Links of singular points of complex hypersurfaces
This page has not been refereed. The information given here might be incomplete or provisional. |
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 -dimensional sphere
.
2 Construction
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
.
Properties of links
\begin{thm}
The space is
-connected.
\end{thm}
The homeomorphism type 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 the following
\begin{thm}
Each fiber is a smooth parallelizable manifold, the closure
has boundary
and 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. Therefore to understand the classification and invariants of the links
one needs to understand the classification and invariants of highly-connected odd dimensional manifolds, for which see.
On the other hand, as the link is closely related to the singular point
of the complex hypersurfaces, some of the topological invariants of
are computable from the polynomial.
Let be a complex hypersurface defined by
,
be an isolated singular point of
. Let
. By putting all these
's together we get the gradient field of
, which can be viewed as a map
,
. If
is an isolated singular point, then
is a well-defined map from a small sphere
centered at
to the unit sphere
of
. The mapping degree
is called the multiplicity of the isolated singular point
. (
is also called the Milnor number of
.)
\begin{thm}
The middle homology group is a free abelian group of rank
.
\end{thm}
Furthermore, the homology groups of the link 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](/images/math/f/0/2/f021cd5d7984fb9f74fe1dffdd634461.png)
of the pair . The map
is the adjoint of the intersection pairing on
![\displaystyle s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z.](/images/math/b/d/0/bd0eb33afcbd9a713455e77eb7f06737.png)
A symmetric or skew symmetric bilinear form on a free abelian group is usually referred to as a lattice. is called the Milnor lattice of the singular point. Thus the homology groups of the link
is completely determined by the Milnor lattice of the singular point.
Topological spheres as links of singular points
Especially, the link is an integral homology sphere if and only if the intersection form
is unimodular, i.~e.~the matrix of
has determinant
. If
, the Generalized Poincare Conjecture implies that
is a topological sphere.
By Theorem 4, there is a smooth fiber bundle over with fiber
. The natural action of a generator of
induces the characteristic homeomorphism
of the fiber
.
is the induced isomorphism on homology. Let
be the characteristic polynomial of the linear transformation
.
Lemma 7.1.
For the manifolds
is a topological sphere is and only if the interger
equals to
.
When is a topological sphere, as it is the boundary of an
-connected parallelisable
-manifold
, our knowledge of exotic spheres allows us to determine the diffeomorphism class of
completely:
- if
is even, the diffeomorphism class of
is determined by the signature of the intersection pairing
![\displaystyle s \colon H_n(\overline{F_0}) \otimes H_n(\overline{F_0}) \to \mathbb Z](/images/math/e/e/2/ee254071649e5ddfaf1a2bbaf8f77e34.png)
- if
is odd, the diffeomorphism class of
is determined by the Kervaire invariant
![\displaystyle c(\overline{F_0}) \in \mathbb Z_2](/images/math/6/9/7/697f13d9e5a5400406dabd5fd13927ad.png)
which was computed in Levine1966
![\displaystyle c(\overline{F_0})=0 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 1 \pmod 8](/images/math/b/c/6/bc69a14e07767bf7d821657ed7154edc.png)
![\displaystyle c(\overline{F_0})=1 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 3 \pmod 8](/images/math/2/e/b/2eb3d4b471c1a7a3f67f0c682e6fde67.png)
4 Examples
A class of singular points which were studies extensively is the Brieskorn singularitites. These are defined by the polynomials of the form
![\displaystyle f(z_1, \dots, z_{n+1})=(z_1)^{a_1}+\dots +(z_{n+1})^{a_{n+1}}](/images/math/a/6/5/a65a9dc9e19f77ebfd05a590a599fd9c.png)
where are intergers
. The origin is an isolated singular point of
.
\begin{thm}(Brieskorn-Pham)
The group is free abelian of rank
![\displaystyle \mu = (a_1-1)(a_2-1)\cdots (a_{n+1}-1.)](/images/math/f/7/5/f756989e538b8d04aeb29cdef8a11478.png)
The characteristic polynomial is
![\displaystyle \Delta(t)=\prod(t-\omega_1 \omega_2 \cdots \omega_{n+1}),](/images/math/b/2/4/b24533dbf8df8000a9593c6e8671f2ec.png)
where each ranges over all
-th root of unit other than
.
