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Sheaves associated to holomorphic first integrals. (English) Zbl 1063.37044

Summary: Let \({\mathcal F}: L \to TS\) be a foliation on a complex, smooth and irreducible projective surface \(S\), assume \({\mathcal F}\) admits a holomorphic first integral \(f:S \rightarrow {\mathbb{P}}^1\). If \(h^0(S,{\mathcal O}_S(-n{\mathcal K}_S))>0\) for some \(n\geq 1\) we prove the inequality: \((2n-1)(g-1) \leq h^1(S, {\mathcal L}^{\prime -1}(-(n-1)K_S)) +h^0 (S, {\mathcal L}^\prime) +1\). If \(S\) is rational we prove that the direct image sheaves of the co-normal sheaf of \({\mathcal F}\) under \(f\) are locally free; and give some information on the nature of their decomposition as direct sum of invertible sheaves.

MSC:

37F75 Dynamical aspects of holomorphic foliations and vector fields
34A26 Geometric methods in ordinary differential equations
14H99 Curves in algebraic geometry
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