Links of singular points of complex hypersurfaces

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 $3$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex> 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 $S^3$. </wikitex> == Construction and examples == <wikitex>; 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 n$; 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$. \begin{thm} The space $K=V\cap S_{\epsilon}$ is $(n-2)$-connected. \end{thm} The topology 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 \begin{thm} Each fiber is a smooth parallelizable manifold with boundary, having the homotopy type of a bouquet of $n$-spheres $S^n\vee \cdots \vee S^n$. \end{thm} </wikitex> == Invariants == <wikitex>; 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. </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}3$-dimensional sphere $S^3$$S^3$.

2 Construction and examples

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

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

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

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

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

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

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

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 $n$$n$-spheres $S^n\vee \cdots \vee S^n$$S^n\vee \cdots \vee S^n$. \end{thm}

3 Invariants

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

4 Classification/Characterization

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5 Further discussion

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6 References

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