\end{thm}
The simplest nontrivial example is ,
. Then
,
. The characteristic polynomial is
![\displaystyle \Delta(t)=t^2-t+1 \ \ \textrm{for} \ \ n \ \ \textrm{odd}](/images/math/b/d/a/bdacd5189bdd5ee7bb3ed2b27d62b83a.png)
![\displaystyle \Delta(t)=t^2+t+1 \ \ \textrm{for} \ \ n \ \ \textrm{even}.](/images/math/e/6/f/e6fce7c900e4fef00ae09dc0a1b1574c.png)
For we have
so the link
is a topological sphere of dimension
;
, thus by Levine
has nontrivial Kervaire invariant.
Especially for
is the Kervaire sphere.
The above example is a special case the -singularities, whose defining polynomial is
![\displaystyle f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}](/images/math/c/d/4/cd4ce5ee1f724cba906ef0dec2f6686d.png)
being an interger
.
5 Further discussion
...
6 References
- [Dimca1992] A. Dimca, Singularities and topology of hypersurfaces, Springer-Verlag, New York, 1992. MR1194180 (94b:32058) Zbl 0753.57001
- [Hirzebruch&Mayer1968] F. Hirzebruch and K. H. Mayer,
-Mannigfaltigkeiten, exotische Sphären und Singularitäten, Springer-Verlag, Berlin, 1968. MR0229251 (37 #4825) Zbl 0172.25304
- [Milnor1968] J. Milnor, Singular points of complex hypersurfaces, Princeton University Press, Princeton, N.J., 1968. MR0239612 (39 #969) Zbl 0224.57014
![S^3](/images/math/0/e/b/0ebf327477419ed9318cd89c957c1f9c.png)
2 Construction
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
.
Properties of links
\begin{thm}
The space is
-connected.
\end{thm}
The homeomorphism type 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 the following
\begin{thm}
Each fiber is a smooth parallelizable manifold, the closure
has boundary
and 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. Therefore to understand the classification and invariants of the links
one needs to understand the classification and invariants of highly-connected odd dimensional manifolds, for which see.
On the other hand, as the link is closely related to the singular point
of the complex hypersurfaces, some of the topological invariants of
are computable from the polynomial.
Let be a complex hypersurface defined by
,
be an isolated singular point of
. Let
. By putting all these
's together we get the gradient field of
, which can be viewed as a map
,
. If
is an isolated singular point, then
is a well-defined map from a small sphere
centered at
to the unit sphere
of
. The mapping degree
is called the multiplicity of the isolated singular point
. (
is also called the Milnor number of
.)
\begin{thm}
The middle homology group is a free abelian group of rank
.
\end{thm}
Furthermore, the homology groups of the link 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](/images/math/f/0/2/f021cd5d7984fb9f74fe1dffdd634461.png)
of the pair . The map
is the adjoint of the intersection pairing on
![\displaystyle s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z.](/images/math/b/d/0/bd0eb33afcbd9a713455e77eb7f06737.png)
A symmetric or skew symmetric bilinear form on a free abelian group is usually referred to as a lattice. is called the Milnor lattice of the singular point. Thus the homology groups of the link
is completely determined by the Milnor lattice of the singular point.
Topological spheres as links of singular points
Especially, the link is an integral homology sphere if and only if the intersection form
is unimodular, i.~e.~the matrix of
has determinant
. If
, the Generalized Poincare Conjecture implies that
is a topological sphere.
