On the number of singularities in meromorphic foliations on compact complex surfaces

Ballico, Edoardo

doi:10.1090/S0002-9947-1994-1191609-0

On the number of singularities in meromorphic foliations on compact complex surfaces

Author: Edoardo Ballico
Journal: Trans. Amer. Math. Soc. 343 (1994), 417-432
MSC: Primary 32L30; Secondary 14J60
DOI: https://doi.org/10.1090/S0002-9947-1994-1191609-0
MathSciNet review: 1191609
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Here we study meromorphic foliations with singularities on a smooth compact complex surface. Aim: study the locations of the singularities, using vector bundle techniques and techniques introduced in the cohomological study of projective geometry.

Edoardo Ballico, Generators for the homogeneous ideal of $s$ general points in ${\bf P}_3$, J. Algebra 106 (1987), no. 1, 46–52. MR 878467, DOI https://doi.org/10.1016/0021-8693%2887%2990020-2
Edoardo Ballico, On the homogeneous ideal of projectively normal curves, Ann. Mat. Pura Appl. (4) 154 (1989), 83–90. MR 1043066, DOI https://doi.org/10.1007/BF01790343
W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574
Alessio Corti, Polynomial bounds for the number of automorphisms of a surface of general type, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 1, 113–137. MR 1088272
Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. MR 0354657
Alan Eastwood, Collision de biais et application à l’interpolation, Manuscripta Math. 67 (1990), no. 3, 227–249 (French). MR 1046987, DOI https://doi.org/10.1007/BF02568431
G. Elencwajg and O. Forster, Vector bundles on manifolds without divisors and a theorem on deformations, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 4, 25–51 (1983) (English, with French summary). MR 694127
Xavier Gómez-Mont, The transverse dynamics of a holomorphic flow, Ann. of Math. (2) 127 (1988), no. 1, 49–92. MR 924673, DOI https://doi.org/10.2307/1971416
Xavier Gómez-Mont, Holomorphic foliations in ruled surfaces, Trans. Amer. Math. Soc. 312 (1989), no. 1, 179–201. MR 983870, DOI https://doi.org/10.1090/S0002-9947-1989-0983870-8
Xavier Gómez-Mont, On closed leaves of holomorphic foliations by curves, Algebraic geometry and complex analysis (Pátzcuaro, 1987) Lecture Notes in Math., vol. 1414, Springer, Berlin, 1989, pp. 61–98. MR 1042357, DOI https://doi.org/10.1007/BFb0090252
Xavier Gómez-Mont, Foliations by curves of complex analytic spaces, The Lefschetz centennial conference, Part III (Mexico City, 1984) Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1987, pp. 123–141. MR 893860
Xavier Gómez-Mont, Integrals for holomorphic foliations with singularities having all leaves compact, Ann. Inst. Fourier (Grenoble) 39 (1989), no. 2, 451–458 (English, with French summary). MR 1017286
Xavier Gómez-Mont and George Kempf, Stability of meromorphic vector fields in projective spaces, Comment. Math. Helv. 64 (1989), no. 3, 462–473. MR 998859, DOI https://doi.org/10.1007/BF02564687
Xavier Gómez-Mont and Jesús Muciño, Persistent cycles for holomorphic foliations having a meromorphic first integral, Holomorphic dynamics (Mexico, 1986) Lecture Notes in Math., vol. 1345, Springer, Berlin, 1988, pp. 129–162. MR 980957, DOI https://doi.org/10.1007/BFb0081400
Robin Hartshorne and André Hirschowitz, Droites en position générale dans l’espace projectif, Algebraic geometry (La Rábida, 1981) Lecture Notes in Math., vol. 961, Springer, Berlin, 1982, pp. 169–188 (French). MR 708333, DOI https://doi.org/10.1007/BFb0071282
A. Hirschowitz, Sur la postulation générique des courbes rationnelles, Acta Math. 146 (1981), no. 3-4, 209–230 (French). MR 611384, DOI https://doi.org/10.1007/BF02392464
André Hirschowitz, La méthode d’Horace pour l’interpolation à plusieurs variables, Manuscripta Math. 50 (1985), 337–388 (French, with English summary). MR 784148, DOI https://doi.org/10.1007/BF01168836
Klaus Hulek, Stable rank-$2$ vector bundles on ${\bf P}_{2}$ with $c_{1}$ odd, Math. Ann. 242 (1979), no. 3, 241–266. MR 545217, DOI https://doi.org/10.1007/BF01420729
Shoshichi Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ; Princeton University Press, Princeton, NJ, 1987. Kanô Memorial Lectures, 5. MR 909698
Monica Idà, On the homogeneous ideal of the generic union of lines in ${\bf P}^3$, J. Reine Angew. Math. 403 (1990), 67–153. MR 1030412, DOI https://doi.org/10.1515/crll.1990.403.67
Masaki Maruyama, Elementary transformations in the theory of algebraic vector bundles, Algebraic geometry (La Rábida, 1981) Lecture Notes in Math., vol. 961, Springer, Berlin, 1982, pp. 241–266. MR 708337, DOI https://doi.org/10.1007/BFb0071286
Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. MR 561910

D. Perrin, Courbes passant par m points généraux de ${{\mathbf {P}}^3}$, Mém. Soc. Math. France 28/29 (1987).

Leslie G. Roberts, The ideal generation conjecture for $28$ points in ${\bf P}^3$, Canad. J. Math. 38 (1986), no. 5, 1228–1238. MR 869723, DOI https://doi.org/10.4153/CJM-1986-062-x