# Formes différentielles abéliennes, bornes de Castelnuovo et géométrie des tissus

### Alain Hénaut

Université de Bordeaux I, Talence, France

## Abstract

A $d$-web \( {\Cal W}(d) \) is given by $d$ complex analytic foliations of codimension $n$ in \( ({\sumbbb C}^N,0) \) such that the leaves are in general position. We are interested in the geometry of such configurations. A complex \( ({\Cal A}^{\bullet},\delta) \) of \( {\sumbbb C} \)-vector spaces is defined in which \( {\Cal A}^0 \) corresponds to functions and \( {\Cal A}^p \) to $p$-forms of the web \( {\Cal W}(d) \) for $1≤p≤n$. If $N=kn$ with $k≥2$, it is proved that \( r_p:=\dim_{\,\sumbbb C}{\Cal A}^p \) is a finite analytic invariant of \( {\Cal W}(d) \) with an optimal upper bound $π_{p}(d,k,n)$ for $0≤p≤n$. These bounds generalize the Castelnuovos ones for genus of curves in \( {\sumbbb P}^{k} \) with degree $d$. Some characterization of the the space $H_{0}(V_{n},ω_{V_{n}})$ of abelian differentials to an algebraic variety $V_{n}$ in \( {\sumbbb P}^{n+k-1} \) of pure dimension $n$ with degree $d$ is given. Moreover, using duality and Abels theorem, we investigate how for suitable $V_{n}$ the natural complex $(H_{0}(V_{n},ω_{V_{n}}),d)$ and the abelian relation complex \( ({\Cal A}^{\bullet},\delta) \) of the linear web associated to $V_{n}$ in \( ({\sumbbb C}^{kn},0) \) are related.

## Cite this article

Alain Hénaut, Formes différentielles abéliennes, bornes de Castelnuovo et géométrie des tissus. Comment. Math. Helv. 79 (2004), no. 1, pp. 25–57

DOI 10.1007/S00014-003-0787-4