Sheaves associated to holomorphic first integrals
Annales de l'Institut Fourier, Tome 50 (2000) no. 3, pp. 909-919.

Soit S une surface projective, lisse et irréductible, soit ℱ:L→TS un feuilletage sur S avec une intégrale première holomorphe f:S→ℙ 1 . Si h 0 (S,𝒪 S (-n𝒦 S ))>0 pour n≥1 nous démontrons l’inégalité (2n-1)(g-1)≤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→TS be a foliation on a complex, smooth and irreducible projective surface S, assume ℱ admits a holomorphic first integral f:S→ℙ 1 . If h 0 (S,𝒪 S (-n𝒦 S ))>0 for some n≥1 we prove the inequality: (2n-1)(g-1)≤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.

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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 = {http://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. http://www.numdam.org/articles/10.5802/aif.1778/

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