# ANNALES DE L'INSTITUT FOURIER

A splitting theorem for the Kupka component of a foliation of $ℂ{ℙ}^{n},\phantom{\rule{0.277778em}{0ex}}n\ge 6$. Addendum to an addendum to a paper by Calvo-Andrade and Soares
Annales de l'Institut Fourier, Tome 49 (1999) no. 4, pp. 1423-1425.

On considère ici les feuilletages holomorphes singuliers de codimension 1 dans $ℂ{ℙ}^{n},\phantom{\rule{0.277778em}{0ex}}n\ge 6$ avec une composante de Kupka compacte $K$. On démontre que $K$ est une intersection complète.

Here we show that a Kupka component $K$ of a codimension 1 singular foliation $F$ of $ℂ{ℙ}^{n},\phantom{\rule{0.277778em}{0ex}}n\ge 6$ is a complete intersection. The result implies the existence of a meromorphic first integral of $F$. The result was previously known if $\mathrm{deg}\left(K\right)$ was assumed to be not a square.

@article{AIF_1999__49_4_1423_0,
author = {Ballico, Edoardo},
title = {A splitting theorem for the {Kupka} component of a foliation of ${\mathbb {C}}{\mathbb {P}}^n,\;n\ge 6$. {Addendum} to an addendum to a paper by {Calvo-Andrade} and {Soares}},
journal = {Annales de l'Institut Fourier},
pages = {1423--1425},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {49},
number = {4},
year = {1999},
doi = {10.5802/aif.1723},
zbl = {0959.32037},
mrnumber = {2001b:32059},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1723/}
}
Ballico, Edoardo. A splitting theorem for the Kupka component of a foliation of ${\mathbb {C}}{\mathbb {P}}^n,\;n\ge 6$. Addendum to an addendum to a paper by Calvo-Andrade and Soares. Annales de l'Institut Fourier, Tome 49 (1999) no. 4, pp. 1423-1425. doi : 10.5802/aif.1723. https://aif.centre-mersenne.org/articles/10.5802/aif.1723/

[B] E. Ballico, A splitting theorem for the Kupka component of a foliation of ℂℙn, n ≥ 6. Addendum to a paper by Calvo-Andrade and Soares, Ann. Inst. Fourier, 45-4 (1995), 1119-1121. | Numdam | Zbl 0831.58046

[CS] O. Calvo-Andrade, M. Soares, Chern numbers of a Kupka component, Ann. Inst. Fourier, 44-4 (1994), 1219-1236. | Numdam | MR 95m:32045 | Zbl 0811.32024

[CL] D. Cerveau, A. Lins, Codimension one foliations in ℂℙn n ≥ 3, with Kupka components, in Complex analytic methods in dinamical systems, Astérisque, (1994), 93-133. | Zbl 0823.32014

[F] G. Faltings, Ein Kriterium für vollständige Durchsnitte, Invent. Math., 62 (1981), 393-401. | MR 82f:14050 | Zbl 0456.14027

[FL] W. Fulton, R. Lazarsfeld, Connectivity in algebraic geometry, in Algebraic Geometry, Proceedings Chicago 1980, Lect. Notes in Math. 862, Springer-Verlag (1981), 26-92. | MR 83i:14002 | Zbl 0484.14005

[GN] X. Gomez-Mont, A. Lins-Neto, A structural stability of foliations with a meromorphic firs integral, Topology, 30 (1990), 315-334. | MR 92j:32114 | Zbl 0735.57014

[GH] Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, 1978. | Zbl 0408.14001

[OSS] Ch. Okonek, M. Schneider, H. Spindler, Vector Bundles on Complex Projective spaces, Progress in Math., 3, Birkhäuser, Basel, 1978.