no. 3
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.

@article{AIF_2000__50_3_909_0,
     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/}
}
TY  - JOUR
AU  - Zamora, Alexis García
TI  - Sheaves associated to holomorphic first integrals
JO  - Annales de l'Institut Fourier
PY  - 2000
SP  - 909
EP  - 919
VL  - 50
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1778/
DO  - 10.5802/aif.1778
LA  - en
ID  - AIF_2000__50_3_909_0
ER  - 
%0 Journal Article
%A Zamora, Alexis García
%T Sheaves associated to holomorphic first integrals
%J Annales de l'Institut Fourier
%D 2000
%P 909-919
%V 50
%N 3
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1778/
%R 10.5802/aif.1778
%G en
%F AIF_2000__50_3_909_0
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/

[1] W. Barth, C. Peters, and A. Van De Ven, Compact Complex Surfaces, Springer Verlag, 1984. | MR | Zbl

[2] M. Brunella, Feuilletages holomorphes sur les surfaces complexes compactes, Ann. scient. Ec. Norm. Sup., 30 (1997), 569-594. | Numdam | MR | Zbl

[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. | MR | Zbl

[5] G. Kempf, Algebraic Varieties, Cambridge University Press, 1993. | MR | Zbl

[6] D. Mumford, Abelian Varieties, Oxford University Press, 1970. | MR | Zbl

[7] H. Poincaré, Sur l'intégration algébrique des équations différentielles du primer ordre, Rendiconti del Circolo Matematico di Palermo, 5 (1891), 161-191. | JFM

[8] A. Seidennberg, Reduction of Singularities of the Differential Equation Ady - Bdx, Am. Journal of Math., (1968), 248-269. | Zbl

[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. | MR | Zbl

Cité par Sources :