By Theorem 4, there is a smooth fiber bundle over with fiber
. The natural action of a generator of
induces the characteristic homeomorphism
of the fiber
.
is the induced isomorphism on homology. Let
be the characteristic polynomial of the linear transformation
.
Lemma 7.1.
For the manifolds
is a topological sphere is and only if the interger
equals to
.
When is a topological sphere, as it is the boundary of an
-connected parallelisable
-manifold
, our knowledge of exotic spheres allows us to determine the diffeomorphism class of
completely:
- if
is even, the diffeomorphism class of
is determined by the signature of the intersection pairing
![\displaystyle s \colon H_n(\overline{F_0}) \otimes H_n(\overline{F_0}) \to \mathbb Z](/images/math/e/e/2/ee254071649e5ddfaf1a2bbaf8f77e34.png)
- if
is odd, the diffeomorphism class of
is determined by the Kervaire invariant
![\displaystyle c(\overline{F_0}) \in \mathbb Z_2](/images/math/6/9/7/697f13d9e5a5400406dabd5fd13927ad.png)
which was computed in Levine1966
![\displaystyle c(\overline{F_0})=0 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 1 \pmod 8](/images/math/b/c/6/bc69a14e07767bf7d821657ed7154edc.png)
![\displaystyle c(\overline{F_0})=1 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 3 \pmod 8](/images/math/2/e/b/2eb3d4b471c1a7a3f67f0c682e6fde67.png)
4 Examples
A class of singular points which were studies extensively is the Brieskorn singularitites. These are defined by the polynomials of the form
![\displaystyle f(z_1, \dots, z_{n+1})=(z_1)^{a_1}+\dots +(z_{n+1})^{a_{n+1}}](/images/math/a/6/5/a65a9dc9e19f77ebfd05a590a599fd9c.png)
where are intergers
. The origin is an isolated singular point of
.
\begin{thm}(Brieskorn-Pham)
The group is free abelian of rank
![\displaystyle \mu = (a_1-1)(a_2-1)\cdots (a_{n+1}-1.)](/images/math/f/7/5/f756989e538b8d04aeb29cdef8a11478.png)
The characteristic polynomial is
![\displaystyle \Delta(t)=\prod(t-\omega_1 \omega_2 \cdots \omega_{n+1}),](/images/math/b/2/4/b24533dbf8df8000a9593c6e8671f2ec.png)
where each ranges over all
-th root of unit other than
.
\end{thm}
The simplest nontrivial example is ,
. Then
,
. The characteristic polynomial is
![\displaystyle \Delta(t)=t^2-t+1 \ \ \textrm{for} \ \ n \ \ \textrm{odd}](/images/math/b/d/a/bdacd5189bdd5ee7bb3ed2b27d62b83a.png)
![\displaystyle \Delta(t)=t^2+t+1 \ \ \textrm{for} \ \ n \ \ \textrm{even}.](/images/math/e/6/f/e6fce7c900e4fef00ae09dc0a1b1574c.png)
For we have
so the link
is a topological sphere of dimension
;
, thus by Levine
has nontrivial Kervaire invariant.
Especially for
is the Kervaire sphere.
The above example is a special case the -singularities, whose defining polynomial is
![\displaystyle f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}](/images/math/c/d/4/cd4ce5ee1f724cba906ef0dec2f6686d.png)
being an interger
.
5 Further discussion
...
6 References
- [Dimca1992] A. Dimca, Singularities and topology of hypersurfaces, Springer-Verlag, New York, 1992. MR1194180 (94b:32058) Zbl 0753.57001
- [Hirzebruch&Mayer1968] F. Hirzebruch and K. H. Mayer,
-Mannigfaltigkeiten, exotische Sphären und Singularitäten, Springer-Verlag, Berlin, 1968. MR0229251 (37 #4825) Zbl 0172.25304
- [Milnor1968] J. Milnor, Singular points of complex hypersurfaces, Princeton University Press, Princeton, N.J., 1968. MR0239612 (39 #969) Zbl 0224.57014