Small neighborhoods of singularities on varieties - MathOverflow

Small neighborhoods of singularities on varieties

2010-07-24T13:56:55Z

In Singular points of complex hypersurfaces, John Milnor proves the following theorem:

Let $x \in V$ be a point on a variety $V$ in $\mathbb{R}^n$ or $\mathbb{C}^n$. Assume $x$ is either a smooth point or an isolated singularity. Let $D_{\epsilon}$ be the closed $\epsilon$-ball about $x$, $S_{\epsilon}$ its boundary (the sphere about $x$ of radius $\epsilon$), and $K = V \cap S_{\epsilon}$. Then for $\epsilon$ sufficiently small, the pair $(D_{\epsilon}, V \cap D_{\epsilon})$ is homeomorphic to the pair $(CS_{\epsilon}, CK)$, where $C$ denotes taking the cone. (Theorem 2.10)

In Remark 2.11, Milnor observes that this theorem "likely" holds even if $x$ is a non-isolated singularity; in particular, it is known even in this case that "a suitably chosen neighborhood of any point is homeomorphic to the cone over something."

This book was written in 1968. What is the current status of this problem?

Answer by Greg Kuperberg for Small neighborhoods of singularities on varieties

2010-07-24T22:50:10Z

There is a good paper of Goresky, "Triangulation of Stratified Objects", that I think reasonably quickly implies Milnor's result and its generalization to non-isolated singularities. The result is that any Whitney-stratified set, and in particular any algebraic variety in $\mathbb{C}^n$, is supported on a smooth triangulation. I think that you just need that and the inverse function theorem.

As I meant to explain in the comments, this theorem has sometimes been regarded as a "chore" theorem. You can look at what Goresky says: "Triangulation theorems for stratified objects have been obtained independently by Hendricks (unpublished), Johnson (unpublished), and Kato (in Japanese)". When Goresky wrote his paper, it was a messy question that did not have a well-defined status. Now the situation is a bit better and I think that this generalization of Milnor's result can be called settled. Sometimes a good author not only proves a chore theorem, but also cleans it up an elevates it to non-chore status. But a lot of chore theorems are never proven in a clean form or are never proven at all.

Answer by A. Lerario for Small neighborhoods of singularities on varieties

2011-07-23T11:25:53Z

Indeed the following theorem to me seems exactly you were looking for (see J. Bochnak, M. Coste, M-F. Roy, "Real algebraic geometry", Theorem 9.3.6 [Local conic structure]):

Let $E$ be a semialgebraic susbet of $\mathbb{R}^n$ and $x$ be a nonisolated point of $E.$ Let also $D_\epsilon$ be the closed $\epsilon$-ball around $x$ and $S_\epsilon$ its boundary. Set $K=S_\epsilon \cap E$. Then there for $\epsilon>0$ small enough the pair $(D_\epsilon,E∩D_\epsilon)$ is semialgebraically homeomorphic to the pair $(CS_\epsilon,CK)$, where $C$ denotes taking the cone. Moreover the semialgebraic homeomorphism can be chosen as to preserve the distance from $x.$

Two words of remarks on the previous statement:

Every real or complex algebraic set in $\mathbb{R}^n$ or in $\mathbb{C}^n\simeq \mathbb{R}^{2n}$ is a semialgebraic set.
The point $x$ is any nonisolated point of $E$ (no matter singular - in whatever meaning this word has for a general semialgebraic set - or regular).