Modules of derivations, logarithmic ideals and singularities of maps on analytic varieties

Abstract

We introduce the module of derivations ${\Theta }_{h,M}$ attached to a given analytic map $h:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^p,0)$ and a submodule $M\subseteq {\mathcal{O}}_n^p$ and analyse several exact sequences related to ${\Theta }_{h,M}$. Moreover, we obtain formulas for several numerical invariants associated to the pair $(h,M)$ and a given analytic map germ $f:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^q,0)$. In particular, if $X$ is an analytic subvariety of ${\mathbb{C}}^n$, we derive expressions for analytic invariants defined in terms of the module ${\Theta }_X$ of logarithmic vector fields of $X$.