Rank of ordinary webs in codimension one. An effective method

Abstract

We are interested by holomorphic $d$-webs $W$ of codimension one in a complex $n$-dimensional manifold $M$. If they are ordinary, i.e. if they satisfy to some condition of genericity (whose precise definition is recalled), we proved in [CL] that their rank $\rho (W)$ is upper-bounded by a certain number ${\pi }^{\prime }(n,d)($which, for $n\ge 3$, is stictly smaller than the Castelnuovo-Chern's bound $\pi (n,d))$. In fact, denoting by $c(n,h)$ the dimension of the space of homogeneous polynomials of degree $h$ with $n$ unknowns, and by $h_0$ the integer such that $$c(n,h_0-1)<d\le c(n,h_0),$$ ${\pi }^{\prime }(n,d)$ is just the first number of a decreasing sequence of positive integers $${\pi }^{\prime }(n,d)={\rho }_{h_0-2}\ge {\rho }_{h_0-1}\ge \cdots \ge {\rho }_h\ge {\rho }_{h+1}\ge \cdots \ge {\rho }_{\infty }=\rho (W)\ge 0$$ becoming stationary equal to $\rho (W)$ after a finite number of steps. This sequence is an interesting invariant of the web, refining the data of the only rank. The method is effective : theoretically, we can compute ${\rho }_h$ for any given $h$ ; and, as soon as two consecutive such numbers are equal (${\rho }_h={\rho }_{h+1},h\ge h_0-2$), we can construct a holomorphic vector bundle $R_h\to M$ of rank ${\rho }_h$, equipped with a tautological holomorphic connection ${\nabla }^h$ whose curvature $K^h$ vanishes iff the above sequence is stationary from there. Thus, we may stop the process at the first step where the curvature vanishes. Examples will be given.

13 pages, no figures