Formes différentielles abéliennes, bornes de Castelnuovo et géométrie des tissus

Abstract

A $d$-web $\mathcal{W}(d)$ is given by $d$ complex analytic foliations of codimension $n$ in $({\mathbb{C}}^N,0)$ such that the leaves are in general position. We are interested in the geometry of such configurations. A complex $({\mathcal{A}}^{\bullet },\delta )$ of $\mathbb{C}$-vector spaces is defined in which ${\mathcal{A}}^0$ corresponds to functions and ${\mathcal{A}}^p$ to $p$-forms of the web $\mathcal{W}(d)$ for $1\le p\le n$. If $N=kn$ with $k\ge 2$, it is proved that $r_p:={\dim }_{\mathbb{C}}{\mathcal{A}}^p$ is a finite analytic invariant of $\mathcal{W}(d)$ 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 ${\mathbb{P}}^k$ 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 ${\mathbb{P}}^{n+k-1}$ 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 $({\mathcal{A}}^{\bullet },\delta )$ of the linear web associated to $V_n$ in $({\mathbb{C}}^{kn},0)$ are related.