Exterior multiplication with singularities: a Saito's theorem on vector bundles

TL;DR

This paper extends Saito's algebraic theorem to differential geometry, providing conditions under which a section of an exterior power bundle can be expressed as a wedge product sum, considering singularities and regularity.

Contribution

It introduces a geometric version of Saito's theorem, establishing sufficiency conditions involving singularities for expressing sections as wedge products in smooth, analytic, and holomorphic categories.

Findings

Condition involving the depth of the ideal is sufficient for the wedge decomposition.

Results hold in smooth, real analytic, and holomorphic categories.

The theorem applies to bundles over closed subsets of manifolds.

Abstract

Let $E$ be a vector bundle over a suitable differential manifold $M$ and let $\wedge^p E$ denote $p$-exterior product of $E$. Given sections $\omega_1,\dots,\omega_k$ of $E$ and a section $\eta$ of $\wedge^p E$, we consider the problem if $\eta$ can be written in the form $$\eta=\sum \omega_i\wedge\gamma_i,$$ where $\gamma_i$ are sections of $\wedge^{p-1}E$. An obvious necessary condition $\Omega\wedge\eta=0$, where $\Omega=\omega_1\wedge\cdots\wedge\omega_k$, has to be supplemented with a condition that the form $\Omega$ has sufficiently regular singularities at points where $\Omega(x)=0$. Such a local condition is suggested by an algebraic theorem of K. Saito and is given in terms of the depth of the ideal defined by coefficients of $\Omega$. Working in the smooth, real analytic and holomorphic (with $M$ Stein manifold) categories, we show that the condition is sufficient for the above property to hold. Moreover, in the smooth category it is sufficient for existence of a continuous right inverse to the operator defined by $(\gamma_1,\dots,\gamma_k)\mapsto\sum \omega_i\wedge\gamma_i$. All these results are also proven in the case where $E$ is a bundle over a suitable closed subset of $M$.