Ordinary and calibrated differential operators Application to curvilinear webs

TL;DR

The article develops a general framework for solution spaces of linear PDE systems defined by ordinary differential operators between vector bundles on an $n$-manifold. It derives a universal finite upper bound $\pi(n,k,p,q)$ on the dimension of germs of solutions and, when calibration holds, builds a vector bundle $\mathcal E$ with a tautological connection whose curvature obstructs achieving the bound, plus a curvature-concentration phenomenon. Specializing to curvilinear webs, the paper yields an order-1 ordinary calibrated operator whose solutions are abelian relations, recovers Damiano's rank bound, and provides a curvature-based criterion for maximal web rank. Overall, the work unifies dimension bounds for PDE solution spaces with a geometric curvature framework, and applies it to web geometry to quantify obstructions to maximal abelian relation spaces.

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 $π(n,k,p,q)$ 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 $π(n,k,p,q)$ 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.