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 $d\in \{k,\dots ,n-1\}$ which is a generalization of both the Nash modification and the conormal variety of $(X,0)$.