To any -web of codimension one on a holomorphic -dimensional manifold (), we associate an analytic subset of . We call ordinary the webs for which has a dimension at most or is empty. This condition is generically satisfied, at least at the level of germs.
We prove that the rank of an ordinary -web has an upper-bound which, for , is strictly smaller than the bound proved by Chern, denoting the Castelnuovo’s number. This bound is optimal. Setting , let be the integer such that . The number is then equal
- to 0 for ,
- and to for .
Moreover, if is precisely equal to , we define off a holomorphic connection on a holomorphic bundle of rank , such that the set of Abelian relations off 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 .
When , is always empty so that any web is ordinary, , and any may be written : we recover the results given in [9].
@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}, pages = {197--214}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 11}, number = {1}, year = {2012}, mrnumber = {2953049}, zbl = {1244.53014}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2012_5_11_1_197_0/} }
TY - JOUR AU - Cavalier, Vincent AU - Lehmann, Daniel TI - Ordinary holomorphic webs of codimension one JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2012 SP - 197 EP - 214 VL - 11 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2012_5_11_1_197_0/ LA - en ID - ASNSP_2012_5_11_1_197_0 ER -
%0 Journal Article %A Cavalier, Vincent %A Lehmann, Daniel %T Ordinary holomorphic webs of codimension one %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2012 %P 197-214 %V 11 %N 1 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2012_5_11_1_197_0/ %G en %F 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://archive.numdam.org/item/ASNSP_2012_5_11_1_197_0/
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