Most of the material of this section can be found in the books [Dim92], [GLS07]
and [Mil68].
Following standard conventions we denote byV(I) the vanishing locus of an ideal of functions and byp
Ithe radical ideal ofI. Since we usually work on germs of functions,V(I) is a set germ. Occasionally we abbreviate the partial derivatives∂1, . . . ,∂nsimply by a single∂.
LetC{x1, . . . ,xn} be the power series ring inn complex variables. Some-times this ring is denoted asOCn,0 emphasizing that it is the stalk of the sheaf of germs of holomorphic functions on then-dimensional complex vector space. Its maximal ideal is denoted bymCn,0. An element ofC{x}, f: (Cn,0)→Cis nothing but a holomorphic function germ. Iff ∈mCn,0, then byV(f) we denote the germ of a set, whereas by (f =0) we will denote the set of zeros off not as a germ but for a fixed representative off.
We say that f has an isolated singularity at the origin (orisan isolated singularity) if its critical locus Crit(f)={x∈(Cn,0)|∂1f(x)=0, . . . ,∂nf(x)= 0} is either void or the origin itself. The ringC{x1, . . . ,xn} is factorial and a non-unitf is called reduced if in its prime factor decomposition no multiple factor occurs. We have
Proposition 1.1. Let f: (Cn,0)→(C, 0)be a holomorphic germ.
1. If f has an isolated singularity at the origin, then it is reduced.
2. If n=2and f is reduced, then f has an isolated singularity at the origin.
3. On the level of germs of sets we have V(∂f)⊂V(f).
To two plane germsf,g: (C2,0)→Cwe can assign their intersection num-beri(f,g). It can be defined as the dimension of the vector spaceC{x,y}〈f,g〉. With this definition it follows thati(f,g) is zero if and only iff(0)6=0 org(0)6=0.
Furthermorei(f,g)< ∞if and only iff andghave no common factor. Ifg is irreducible andγ(t) is a parametrization ofV(g), theni(f,g) can be com-puted as the order intoff ◦γ(t).
To any holomorphic germf: (Cn,0)→Cwe can assign the Milnor num-berµ(f) which can be given as the dimension of the vector space〈∂C{x1,...,xn}
1f,...,∂nf〉. This number is finite if and only iff has an isolated singularity at the origin.
If f: (Cn,0)→(C, 0) is a holomorphic germ, we can take a sufficiently small sphereS2n−1² (of radius 0<²¿1 around the origin0) and intersect it with the vanishing locus off. This defines a linkL:=S2n² −1∩(f =0),→S2n² −1 whose isotopy type is independent of the choice of². If we remove this ex-ceptional set from the sphere we obtain the classical (knottheoretic) Milnor fibration:
Theorem 1.2(Milnor fibration).
Let f: (Cn,0)→(C, 0)be a holomorphic germ. Then for any sufficiently small
²>0, the map
f
|f|:S2n−1² \L→S1 is a smooth locally trivial fibration.
The (diffeomorphism type of the) fibre of this fibration is called the Mil-nor fibre of f. Since the topology of hypersurface singularities is so closely related with knots resp. links we must dwell a little on that subject. One has the following theorem of Milnor and Burghela-Verena:
Proposition 1.3(Conic Structure Lemma).
Let f: (Cn,0)→(C, 0)be a holomorphic germ. Then the vanishing locus of f is homeomorphic to the cone over the link of f with apex at0. This home-omorphism extends to the natural homehome-omorphism of the closed ball with the cone over the sphere. To be precise, if we denote by B²2nthe closed²-ball
1.1. Topological and Algebraic Properties of Singularities
around0, then for sufficiently small²>0we have a homeomorphism of pairs (B²2n,B²2n∩(f =0))≈Cone(S2n−1² ,S2n−1² ∩(f =0)).
In particularV(f) is contractible, whereasV(f) \ {0} is homotopy equiva-lent to a disjoint union of circles with the same number of components as the link has. Furthermore, the complementB²2n\ (f =0) has the same ho-mology as that stated in the next proposition.
Proposition 1.4. Let L,→S3be a link with m components. Then H0(S3\L;Z)=Z
H1(S3\L;Z)=Zm H2(S3\L;Z)=Zm−1 Hi(S3\L;Z)=0,i≥3.
It is possible to describe explicitely a basis for the first homology of the link complement. On each connected component of the link fix a point and choose a small transversal slice through this point. Then in the transversal slice take a loop encircling the distinguished point. When we do this for all the distinguished points we get a collection of loops which provides a basis forH1(S3\L;Z). The following theorem is valid with modifications in higher dimensions as well, the proof uses however the Gysin sequence, we refer to [Dim92].
Theorem 1.5. Let f =f1m1. . .frmr be the prime factor decomposition of f ∈ mC2,0. Then the Milnor fibre F of f has exactly m:=gcd(m1, . . . ,mr) con-nected components and for any base point z∈F we have an exact sequence
0→[π1(S²3\L,z),π1(S3²\L,z)]→π1(F,z)→Zr−1→0 (1.1) Proof. The Milnor fibrationF→S3²\L→S1yields the exact homotopy se-quence
0→π1(F,z)→π1(S3²\L,z)→π1(S1,f(z))→π0(F)→π0(S3²\L)→π0(S1)=0.
First recall that the link complement is connected. Therefore by the univer-sal property of the abelianization we can factor one of the involved maps by
the Hurewicz homomorphismH
0 π1(F,z) π1(S3²\L,z) π1(S1,f(z)) π0(F) 0
H1(S3²\L;Z)
φ
H ψ
The mapψsends a small transversal loopγi corresponding to the link componentV(fi)∩Sto its image inS1by composition withf/|f|. Choose a sufficiently small neighbourhood of a fixed point on (fi =0)∩S3² so that when we write
f
|f|= fimi
|fimi|
r
Y
j=1,j6=i
fjmj
|fjmj|
the product term is nonzero and such that the point is given in local coor-dinates by (x,y)=(0, 0), the function is fi =x and the transversal loop is γ(t)=(δei2πt, 0) wheret∈[0, 1] andδ<<1. Then after a coordinate trans-formation we have
f
|f|◦γ(t)= xmi
|xmi|◦γ(t)=ei2πmit.
HenceΨmaps the basisγ1, . . . ,γr ofH1(S²3\L;Z) tom1[S1], . . . ,mr[S1] mod-ulo orientation. The image ofΨis thereforemZwherem=gcd(m1, . . . ,mr) and sinceHis surjective we getπ0(F)∼=Z/mZ, hence the Milnor fibre has mconnected components. The final assertion about the exact sequence is now easily seen.
Corollary 1.6. A reduced and a nonreduced holomorphic germ cannot be right equivalent under a homeomorphism.
The following remarkable theorem is due to various people. It states that a certain module is free and of finite rank. In 1970, E. Brieskorn and P. Deligne proved the finiteness of the module and M. Sebastiani proved its torsionfree-ness. A nice proof of the latter has also been given by B. Malgrange.
Theorem 1.7. ([Bri70], [Seb70], [Mal74])
If f: (Cn,0)→(C, 0)has an isolated singularity at the origin, then the Brieskorn