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

### Alain Hénaut

Université de Bordeaux I, Talence, France

## Abstract

A $d$-web $W(d)$ is given by $d$ complex analytic foliations of codimension $n$ in $(C_{N},0)$ such that the leaves are in general position. We are interested in the geometry of such configurations. A complex $(A_{∙},δ)$ of $C$-vector spaces is defined in which $A_{0}$ corresponds to functions and $A_{p}$ to $p$-forms of the web $W(d)$ for $1≤p≤n$. If $N=kn$ with $k≥2$, it is proved that $r_{p}:=dim_{C}A_{p}$ is a finite analytic invariant of $W(d)$ 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 $P_{k}$ 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 $P_{n+k−1}$ 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 $(A_{∙},δ)$ of the linear web associated to $V_{n}$ in $(C_{kn},0)$ 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