Holomorphic Tensors and Vector Bundles on Projective Varieties

Bogomolov, Fedor

doi:10.1070/im1979v013n03abeh002076

Cited by 250 publications

(236 citation statements)

References 20 publications

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“…In [3,Theorem 5] Bogomolov proved that on a complex projective surface of general type we have 4c 2 ≥ c 2 1 . Later, Miyaoka [27] and Yau [38], proved the optimal inequality 3c 2 ≥ c 2 1 .…”

Section: Theorem 2 Assume That X Be Can Be Lifted To the Ring W 2 (K)mentioning

confidence: 99%

“…It is well known that both these inequalities fail in positive characteristic (see, e.g., [7,15,30,33,37]). In 1989 Shepherd-Barron conjectured (see [34, p. 244]) that the crucial theorem in algebraic proofs of the Bogomolov-Miyaoka-Yau inequality (see [3,Theorem 4] Theorem 13). This is a strong version of the inequality conjectured by Shepherd-Barron (see [34, p. 244]).…”

Section: Theorem 2 Assume That X Be Can Be Lifted To the Ring W 2 (K)mentioning

confidence: 99%

“…In [3] Bogomolov proved a celebrated inequality saying that in case n = 2 and k = C, the discriminant (E) = 2rc 2 (E) − (r − 1)c 2 1 (E) of any slope H -semistable rank r vector bundle E is non-negative. Together with the Mehta-Ramanathan restriction theorem this implies that for k = C and any n ≥ 2 we have (E)H n−2 ≥ 0 for any slope H -semistable torsion free sheaf E of rank r on X .…”

Section: Introductionmentioning

confidence: 99%

“…Roughly, there are three different proofs of this inequality. The first one, due to Bogomolov (see [3,12]), is algebraic, using the fact that symmetric powers of a semistable bundle have few sections. The second one, due to Gieseker (see [11]), is also algebraic, using reduction to positive characteristic and studying sections of the Frobenius pull back of the bundle.…”

Section: Introductionmentioning

confidence: 99%

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Bogomolov’s inequality for Higgs sheaves in positive characteristic

Langer

2014

Invent. math.

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We prove Bogomolov's inequality for Higgs sheaves on varieties in positive characteristic p that can be lifted modulo p 2 . This implies the Miyaoka-Yau inequality on surfaces of non-negative Kodaira dimension liftable modulo p 2 . This result is a strong version of Shepherd-Barron's conjecture. Our inequality also gives the first algebraic proof of Bogomolov's inequality for Higgs sheaves in characteristic zero, solving the problem posed by Narasimhan.

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Section: Theorem 2 Assume That X Be Can Be Lifted To the Ring W 2 (K)mentioning

confidence: 99%

Section: Theorem 2 Assume That X Be Can Be Lifted To the Ring W 2 (K)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bogomolov’s inequality for Higgs sheaves in positive characteristic

Langer

2014

Invent. math.

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“…(This conjecture predicts the finiteness of curves of geometric genus ≤ 1 on every surface of general type. By means of a different approach, Bogomolov himself settled a weaker form of the conjecture in [3], covering remarkably general cases of it).…”

Section: χ = 2g − 2 + (S)mentioning

confidence: 99%

An $abcd$ theorem over function fields and applications

Corvaja¹,

Zannier²

2011

Bul. Soc. Math. France

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Abstract. -We provide a lower bound for the number of distinct zeros of a sum 1 + u + v for two rational functions u, v, in term of the degree of u, v, which is sharp whenever u, v have few distinct zeros and poles compared to their degree. This sharpens the "abcd-theorem" of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface x a + y a + z c = 1 contains only finitely many rational or elliptic curves, provided a ≥ 10 4 and c ≥ 2; this provides special cases of a known conjecture of Bogomolov. Résumé (Un théorème abcd sur les corps de fonctions et applications)Nous démontrons une minoration pour le nombre de zéros distincts d'une somme 1 + u + v, u, v étant deux fonctions rationnelles, en fonction du degré de u et v; cette minoration est forte si le nombre de zéros et poles de u, v est suffisament petit par rapport à leur degré. Dans certains cas, on obtient une amélioration de l'inégalité de Voloch et Brownawell-Masser, qui entraîne des nouveaux résultats de finitude sur les équations diophantiennes sur les corps de fonctions.Par exemple, nous démontrons que la surface de type Fermat définie par l'équation x a +y a +z c = 1 ne contient qu'un nombre fini de courbes rationnelles ou elliptiques, dès que a ≥ 10 4 et c ≥ 2. Ce résultat constitue un cas particulier d'une célèbre conjecture de Bogomolov.

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Fujita's conjecture for quasi‐elliptic surfaces

Chen

2021

Mathematische Nachrichten

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Holomorphic Tensors and Vector Bundles on Projective Varieties

Cited by 250 publications

References 20 publications

Bogomolov’s inequality for Higgs sheaves in positive characteristic

Bogomolov’s inequality for Higgs sheaves in positive characteristic

An $abcd$ theorem over function fields and applications

Fujita's conjecture for quasi‐elliptic surfaces

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