth projective curve, and(ii) f :C → EG/P is a map such that the morphismφ ◦ f is separable (the mapφ is

defined in (2))

the inequality degree(f ∗ξ) � 0 hold.

In view of [3, Theorem 3.1] it is natural to ask the following question:

Question 1.2. Fix a proper parabolic subgroupP of a simple groupG and also fix anontrivial anti-dominant characterλ of P . Is it true that a principalG-bundleEG overXis semistable if any only if the line bundleEG(λ) over EG/P is separably numericalleffective?

The following lemma shows that ifEG is semistable thenEG(λ) is separably numerically effective.

I. Biswas, A.J. Parameswaran / Bull. Sci. math. 129 (2005) 267–273 269

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Lemma 1.3. As in Question1.2, fix a proper parabolic subgroupP of a simple groupGand also fix a nontrivial anti-dominant characterλ of P . Let EG be a semistable principal G-bundle overX. Then the line bundleEG(λ) over EG/P is separably numericallyeffective.

Proof. Take any morphismf :C → EG/P as in Definition 1.1, whereC is a smoothprojective curve, such that the morphismφ ◦ f is separable. SinceEG is a semistableG-bundle andφ ◦ f is separable, the pull-back

ECG := (φ ◦ f )∗EG

is a semistable principalG-bundle overC. That ECG is semistable follows from the ex

istence and uniqueness of the Harder–Narasimhan reduction of aG-bundle (see [1,2])Indeed, using the Galois descent, the uniqueness of the Harder–Narasimhan redu(φ ◦ f )∗EG descents to the Harder–Narasimhan reduction ofEG.

The morphismf defines a section

f̂ :C → ECG/P = (φ ◦ f )∗EG/P.

The line bundlef ∗EG(λ) over C is identified with the pull-backf̂ ∗ECG(λ), where

ECG(λ) = (EC

G × kλ)/P is the line bundle overECG/P associated to the principalP -bundle

ECG → EC

G/P for the characterλ. In particular, we have

degree(f ∗EG(λ)

) = degree(f̂ ∗EC

G(λ)). (3)

Since the principalG-bundleECG is semistable, we have degree(f̂ ∗EC

G(λ)) � 0. Nowusing (3) we conclude that the line bundleEG(λ) over EG/P is separably numericalleffective. This completes the proof of the lemma.�

The above lemma justifies Question 1.2. In the next section we will give exampshow that Question 1.2 has a negative answer.

2. The case of vector bundles

In this section we setG = PGL(n, k).Let E be a vector bundle overX. Let P(E) denote the projective bundle overX para-

metrizing one-dimensional quotients of the fibers ofE. The tautological line bundle oveP(E) will be denoted byOP(E)(1). Let

π :P(E) → X (4)

be the natural projection.

Definition 2.1. A vector bundleE overX will be calledseparably numerically effectiveiffor every pair(C,f ), whereC is a smooth projective curve andf :C → P(E) a morphismwith π ◦ f separable (the projectionπ is defined in (4)), the line bundlef ∗OP(E)(1) overC is of nonnegative degree.

The following corollary is a consequence of Lemma 1.3.


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Corollary 2.2. Let E be a semistable vector bundle overX of degree zero. ThenE isseparably numerically effective.

Proof. In Lemma 1.3, setG = PGL(n, k), wheren = rank(E) and setEG to be the princi-pal PGL(n, k)-bundle overX defined byE. Also, in Lemma 1.3, setP to be the parabolicsubgroup of PGL(n, k) that preserves a hyperplane ink⊕n, and setλ to be the unique generator of the anti-dominant characters. (Note that the character group ofP is isomorphicto Z.) SoEG/P = P(E).

Take any pair(C,f ) as in Definition 2.1, whereC is a smooth projective curve anf :C → P(E) a morphism withπ ◦ f separable. It is easy to see that

OP(E)(n) ⊗ π∗n∧

E∗ = EG(λ) (5)

over EG/P = P(E), whereEG(λ) is the line bundle overEG/P = P(E) defined by thecharacterλ andπ is the projection in (4). Indeed, the line bundleEG(λ) is the relativeanticanonical line bundle for the projectionπ . Therefore, we have

f ∗OP(E)(n) ⊗ (π ◦ f )∗n∧

E∗ = f ∗EG(λ) (6)

over the curveC.Since the vector bundleE is semistable, the corresponding PGL(n, k)-bundle is also

semistable. Therefore, from Lemma 1.3 we conclude that

degree(f ∗EG(λ)

)� 0.

Since degree(E) = 0, from the above inequality and (6) we conclude that

n · degree(f ∗OP(E)(1)

) = degree(f ∗OP(E)(n)

) = degree(f ∗EG(λ)

)� 0.

This completes the proof of the corollary.�Remark 2.3. Let E be a vector bundle overX of rankn and degree zero. As in the proofCorollary 2.2, setP ⊂ PGL(n, k) to be the parabolic subgroup that preserves a hyperpandλ to be the generator of the anti-dominant characters. From (5) it follows immedthat the vector bundleE is separably numerically effective if and only if the line bundEG(λ) overEG/P = P(E) is separably numerically effective.

Proposition 2.4. LetV be a semistable vector bundle overX of rank two with

degree(V ) > 0.

Assume thatF ∗XV has a quotient line bundle of negative degree(the Frobenius mapFX

is defined in(1)). Let V ′ be any semistable vector bundle overX with degree(V ′) =degree(V ). Then the nonsemistable vector bundleW := V ′ ⊕ V ∗ over X is separablynumerically effective.

