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

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.

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.

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
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