Link of singularities

For an isolated plane curve singularity, given by homogeneous equation $F=0 \subset \mathbb{C}^2$, one consider the curve $(F=0) \cap S^3 \subset S^3$, and we call it the link of singularity. some properties of the singularity are embedded in the corresponding link and one can start studying the singularity by studyng the invariants of the corresponding link, like Jones polynomial ...

Question: Is there any similar construction for more general singularities? Like when $X$ is an irreducible singular variety and the singular locus $X^{sing}$ is isomorphic to a smooth variety $V$?

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Yes, these sorts of things have been considered. For example, if $X$ is locally embedded into $\mathbb{C}^n$, the intersection of $X$ with a small sphere centred at an isolated singularity $p\in X$ is again called the link. Of course, there is a much longer story which someone might answer. –  Donu Arapura Apr 4 '11 at 16:33
I think this method of studying algebraic varieties goes back at least as far as Wirtinger's work in 1895. –  Ryan Budney Apr 4 '11 at 18:18

1 Answer

I think you will be interested in reading about the so-called Milnor map, and the so-called Minor Fibration Theorem. The standard reference is John Milnor's book "Singular points of complex hypersurfaces" from 1968. Although people ususally assume that the hypersurface has an isolated singularity. That means that the singularity has finite multiplicity. The non-isolated case (where the singular set is a smooth variety) is much more complicated. Nevertheless, as a starting point, I recommend looking at the Milnor Map and the Milnor Fibration Theorem.

There are lots of results on the topology of the Milnor fibre. For example if $f : (\mathbb{C}^n,0) \to (\mathbb{C},0)$ is a holomorphic map germ with an isolated critical point at $0 \in \mathbb{C}^n$ then the Milnor fibre is homotopy equivalent to the bouquet of $\mu$-spheres, where $\mu$ denotes the Milnor number. The Milnor number is given by the absolute value of the Poincaré-Hopf index of the gradient vector field $\nabla f$ at $0 \in \mathbb{C}^n.$

Take a look at the introduction to this paper for a little more detail and some references. Take a look at this paper for the non-isolated case. Both papers include information about the links.

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The second link is now dead. Do you have an updated link or a bibliographic reference we can search? –  Sam Lisi Sep 17 '12 at 20:45
@Sam Lisi: I'm guessing it was one of the several papers co-authored by Anne Pichon with "boundary of the Milnor fibre of nonisolated singularities" in the title (do a MathSciNet search). –  Joshua Grochow Nov 7 '12 at 3:40