Ordinary holomorphic webs of codimension one

Cavalier, Vincent; Lehmann, Daniel

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 1, pp. 197-214.

To any $d$-web of codimension one on a holomorphic $n$-dimensional manifold $M$ ($d>n$), we associate an analytic subset $S$ of $M$. We call ordinary the webs for which $S$ has a dimension at most $n-1$ or is empty. This condition is generically satisfied, at least at the level of germs.

We prove that the rank of an ordinary $d$-web has an upper-bound ${\pi }^{\text{'}}(n,d)$ which, for $n\ge 3$, is strictly smaller than the bound $\pi (n,d)$ proved by Chern, $\pi (n,d)$ denoting the Castelnuovo’s number. This bound is optimal. Setting $c(n,h)=\left(\begin{array}{c}n-1+hh\end{array}\right)$, let $k_0$ be the integer such that $c(n,k_0)\le d<c(n,k_0+1)$. The number ${\pi }^{\text{'}}(n,d)$ is then equal

• to 0 for $d<c(n,2)$,
• and to ${\sum }_{h=1}^{k_0}\left(d-c(n,h)\right)$ for $d\ge c(n,2)$.

Moreover, if $d$ is precisely equal to $c(n,k_0)$, we define off $S$ a holomorphic connection on a holomorphic bundle $ℰ$ of rank ${\pi }^{\text{'}}(n,d)$, such that the set of Abelian relations off $S$ is isomorphic to the set of holomorphic sections of $ℰ$ with vanishing covariant derivative: the curvature of this connection, which generalizes the Blaschke curvature, is then an obstruction for the rank of the web to reach the value ${\pi }^{\text{'}}(n,d)$.

When $n=2$, $S$ is always empty so that any web is ordinary, ${\pi }^{\text{'}}(2,d)=\pi (2,d)$, and any $d$ may be written $c(2,k_0)$: we recover the results given in [9].

Classification : 53A60, 14C21, 32S65

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