Ordinary and calibrated differential operators Application to curvilinear webs

TL;DR

This paper investigates the solution spaces of linear differential systems, introduces the concept of calibrated operators, and applies these ideas to curvilinear webs to understand their rank and curvature obstructions.

Contribution

It develops a framework for analyzing solution spaces of differential operators, introduces calibration conditions, and applies these to curvilinear webs to define web curvature and rank bounds.

Findings

Provides an upper bound for solution space dimension

Constructs a vector bundle with a tautological connection related to solutions

Defines web curvature as an obstruction to maximum rank

Abstract

We study the space of the solutions $s$ of any system of partial differential equations $ D(j^ks)=0 $ defined by a linear and homogeneous differential operator $ D:J^kE\to F $ of any order $k\geq 1$, which is ``ordinary" (i.e. which is generic in some sense among all $D$'s), $E$ and $F$ being vector bundles over a $n$-dimensional manifold $V$, and $D$ being assumed to be surjective at any point of $V$. In some range of the ranks $p$ and $q$ of these bundles ($p < q\leq np$ in the case $k=1$), we first give an upper-bound $\pi(n,k,p,q)$ for the dimension of the space ${\mathcal S}_m$ of the germs of solutions at a generic point $m$ of the ambiant manifold. If these ranks satisfy moreover to some condition of integrality (in the case $k=1$, $\frac{p(n-1)}{q-p}$ must be an integer), and we then say that $D$ is ``calibrated", we build a vector bundle $\mathcal E$ of rank $\pi(n,k,p,q)$ on $V$, provided with a tautological connection $\nabla$, whose curvature is an obstruction for the dimension of ${\mathcal S}_m$ to reach its maximal value. We also prove a ``theorem of concentration'' : relatively to some convenient trivialization of $\mathcal E$, some coefficients of this curvature vanish systematically. As an example, we provide, for any curvilinear $d$-web on $V$, a differential operator $D$ of order one, which is always ordinary and calibrated, and for which ${\mathcal S}_m$ is the space of germs of abelian relations ([L]). Thus, we recover the Damiano's upper-bound ([D1]) for the rank of such a web, and we can define in the most general case the ``curvature'' of such a web, already known for $n=2$ (see [BB] if $d=3$, and [Pa],[H1],[Pi1] for arbitrary $d$), obstruction for this rank to be maximum.