Links of singular points of complex hypersurfaces
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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
of the pair . The map is the adjoint of the intersection pairing on
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
- if is odd, the diffeomorphism class of is determined by the Kervaire invariant
which was computed in Levine1966
4 Examples
A class of singular points which were studies extensively is the Brieskorn singularitites. These are defined by the polynomials of the form
where are intergers . The origin is an isolated singular point of .
\begin{thm}(Brieskorn-Pham) The group is free abelian of rank
The characteristic polynomial is
where each ranges over all -th root of unit other than . \end{thm}
The simplest nontrivial example is , . Then , . The characteristic polynomial is
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 of the -singularities, whose defining polynomial is
being an interger . The Milnor lattice of an -singularity is represented by the Dynkin diagram of the simple Lie algebra . When , the diffeomorphism class of the link is obtained from its Milnor lattice and the classification of simple-connected 5-manifolds (see 5-manifolds: 1-connected):
- is diffeomorphic to if is odd;
- is diffeomorphic to if is even.
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
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
of the pair . The map is the adjoint of the intersection pairing on
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
- if is odd, the diffeomorphism class of is determined by the Kervaire invariant
which was computed in Levine1966
4 Examples
A class of singular points which were studies extensively is the Brieskorn singularitites. These are defined by the polynomials of the form
where are intergers . The origin is an isolated singular point of .
\begin{thm}(Brieskorn-Pham) The group is free abelian of rank
The characteristic polynomial is
where each ranges over all -th root of unit other than . \end{thm}
The simplest nontrivial example is , . Then , . The characteristic polynomial is
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 of the -singularities, whose defining polynomial is
being an interger . The Milnor lattice of an -singularity is represented by the Dynkin diagram of the simple Lie algebra . When , the diffeomorphism class of the link is obtained from its Milnor lattice and the classification of simple-connected 5-manifolds (see 5-manifolds: 1-connected):
- is diffeomorphic to if is odd;
- is diffeomorphic to if is even.
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