Let X be a smooth projective curve of genus $g\geq 2$ defined over an
algebraically closed field k of characteristic $p>0$ and let $F:X\rightarrow
X_{1}$ be the relative k-linear Frobenius map. We prove (Theorem 1.1) E is a
stable bundle on $X_{1}$ with $I(E)= (p-1)(2g-2)$ if and only if E is the
direct image of some stable bundle W on $X$.Comment: 4 page

Liu, Congjun

Zhou, Mingshuo

Comptes Rendus Mathematique

English

arXiv

Congjun Liu

Mingshuo Zhou

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 351 (2013) 381–383Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comAlgebraic GeometryStable bundles as Frobenius morphism direct imageFaisceaux stables en tant qu’images directes par le morphisme de FrobeniusCongjun Liu, Mingshuo ZhouInstitute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, PR Chinaa r t i c l e i n f o a b s t r a c tArticle history:Received 27 January 2013Accepted after revision 26 April 2013Available online 4 June 2013Presented by Claire VoisinLet X be a smooth projective curve of genus g  2 over an algebraically closed field k ofcharacteristic p > 0, and let F : X → X1 be the relative Frobenius morphism. We show thata vector bundle E on X1 is the direct image under F of some stable bundle on X if andonly if the instability of F ∗ E is equal to (p − 1)(2g − 2).© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éSoient X une courbe projective lisse de genre g  2 définie sur un corps k algébriquementclos de caractéristique p > 0, et F : X → X1 le morphisme de Frobenius relatif. On montrequ’un fibré vectoriel E sur X1 est l’image directe sous F d’un certain fibré stable sur X siet seulement si l’instabilité de F ∗ E est égale à (p − 1)(2g − 2).© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionLet X be a smooth projective curve of genus g  2 defined over an algebraically closed field k of characteristic p > 0.The absolute Frobenius morphism F X : X → X is induced by OX → OX , f → f p . Let F : X → X1 := X ×k k denote therelative Frobenius morphism over k. One of the themes is to study its action on the geometric objects on X . Recall that avector bundle E on a smooth projective curve is called semi-stable (resp. stable) if μ(E ′)  μ(E) (resp. μ(E ′) < μ(E)) forany nontrivial proper subbundle E ′ ⊂ E , where μ(E) is the slope of E . It is known that F∗ preserves the stability of vectorbundles (cf. [5]), but F ∗ does not preserve the semi-stability of vector bundles (cf. [1] for example).Semi-stable bundles are basic constituents of vector bundles in the sense that any bundle E admits a unique filtration:HN•(E): 0 = HN0(E) ⊂ HN1(E) ⊂ · · · ⊂ HN(E) = E,which is the so-called Harder–Narasimhan filtration, such that:(1) grHNi (E) := HNi(E)/HNi−1(E) (1  i  ) are semi-stable;(2) μ(grHN1 (E)) > μ(grHN2 (E)) > · · · > μ(grHN (E)).The rational number I(E) := μ(grHN1 (E)) − μ(grHN (E)), which measures how far a vector bundle is from being semi-stable,is called the instability of E . It is clear that E is semi-stable if and only if I(E) = 0.E-mail addresses: liucongjun@amss.ac.cn (C. Liu), zhoumingshuo@amss.ac.cn (M. Zhou).1631-073X/$ – see front matter © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.crma.2013.04.021382 C. Liu, M. Zhou / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 381–383Given a semi-stable bundle E on X1, then F ∗E may not be semi-stable, so it is natural to consider the instability I(F ∗E).In [4, Theorem 3.1], the author proves I(F ∗E)  ( − 1)(2g − 2), where  is the length of Harder–Narasimhan filtrationof F ∗E . If E = F∗W where W is a stable bundle on X , we know, by Sun’s theorem [5, Theorem 2.2], that E is stable,the length of Harder–Narasimhan filtration of F ∗E is p and I(F ∗E) = (p − 1)(2g − 2). Thus I(F ∗E) = (p − 1)(2g − 2) is anecessary condition for E to be a direct image under Frobenius. In this short note, we show the following theorem:Theorem 1. Let E be a stable vector bundle on X. Then the following statements are equivalent:(1) There exists a stable bundle W such that E = F∗W ;(2) I(F ∗E) = (p − 1)(2g − 2).The case rk E = p was proved in [3]. Our observation is that the arguments in [3] together with Sun’s theorem imply thegeneral case.2. Proof of the theoremLet X be a smooth projective curve over an algebraically closed field k with char(k) = p > 0. The absolute Frobeniusmorphism F X : X → X is induced by the following homomorphism:OX → OX , f → f p.Let F : X → X1 := X ×k k denote the relative Frobenius morphism over k that satisfies the following commutative diagram:XF XF X1 XSpec(k)Fk Spec(k).For a vector bundle E on X , the slope of E is defined asμ(E) := deg Erk Ewhere rk E (resp. deg E) denotes the rank (resp. degree) of E . Then:Definition 1. A vector bundle E on X is called semi-stable (resp. stable) if for any nontrivial proper subbundle E ′ ⊂ E , wehaveμ(E ′) (resp. <)μ(E).Theorem 2 (Harder–Narasimhan filtration). For any vector bundle E, there is a unique filtration:HN•(E): 0 = HN0(E) ⊂ HN1(E) ⊂ · · · ⊂ HN(E) = E,which is called Harder–Narasimhan filtration, such that:(1) grHNi (E) := HNi(E)/HNi−1(E) (1  i  ) are semi-stable;(2) μ(grHN1 (E)) > μ(grHN2 (E)) > · · · > μ(grHN (E)).By using this unique filtration of E , an invariant I(E) of E , which is called the instability of E was introduced (see [5]and [4]). It is a rational number and measures how far is E from being semi-stable.Definition 2. Let μmax(E) = μ(grHN1 (E)), μmin(E) = μ(grHN (E)). Then the instability of E is defined to beI(E) := μmax(E) − μmin(E).It is easy to see that a vector bundle E is semi-stable if and only if I(E) = 0.For any semi-stable bundle E , letHN•(F ∗E): 0 = HN0(F ∗E) ⊂ HN1(F ∗E) ⊂ · · · ⊂ HN(F ∗E) = F ∗Ebe the Harder–Narasimhan filtration of F ∗E . Then we have the following lemma, which is implicit in [3].C. Liu, M. Zhou / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 381–383 383Lemma 1. For any semi-stable bundle E, we haveμmax(F ∗E) p · μ(E) + (p − 1)(g − 1);μmin(F ∗E) p · μ(E) − (p − 1)(g − 1),and if I(F ∗E) = μmax(F ∗E) − μmin(F ∗E) = (p − 1)(2g − 2). Thenμmax(F ∗E) = p · μ(E) + (p − 1)(g − 1);μmin(F ∗E) = p · μ(E) − (p − 1)(g − 1).Now we prove our theorem by using this Lemma 1, the canonical filtration on the vector bundle V = F ∗ F∗W and Sun’stheorem on the stability of Frobenius’ direct images.Proof of Theorem 1. (1) ⇒ (2). In [2, Section 5.3], there is a canonical filtration on the vector bundle V = F ∗ F∗W :0 = V 0 ⊂ V 1 ⊂ · · · ⊂ V−1 ⊂ V ⊂ · · · ⊂ V p−1 ⊂ V p = V ,which is indeed the Harder–Narasimhan filtration on V , and satisfiesV/V−1 ∼= (V+1/V) ⊗ Ω1Xfor 1   p − 1, and V p/V p−1 ∼= W . So μ(V p/V p−1) = μ(W ), μ(V 0/V 1) = μ(W ) + (p − 1)(2g − 2), and now the result isclear.(2) ⇒ (1). Since I(F ∗E) = (p − 1)(2g − 2), we have μmax(F ∗E) = p · μ(E) + (p − 1)(g − 1), μmin(F ∗E) = p · μ(E) −(p − 1)(g − 1) by Lemma 1. We consider the surjection:F ∗E → grHN(F ∗E).The bundle grHN (F∗E) is semi-stable of slope μmin(F ∗E). Replacing grHN (F ∗E) by a stable graded piece W in the Jordan–Hölder filtration of grHN (F∗E), we have a surjection:F ∗E → W ,where W is a stable bundle of slope μ(W ) = μmin(F ∗E) = p · μ(E) − (p − 1)(g − 1). By adjunction, we have a nontrivialmorphism:ψ : E → F∗W .By Sun’s theorem (cf. [5, Theorem 2.2]), we know that F∗W is a stable bundle of slope:μ(F∗W ) = μ(W )p+ (p − 1)(g − 1)p= μ(E).Thus ψ induce an isomorphism:E ∼= F∗W . AcknowledgementThe authors would like to thank their advisor Professor Xiaotao Sun for encouragements and many useful discussions.References[1] D. Gieseker, Stable vector bundles and the Frobenius morphism, Ann. Sci. Éc. Norm. Super. (4) 6 (1973) 95–101.[2] K. Joshi, S. Ramanan, E. Xia, J.-K. Yu, On vector bundles destabilized by Frobenius pull-back, Compos. Math. 142 (3) (2006) 616–630.[3] V. Mehta, C. Pauly, Semistability of Frobenius direct images over curves, Bull. Soc. Math. Fr. 135 (2007) 105–117.[4] X. Sun, Remarks on semistability of G-bundles in positive characteristic, Compos. Math. 119 (1999) 41–52.[5] X. Sun, Direct images of bundles under Frobenius morphism, Invent. Math. 173 (2008) 427–447.

Stable bundles as Frobenius morphism direct image

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