Formes de Whitney et primitives relatives de formes diff\'erentielles sous-analytiques

TL;DR

This paper proves that under certain conditions, subanalytic differential forms on real-analytic manifolds have relative primitives, using Whitney forms and subanalytic triangulations to solve a linear PDE.

Contribution

It introduces a method to construct relative primitives of subanalytic forms via Whitney forms and triangulations, extending previous results to a subanalytic setting.

Findings

Existence of relative primitives for subanalytic forms under fiberwise exactness.

Use of Whitney forms and triangulations to translate the problem.

Solution relies on subanalytic solutions to a linear PDE.

Abstract

Let $X$ be a real-analytic manifold and $g\colon X\to{\mathbf R}^n$ a proper triangulable subanalytic map. Given a subanalytic $r$-form $\omega$ on $X$ whose pull-back to every non singular fiber of $g$ is exact, we show tha $\omega$ has a relative primitive: there is a subanalytic $(r-1)$-form $\Omega$ such that $dg\Lambda (\omega-d\Omega)=0$. The proof uses a subanalytic triangulation to translate the problem in terms of "relative Whitney forms" associated to prisms. Using the combinatorics of Whitney forms, we show that the result ultimately follows from the subanaliticity of solutions of a special linear partial differential equation. The work was inspired by a question of Fran\c{c}ois Treves.