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

TL;DR

This paper introduces modules of derivations linked to analytic maps and submodules, providing formulas for numerical invariants and insights into singularities of maps on analytic varieties.

Contribution

It develops a new framework for modules of derivations associated with analytic maps and submodules, deriving formulas for invariants and analyzing singularities.

Findings

Formulas for numerical invariants of pairs (h, M)

Expressions for invariants of logarithmic vector fields

Analysis of exact sequences related to derivations

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 $X$.