Links of singular points of complex hypersurfaces
Contents |
1 Introduction
The links of singular points of complex hypersurfaces provides a large class of examples of highly-connected odd dimensional manifold which are boundary highly-connected, stably parallelisable even dimensional manifolds. In the case of singular points of complex curves, the link of such a singular point is a fibered link in the -dimensional sphere .
2 Construction and examples
Let be a non-constant polynomial in complex variables. A complex hypersurface is the algebraic set consisting of points such that . A regular point is a point at which some partial derivative does not vanish; if at a point all the partial derivatives vanish, is called a singular point of .
Near a regular point , the complex hypersurface is a smooth manifold of real dimension ; in a small neighborhood of a singular point , the topology of the complex hypersurface is more complicated. One way to look at the topology near , due to Brauner, is to look at the intersection of with a -dimensioanl sphere of small radius centered at .
\begin{thm} The space is -connected. \end{thm}
The topology of is independent of the small paremeter , it is called the link of the singular point .
\begin{thm}(Fibration Theorem) For sufficiently small, the space is a smooth fiber bundle over , with projection map , . Each fiber is parallelizable and has the homotopy type of a finite CW-complex of dimension . \end{thm}
The fiber is usually called the Milnor fiber of the singular point .
A singular point is isolated if there is no other singular point in some small neighborhood of .
In this special situation, the above theorems are strengthened to
\begin{thm} Each fiber is a smooth parallelizable manifold with boundary, having the homotopy type of a bouquet of -spheres . \end{thm}
3 Invariants
Seen from the above section, the link of an isolated singular point of a complex hypersurface of complex dimension is a -connected -dimensional closed smooth manifold. In high dimensional topology, these are called highly connected manifolds, since for a -dimensional closed manifold which is not a homotopy sphere, is the highest connectivity could have.
4 Classification/Characterization
...
5 Further discussion
...
6 References
This page has not been refereed. The information given here might be incomplete or provisional. |