## Pull-back components of the space of holomorphic foliations on CP(N), N at least 3.

With D. Cerveau and A. Lins Neto. To appear in Journal of Algebraic Geometry.

### Summary

My contribution to this article was an Appendix, to be found in various formats below. This appendix proves the following result. Let $S$ be a complex analytic variety, with a given point~$s$. Let $n\geq 1$ be an integer, and $X\to S$ be a smooth analytic family of projective complex analytic varieties of dimension $d>0$, embedded in $\PP^n_S:=\PP^n(\CC)\times S$. Suppose that the fibre $X_s$ at $s$ is a complete intersection. Then there is an open neighborhood of $s$ in $S$ over which $X$ is a complete intersection.

As already written in the appendix, it was very probable that this result is not new. In fact, the editor of the Journal of Algebraic Geometry says that it can be found in Sernesi, E., Small deformations of global complete intersections. Boll. Un. Mat. Ital. (4) 12 (1975), no. 1-2, 138--146'', and in Catanese, F., Moduli of algebraic surfaces. Theory of moduli (Montecatini Terme, 1985), 1--83, Lecture Notes in Math., 1337, Springer, Berlin-New York, 1988''.

As a consequence, the appendix has been replaced by these two references. It might be that the appendix can still be useful, because it addresses exactly the cases needed in the article, and not more (or less).

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###### Dernière modification: 09 novembre 1999
edix@maths.univ-rennes1.fr