# 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 $1\leq p\leq n$. If $N=kn$ with $k\geq 2$, 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 $\pi_{p}(d,k,n)$ for $0\leq p\leq n$. These bounds generalize the Castelnuovos ones for genus of curves in ${\sumbbb P}^{k}$ with degree $d$. Some characterization of the the space $H^0(V_n,\omega^p_{V_n})$ 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 $\bigr(H^0(V_n,\omega^{\bullet}_{V_n}),d\,\bigr)$ and the abelian relation complex $({\Cal A}^{\bullet},\delta)$ of the linear web associated to $V_n$ in $({\sumbbb C}^{kn},0)$ are related.