Relations ab\'eliennes des tissus ordinaires de codimension arbitraire

Abstract

We generalize to webs of any codimension results already known in codimension one. Given a holomorphic $d$-web $\cal W$ of codimension $q$ $(q\leq n-1)$ in an ambiant $n$-dimensional holomorphic manifold $U$, we define for any integer $p$ $(1\leq p\leq q)$ the condition for such a web to be \emph{$p$-ordinary} $($resp. \emph{strongly $p$-ordinary}$)$. If this condition is satisfied, we then prove that its $p$-rank $r_p({\cal W})$ $\bigl($resp. its closed $p$-rank $\widetilde r_p({\cal W})\bigr)$, 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 $\pi_p^0(n,d,q)$ $\bigl($resp. $\pi'_p(n,d,q)\bigr)$ for these ranks. Moreover, for some values of $d$, and we then say then that the web is \emph{$p$-calibrated} $($resp. \emph{strongly $p$-calibrated}$)$, we define a tautological holomorphic connection on a holomorphic vector bundle of rank $\pi_p^0(n,d,q)$ $\bigl($resp. $\pi'_p(n,d,q)\bigr)$, 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 $r_p({\cal W})$ $\bigl($resp. $\widetilde r_p({\cal W})\bigr)$ 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.