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.


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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