Rank of ordinary webs in codimension one an effective method

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 below), we proved in [CL] that their rank $\rho (W)$ is upper-bounded by a certain number $\pi^\prime (n, d)$ (which, for $n \geq 3$, is strictly 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\[c(n, h_0 - 1) \lt d \leq c(n, h_0),\]$\pi^\prime (n, d)$ is just the first number of a decreasing sequence of positive integers\[\pi^\prime (n, d) = \rho_{h_0 - 2} \geq \rho_{h_0 - 1} \geq \dotsc \geq \rho_h \geq \rho_{h+1} \geq \dotsc \geq \rho_\infty = \rho (W) \geq 0\]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, and compute the rank without to have to exhibit explicitly independant abelian relations.

Examples will be given.

Keywords

ordinary webs, abelian relation, rank, connection, curvature

2010 Mathematics Subject Classification

Primary 53A60. Secondary 14C21, 14H45, 53C05.

Full Text (PDF format)

Received 24 January 2018

Accepted 21 December 2018

Published 17 February 2021