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.

Donu's right about the link. You usually consider a function germ $f : (\mathbb{C}^n,0) \to (\mathbb{C},0).$ There are lots of results on the topology of the link. For example, it's homotopy equivalent to a bouquet (wedge product) of $\mu$-spheres where $\mu$ is the Milnor number of $f$; which is finite if and only if $f$ has an isolated critical point at $0 \in \mathbb{C}^n$. In the complex case, the Milnor number is the absolute value of the Poicaré-Hopf index of the gradient vector field $\nabla f$ at $0 \in \mathbb{C}^n.$

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.