2017/12/04 by Daniel Lehmann, Lehmann, Daniel
Mathematics · #14H45) #53A60 (14C21 #53C05 #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Meromorphic and Entire Functions #math.DG #msc:53A60 #msc:53C05
paper · pdf · doi:10.48550/arxiv.1712.00997
20 pages, in French
openalex publication_date 2017/12/04 · arxiv created 2021/12/30 · arxiv updated 2022/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We generalize to webs of any codimension results already known in codimension one. Given a holomorphic d-web \cal W of codimension q (q≤ n-1) in an ambiant n-dimensional holomorphic manifold U, we define for any integer p (1≤ p≤ q) the condition for such a web to be p-ordinary (resp. strongly p-ordinary). If this condition is satisfied, we then prove that its p-rank rp(\cal W) (resp. its closed p-rank \widetilde rp(\cal W)), i.e. the maximal dimension of the vector space of the germs of p-abelian relations (resp. of closed p-abelian relations) at a point m of U, is finite. We then give an upper-bound πp0(n,d,q) (resp. π'p(n,d,q)) for these ranks. Moreover, for some values of d, and we then say then that the web is p-calibrated (resp. strongly p-calibrated), we define a tautological holomorphic connection on a holomorphic vector bundle of rank πp0(n,d,q) (resp. π'p(n,d,q)), for which the sections with vanishing covariant derivative may be identified with p-abelian relations (resp. closed p-abelian relations). The curvature of this connection is then an obstruction for the rank rp(\cal W) (resp. \widetilde rp(\cal W)) to be maximal. The main change is the correction of a mistake (proposition 4, section 6-5) in the first version : the 1-rank of the concerned web is not 0 as we claimed, but 1. However, the important corollary remains true : even at the level of germs, some 2-abelian relation exhibited by Goldberg in [G] on some web of codimension 2 in an ambiant space of dimension 4, is the coboundary of none 1-abelian relation. The section 7, devoted to this correction, is self content, not depending on the previous results of the paper.