Local Resolution of Ideals Subordinated to a Foliation

TL;DR

This paper establishes a method for locally resolving singularities of ideals on manifolds while maintaining the nature of a given singular distribution, especially when the distribution is monomial, extending resolution techniques in singularity theory.

Contribution

It introduces a resolution process that preserves the class of singularities of a distribution under blowings-up, including the case of monomial singular distributions.

Findings

Existence of local resolutions preserving singularity classes.

Resolution preserves monomiality of singular distributions.

Applicable to complex and real-analytic manifolds.

Abstract

Let $M$ be a complex- or real-analytic manifold, $\theta$ be a singular distribution and $\mathcal{I}$ a coherent ideal sheaf defined on $M$. We prove the existence of a local resolution of singularities of $\mathcal{I}$ that preserves the class of singularities of $\theta$, under the hypothesis that the considered class of singularities is invariant by $\theta$-admissible blowings-up. In particular, if $\theta$ is monomial, we prove the existence of a local resolution of singularities of $\mathcal{I}$ that preserves the monomiality of the singular distribution $\theta$.