Links of singular points of complex hypersurfaces

Revision as of 17:18, 7 June 2010

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].

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 ${{Stub}} == Introduction == <wikitex>; The links of singular points of complex hypersurfaces provide a large class of examples of highly-connected odd dimensional manifolds. A standard reference is {{cite|Milnor1968}}. 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 $-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 the following \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. 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 [[highly-connected_manifolds|highly-connected manifolds]]. 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 $f(z_1, \dots, z_{n+1})$, <script type="math/tex">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 $z$.

\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 the following

\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}

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 $f(z_1, \dots, z_{n+1})$, $0$ 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 $g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$, $z \mapsto (g_1(z), \dots, g_{n+1}(z))$.

4 Classification

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5 Further discussion

...

6 References

• [Milnor1968] J. Milnor, Singular points of complex hypersurfaces, Princeton University Press, Princeton, N.J., 1968. MR0239612 (39 #969) Zbl 0224.57014

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 $z$.

\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 the following

\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}

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 $f(z_1, \dots, z_{n+1})$, $0$ 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 $g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$, $z \mapsto (g_1(z), \dots, g_{n+1}(z))$.

4 Classification

...

5 Further discussion

...

6 References

• [Milnor1968] J. Milnor, Singular points of complex hypersurfaces, Princeton University Press, Princeton, N.J., 1968. MR0239612 (39 #969) Zbl 0224.57014