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 $\pi'(n,d)\ \bigl($which, for $n\geq 3$, is stictly smaller than the Castelnuovo-Chern's bound $\pi(n,d)\bigr)$. 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 $$c(n,h_0-1)<d\leq c(n,h_0),$$ $\pi'(n,d)$ is just the first number of a decreasing sequence of positive integers $$\pi'(n,d)=\rho_{h_0-2}\geq \rho_{h_0-1}\geq \cdots\geq \rho_{h}\geq \rho_{h+1}\geq\cdots\geq \rho_{\infty}=\rho(W)\geq 0 $$ 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 ($\rho_h=\rho_{h+1}, \ h\geq h_0-2$), 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.