Links of singular points of complex hypersurfaces

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The homeomorphism type of $K$ is independent of the small paremeter $\epsilon$, it is called the link of the singular point $z$.
The homeomorphism type of $K$ is independent of the small paremeter $\epsilon$, it is called the link of the singular point $z$.
\begin{thm}(Fibration Theorem)
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\begin{thm}(Fibration Theorem)\label{fibration}
For $\epsilon$ sufficiently small, the space $S_{\epsilon}-K$ is a smooth fiber bundle over $S^1$, with projection map $\phi \colon S_{\epsilon}-K \to S^1$, $z \mapsto f(z)/|f(z)|$. Each fiber $F_{\theta}$ is parallelizable and has the homotopy type of a finite CW-complex of dimension $n$.
For $\epsilon$ sufficiently small, the space $S_{\epsilon}-K$ is a smooth fiber bundle over $S^1$, with projection map $\phi \colon S_{\epsilon}-K \to S^1$, $z \mapsto f(z)/|f(z)|$. Each fiber $F_{\theta}$ is parallelizable and has the homotopy type of a finite CW-complex of dimension $n$.
\end{thm}
\end{thm}
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Especially, the link $K$ is an integral [[homology sphere]] if and only if the intersection form $s$ is unimodular, i.~e.~the matrix of $s$ has determinant $\pm1$. If $n\ge 3$, the [[Generalized Poincare Conjecture]] implies that $K$ is a topological sphere.
Especially, the link $K$ is an integral [[homology sphere]] if and only if the intersection form $s$ is unimodular, i.~e.~the matrix of $s$ has determinant $\pm1$. If $n\ge 3$, the [[Generalized Poincare Conjecture]] implies that $K$ is a topological sphere.
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There is another important invariant associated to an isolated singular point of complex hypersurfaces. By Theorem \ref{fibration}, there is a smooth fiber bundle over $S^1$ with fiber $F_{\theta}$. The natural action of a generator of $\pi_1(S^1)$ induces the characteristic homeomorphism $h$ of the fiber $F_0=\phi^{-1}$. $h_* \colon H_n(F_0) \to H_n(F_0)$ is the induced isomorphism on homology. Let $\Delta(t)=\det(tI-h_*)$ be the characteristic polynomial of the linear transformation $h_*$.
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Revision as of 11:03, 8 June 2010

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Contents

1 Introduction

The links of singular points of complex hypersurfaces provide a large class of examples of highly-connected odd dimensional manifolds. A standard reference is [Milnor1968]. See also [Hirzebruch&Mayer1968] and [Dimca1992].

These manifolds are the boundaries of highly-connected, stably parallelisable even dimensional manifolds and hence stably parallelisable themselves. In the case of singular points of complex curves, the link of such a singular point is a fibered link in the 3-dimensional sphere S^3.

2 Construction and examples

Let f(z_1, \dots, z_{n+1}) be a non-constant polynomial in n+1 complex variables. A complex hypersurface V is the algebraic set consisting of points z=(z_1, \dots, z_{n+1}) such that f(z)=0. A regular point z \in V is a point at which some partial derivative \partial f /\partial z_j does not vanish; if at a point z \in V all the partial derivatives \partial f / \partial z_j vanish, z is called a singular point of V.

Near a regular point z, the complex hypersurface V is a smooth manifold of real dimension 2n; in a small neighborhood of a singular point z, the topology of the complex hypersurface V is more complicated. One way to look at the topology near z, due to Brauner, is to look at the intersection of V with a (2n+1)-dimensioanl sphere of small radius \epsilon S_{\epsilon} centered at z.

Properties of links

\begin{thm} The space K=V\cap S_{\epsilon} is (n-2)-connected. \end{thm}

The homeomorphism type of K is independent of the small paremeter \epsilon, it is called the link of the singular point z.

\begin{thm}(Fibration Theorem) For \epsilon sufficiently small, the space S_{\epsilon}-K is a smooth fiber bundle over S^1, with projection map \phi \colon S_{\epsilon}-K \to S^1, z \mapsto f(z)/|f(z)|. Each fiber F_{\theta} is parallelizable and has the homotopy type of a finite CW-complex of dimension n. \end{thm}

The fiber F_{\theta} is usually called the Milnor fiber of the singular point z.

A singular point z is isolated if there is no other singular point in some small neighborhood of z.

In this special situation, the above theorems are strengthened to the following

\begin{thm} Each fiber F_{\theta} is a smooth parallelizable manifold, the closure \overline{F_{\theta}} has boundary K and the homotopy type of a bouquet of n-spheres S^n\vee \cdots \vee S^n. \end{thm}

3 Invariants

Seen from the above section, the link K of an isolated singular point z of a complex hypersurface V of complex dimension n is a (n-2)-connected (2n-1)-dimensional closed smooth manifold. In high dimensional topology, these are called highly connected manifolds, since for a (2n-1)-dimensional closed manifold M which is not a homotopy sphere, (n-2) is the highest connectivity M could have. Therefore to understand the classification and invariants of the links K one needs to understand the classification and invariants of highly-connected odd dimensional manifolds, for which see.

On the other hand, as the link K is closely related to the singular point z of the complex hypersurfaces, some of the topological invariants of K are computable from the polynomial.

Let V be a complex hypersurface defined by f(z_1, \dots, z_{n+1}), z^0 be an isolated singular point of V. Let g_j=\partial f /\partial z_j j=1, \dots, n+1. By putting all these g_j's together we get the gradient field of f, which can be viewed as a map g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}, z \mapsto (g_1(z), \dots, g_{n+1}(z)). If z^0 is an isolated singular point, then z \mapsto g(z)/||g(z)|| is a well-defined map from a small sphere S_{\epsilon} centered at z^0 to the unit sphere S^{2n+1} of \mathbb C^m. The mapping degree \mu is called the multiplicity of the isolated singular point z^0. (\mu is also called the Milnor number of z^0.)

\begin{thm} The middle homology group H_n(F_{\theta}) is a free abelian group of rank \mu. \end{thm}

Furthermore, the homology groups of the link K are determined from the long homology exact sequence

\displaystyle  \cdots \to H_n(\overline{F_{\theta}}) \stackrel{j_*}{\rightarrow} H_n(\overline{F_{\theta}}, K) \stackrel{\partial}{\rightarrow} \widetilde{H}_{n-1}(K) \to 0

of the pair (\overline{F_{\theta}},K). The map j_* is the adjoint of the intersection pairing on \overline{F_{\theta}}

\displaystyle s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z.

Especially, the link K is an integral homology sphere if and only if the intersection form s is unimodular, i.~e.~the matrix of s has determinant \pm1. If n\ge 3, the Generalized Poincare Conjecture implies that K is a topological sphere.

There is another important invariant associated to an isolated singular point of complex hypersurfaces. By Theorem 4, there is a smooth fiber bundle over S^1 with fiber F_{\theta}. The natural action of a generator of \pi_1(S^1) induces the characteristic homeomorphism h of the fiber F_0=\phi^{-1}. h_* \colon H_n(F_0) \to H_n(F_0) is the induced isomorphism on homology. Let \Delta(t)=\det(tI-h_*) be the characteristic polynomial of the linear transformation h_*.



4 Classification

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5 Further discussion

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6 References

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