Ordinary holomorphic webs in codimension one

Abstract

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 π ′(n, d) which, for n ≥ 3, is strictly smaller than the bound π(n, d) proved by Chern, π(n, d) denoting the Castelnuovo’s number. This bound is optimal. Setting c(n,h)= ( n−1+h h ) , let k0 be the integer such that c(n,k0)≤d< c(n, k0+1). The number π ′(n, d) is then equal to 0 for d < c(n, 2), and to ∑k0 h=1 ( d − c(n, h) ) for d ≥ c(n, 2). Moreover, if d is precisely equal to c(n, k0), we define off S a holomorphic connection on a holomorphic bundle E of rank π ′(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 π ′(n, d). When n=2, S is always empty so that any web is ordinary, π ′(2,d)=π(2,d), and any d may be written c(2, k0): we recover the results given in [9]. Mathematics Subject Classification (2010): 53A60 (primary); 14C21, 32S65 (secondary).