On the Nash modification of a germ of complex analytic singularity

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 $(X,0)$.