## 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

### Keywords:

holomorphic foliations; first integrals
Full Text:

### References:

 [1] W. BARTH, C. PETERS, and A. VAN DE VEN, Compact complex surfaces, Springer Verlag, 1984. · Zbl 0718.14023 [2] M. BRUNELLA, Feuilletages holomorphes sur LES surfaces complexes compactes, Ann. scient. Ec. Norm. Sup., 30 (1997), 569-594. · Zbl 0893.32019 [3] X. GOMEZ-MONT, and R. VILA, On meromorphic integrals of holomorphic foliations in surfaces, Unpublished. [4] P. GRIFFITHS, and J. HARRIS, Principles of algebraic geometry, John Wiley & Sons, 1978. · Zbl 0408.14001 [5] G. KEMPF, Algebraic varieties, Cambridge University Press, 1993. · Zbl 0780.14001 [6] D. MUMFORD, Abelian varieties, Oxford University Press, 1970. · Zbl 0223.14022 [7] H. POINCARÉ, Sur l’intégration algébrique des équations différentielles du primer ordre, Rendiconti del Circolo Matematico di Palermo, 5 (1891), 161-191. · JFM 23.0319.01 [8] A. SEIDENNBERG, Reduction of singularities of the differential equation ady - bdx, Am. Journal of Math., (1968), 248-269. · Zbl 0159.33303 [9] A.G. ZAMORA, Foliations on algebraic surfaces having a rational first integral, Public. Mat de la Universitá Aut. de Barcelona, 41 (1997), 357-373. · Zbl 0910.32039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.