Proof. Assume that the above defined vector bundleW is not separably numerically efective. Then there is a pair(C,ψ), whereC is a smooth projective curve andψ :C → X


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a separable morphism such thatψ∗W admits a quotient line bundle of negative degrIndeed, if a pair(C,f ) as in Definition 2.1 violates the condition forW to be separablynumerically effective, then takeψ = π ◦ f , whereπ is defined in (4). Let

ψ∗W → L → 0 (7)

be a quotient line bundle ofψ∗W of negative degree.Since the morphismψ is separable and the vector bundleV ′ is semistable, we conclud

that the vector bundleψ∗V ′ is also semistable [7, p. 278]. Asψ∗V ′ is semistable with

degree(ψ∗V ′) = degree(ψ) · degree(V ′) > 0,

the compositionψ∗V ′ ↪→ ψ∗W → L vanishes (the projection toL is the one in (7)); recalthat degree(L) < 0.

Therefore, there is a projection

h :ψ∗V ∗ → L → 0 (8)

such that the projection in (7) coincides with the composition

ψ∗W = ψ∗V ′ ⊕ ψ∗V ∗ h→ L.

Let

0→ L∗ h∨−→ ψ∗V → (ψ∗V )/L∗ → 0 (9)

be the exact sequence of vector bundles defined by the dual of the homomorphismh in (8).By assumption the vector bundleF ∗

XV admits a quotient line bundle of negative degrLet

f0 :F ∗XV → L0 (10)

be a quotient line bundle of negative degree.Let FC :C → C be the Frobenius morphism of the curveC. We haveψ ◦FC = FX ◦ψ .Consider the composition

F ∗CL∗ F ∗

Ch∨−→ F ∗

Cψ∗V = ψ∗F ∗XV

ψ∗f0−→ ψ∗L0, (11)

where the homomorphismF ∗Ch∨ is the pull-back, byFC , of the homomorphismh∨ in

(9); similarly, ψ∗f0 is the pull-back, byψ , of the homomorphismf0 in (10). Since theline bundleF ∗

CL∗ is of positive degree and the line bundleψ∗L0 is of negative degree thhomomorphism in (11) vanishes. As degree(F ∗

Cψ∗V ) > 0 and degree(ψ∗L0) > 0, fromthe exact sequence

0→ F ∗CL∗ F ∗

Ch∨−→ F ∗

Cψ∗V ψ∗−→ ψ∗L0 → 0

(the exact sequence is obtained from the fact the composition homomorphism ivanishes) we conclude that degree(F ∗

CL∗) > degree(F ∗Cψ∗V )/2. Therefore, the line sub

bundleL∗ h∨−→ ψ∗V in (9) contradicts the semistability condition forψ∗V .On the other hand, asψ is separable andV is semistable, the vector bundleψ∗V in fact

is semistable. Therefore, the vector bundleW is not separably numerically effective. Thcompletes the proof of the proposition.�


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Fix a line bundleζ overX such thatζ⊗2 is isomorphic to the canonical line bundleKX .Such a line bundle is also called atheta characteristic. The unique (up to isomorphismvector bundleV0 that fits in a nontrivial extension

0→ ζ → V0 → ζ ∗ → 0 (12)

is called theGunning bundle.

Example 2.5. Let k be an algebraically closed field of characteristic two. LetX be a smoothprojective curve defined overk of genusg, with g � 4. Fix a closed pointx0 ∈ X. Ifthe genusg is even, then there is a stable vector bundleE0 on X such thatF ∗

XE0 = V0,whereV0 is the Gunning bundle defined in (12) andFX is the Frobenius morphism oX [5, Corollary 4.1]. Ifg is odd then there is a stable vector bundleE′

0 on X such thatF ∗

XE′0 = V0 ⊗OX(x0) [5, Corollary 4.1].

SetV in Proposition 2.4 to be a stable vector bundle overX such that

F ∗XV = E′

0 ⊗OX(x0)

if g is even and

F ∗XV = E′

0

if g is odd (by [5, Corollary 4.1] such a stable vector bundleV exists). Since

degree(ζ ∗ ⊗OX(2x0)

) = 3− g < 0,

the vector bundleV satisfies all the conditions in Proposition 2.4 (the quotient line buof F ∗

XV of negative degree is constructed using the quotientζ ∗ in (12)). Therefore, byProposition 2.4 the vector bundleW := V ′ ⊕ V ∗ is separably numerically effective whiit is not semistable, whereV ′ is any semistable vector bundle overX with degree(V ′) =degree(V ). For example, we can takeV ′ = V or V ′ = ∧2

V . In view of Remark 2.3 thisanswers Question 1.2 negatively.

In fact, there are semistable vector bundles of any given rank and degree. Therefoabove vector bundleV ′ can be chosen to be of arbitrary rank. Hence for any integern � 3there are nonsemistable vector bundles of rankn and degree zero which are not separanumerically effective.

We will give another example giving a negative answer to Question 1.2.

Example 2.6. Let k be any algebraically closed field of characteristicp, with p > 0. LetC be a Mumford curve of genusg (cf. [6]). Assume thatp < g − 1 andp does not divideg − 1. Then there is a stable vector bundleE′

0 on X such thatF ∗XE′

0 is isomorphic tothe Gunning bundleV0; see [4, p. 100, Lemma 5] and [4, p. 99, Proposition 2]. SetV =E′

0 ⊗ OX(x0) in Proposition 2.4, wherex0 is a closed point ofX. It is straightforward tocheck thatV satisfies all the conditions in Proposition 2.4. Therefore, the vector buW := V ′ ⊕ V ∗ is separably numerically effective while it is not semistable, whereV ′ isany semistable vector bundle overX with degree(V ′) = degree(V ).


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101.[5] K. Joshi, S. Ramanan, E.Z. Xia, J.-K. Yu, On vector bundles destabilized by Frobenius pull

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Principal bundles over a curve in positive characteristic, II

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