Ordinary holomorphic webs of codimension one
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 1, p. 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 π ' (n,d) which, for n3, 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+hh, let k 0 be the integer such that c(n,k 0 )d<c(n,k 0 +1). The number π ' (n,d) is then equal

  • to 0 for d<c(n,2),
  • and to h=1 k 0 d - c ( n , h ) for dc(n,2).

Moreover, if d is precisely equal to c(n,k 0 ), we define off S a holomorphic connection on a holomorphic bundle of rank π ' (n,d), such that the set of Abelian relations off S is isomorphic to the set of holomorphic sections of 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,k 0 ): we recover the results given in [9].

Published online : 2018-06-21
Classification:  53A60,  14C21,  32S65
@article{ASNSP_2012_5_11_1_197_0,
     author = {Cavalier, Vincent and Lehmann, Daniel},
     title = {Ordinary holomorphic webs of codimension one},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {1},
     year = {2012},
     pages = {197-214},
     zbl = {1244.53014},
     mrnumber = {2953049},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2012_5_11_1_197_0}
}
Cavalier, Vincent; Lehmann, Daniel. Ordinary holomorphic webs of codimension one. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 1, pp. 197-214. http://www.numdam.org/item/ASNSP_2012_5_11_1_197_0/

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