On the Nash modification of a germ of complex analytic singularity

TL;DR

This paper characterizes the Nash modification of complex analytic singularities as certain subvarieties in a product space, generalizing conormal varieties and introducing the $d$-conormal space concept.

Contribution

It provides a new characterization of Nash modifications for complex singularities and introduces the $d$-conormal space as a generalization of existing concepts.

Findings

Characterization of Nash modifications as subvarieties in a product space.

Generalization of conormal varieties as Legendrian subvarieties.

Introduction of the $d$-conormal space for complex singularities.

Abstract

For a germ $(X,0) \subset (\mathbb{C}^n,0)$ of reduced, equidimensional complex analytic singularity its Nash modification can be constructed as an analytic subvariety $ Z \subset \mathbb{C}^n \times G(k,n)$. We give a characterization of the subvarieties of $\mathbb{C}^n \times G(k,n)$ that are the Nash modification of its image under the projection to $\mathbb{C}^n$. This result generalizes the characterization of conormal varieties as Legendrian subvarieties of $\mathbb{C}^n \times \check{\mathbb{P}}^{n-1}$ with its canonical contact structure. As a by-product we define the $d$-conormal space of $(X,0)$ for any $d \in \{k, \ldots, n-1\}$ which is a generalization of both the Nash modification and the conormal variety of $(X,0)$.