Local Resolution of Ideals Subordinated to a Foliation

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

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