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.

[B] Edoardo Ballico, Generators for the homogeneous ideal of 𝑠 general points in 𝑃₃, J. Algebra 106 (1987), no. 1, 46–52. MR 878467, https://doi.org/10.1016/0021-8693(87)90020-2
[B1] Edoardo Ballico, On the homogeneous ideal of projectively normal curves, Ann. Mat. Pura Appl. (4) 154 (1989), 83–90. MR 1043066, https://doi.org/10.1007/BF01790343
[BPS] 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
[C] 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
[De] Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
[DK] 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
[Ea] Alan Eastwood, Collision de biais et application à l’interpolation, Manuscripta Math. 67 (1990), no. 3, 227–249 (French). MR 1046987, https://doi.org/10.1007/BF02568431
[EF] 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
[G1] Xavier Gómez-Mont, The transverse dynamics of a holomorphic flow, Ann. of Math. (2) 127 (1988), no. 1, 49–92. MR 924673, https://doi.org/10.2307/1971416
[G2] Xavier Gómez-Mont, Holomorphic foliations in ruled surfaces, Trans. Amer. Math. Soc. 312 (1989), no. 1, 179–201. MR 983870, https://doi.org/10.1090/S0002-9947-1989-0983870-8
[G3] 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, https://doi.org/10.1007/BFb0090252
[G4] 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
[G5] 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
[GK] 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, https://doi.org/10.1007/BF02564687
[GM] 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, https://doi.org/10.1007/BFb0081400
[HH] 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, https://doi.org/10.1007/BFb0071282
[H1] A. Hirschowitz, Sur la postulation générique des courbes rationnelles, Acta Math. 146 (1981), no. 3-4, 209–230 (French). MR 611384, https://doi.org/10.1007/BF02392464
[H2] André Hirschowitz, La méthode d’Horace pour l’interpolation à plusieurs variables, Manuscripta Math. 50 (1985), 337–388 (French, with English summary). MR 784148, https://doi.org/10.1007/BF01168836
[Hu] Klaus Hulek, Stable rank-2 vector bundles on 𝑃₂ with 𝑐₁ odd, Math. Ann. 242 (1979), no. 3, 241–266. MR 545217, https://doi.org/10.1007/BF01420729
[K] 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
[I] Monica Idà, On the homogeneous ideal of the generic union of lines in 𝑃³, J. Reine Angew. Math. 403 (1990), 67–153. MR 1030412, https://doi.org/10.1515/crll.1990.403.67
[M] 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, https://doi.org/10.1007/BFb0071286
[OSS] Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. MR 561910
[Pe] D. Perrin, Courbes passant par m points généraux de ${{\mathbf{P}}^3}$ , Mém. Soc. Math. France 28/29 (1987).
[R] Leslie G. Roberts, The ideal generation conjecture for 28 points in 𝑃³, Canad. J. Math. 38 (1986), no. 5, 1228–1238. MR 869723, https://doi.org/10.4153/CJM-1986-062-x