Relations ab\'eliennes des tissus ordinaires de codimension arbitraire

TL;DR

This paper extends results on abelian relations from codimension one webs to webs of arbitrary codimension, establishing conditions for finiteness of ranks, bounds, and connections, with a correction to previous claims about 1-rank.

Contribution

It generalizes the theory of abelian relations to higher codimension webs, introduces new bounds, and constructs tautological connections, correcting earlier misconceptions about 1-rank.

Findings

Finite p-rank and closed p-rank under p-ordinary conditions

Upper bounds for these ranks based on web parameters

Existence of tautological holomorphic connections with curvature as an obstruction

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.