2025/02/10 by Lehmann, Daniel
#53A60 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.06641
We study the space of the solutions s of any system of partial differential equations D(jks)=0 defined by a linear and homogeneous differential operator D:JkE→ F of any order k≥ 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≤ np in the case k=1), we first give an upper-bound π(n,k,p,q) for the dimension of the space \mathcal Sm 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, (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 ∇, whose curvature is an obstruction for the dimension of \mathcal Sm 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 Sm 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.