Sheaves associated to holomorphic first integrals

Zamora, Alexis García

doi:10.5802/aif.1778

Zamora, Alexis García

Annales de l'Institut Fourier, Tome 50 (2000) no. 3, pp. 909-919.

Résumé
Abstract

Soit $S$ une surface projective, lisse et irréductible, soit $ℱ : L \to T S$ un feuilletage sur $S$ avec une intégrale première holomorphe $f : S \to ℙ^{1}$ . Si $h^{0} (S, 𝒪_{S} (- n 𝒦_{S})) > 0$ pour $n \geq 1$ nous démontrons l’inégalité $(2 n - 1) (g - 1) \leq h^{1} (S, ℒ^{' - 1} (- (n - 1) K_{S})) + h_{0} (S, ℒ^{'}) + 1$ . Si $S$ est rationnelle nous démontrons que les images directes du faisceau co-normal sous $f$ sont localement libres et nous donnons des informations sur la nature de leur décomposition comme somme directe des faisceaux inversibles.

Let $ℱ : L \to T S$ be a foliation on a complex, smooth and irreducible projective surface $S$ , assume $ℱ$ admits a holomorphic first integral $f : S \to ℙ^{1}$ . If $h^{0} (S, 𝒪_{S} (- n 𝒦_{S})) > 0$ for some $n \geq 1$ we prove the inequality: $(2 n - 1) (g - 1) \leq h^{1} (S, ℒ^{' - 1} (- (n - 1) K_{S})) + h^{0} (S, ℒ^{'}) + 1$ . If $S$ is rational we prove that the direct image sheaves of the co-normal sheaf of $ℱ$ under $f$ are locally free; and give some information on the nature of their decomposition as direct sum of invertible sheaves.

MR Zbl

DOI : 10.5802/aif.1778

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     author = {Zamora, Alexis Garc{\'\i}a},
     title = {Sheaves associated to holomorphic first integrals},
     journal = {Annales de l'Institut Fourier},
     pages = {909--919},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
     number = {3},
     year = {2000},
     doi = {10.5802/aif.1778},
     mrnumber = {2001g:32075},
     zbl = {01478809},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1778/}
}

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Zamora, Alexis García. Sheaves associated to holomorphic first integrals. Annales de l'Institut Fourier, Tome 50 (2000) no. 3, pp. 909-919. doi : 10.5802/aif.1778. https://www.numdam.org/articles/10.5802/aif.1778/

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