Exterior multiplication with singularities: a Saito theorem in vector bundles
Volume 125 / 2020
Abstract
Let $E$ be a vector bundle over a differentiable manifold $M$ and let $\textstyle\bigwedge ^pE$ denote the $p$th exterior product of $E$. Given sections $\omega _1,\dots ,\omega _k$ of $E$ and a section $\eta $ of $\textstyle\bigwedge ^pE$, we consider the problem of whether $\eta $ can be written in the form \[ \eta =\sum \omega _i\wedge \gamma _i, \] where $\gamma _i$ are sections of $\textstyle\bigwedge ^{p-1}E$. 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 the coefficients of $\Omega $. Working in the smooth, real analytic or holomorphic (with $M$ a Stein manifold) category, we show that the condition is sufficient for the above property to hold. Moreover, in the smooth category it is sufficient for the 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 $M$.
