Formes différentielles abéliennes, bornes de Castelnuovo et géométrie des tissus

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