COMPLEX SINGULARITIES
LÊ DŨNG TRÁNG
Introduction
In the study of complex algebraic varieties, one has to relate the geometry of
the algebraic variety and the algebraic equations with which it is defined. In this
process one is first confronted to distinguish non-singular (or regular) points and
singular points.
When one considers complex hypersurfaces X := {f = 0} defined by a reduced
complex polynomial function f of Cn , non-singular points are the points x of X
where the differential dfx 6= 0. The implicit function theorem implies that, at the
non-singular point x, the hypersurface X is locally isomorphic to an open subspace
of Cn−1 and one can define a tangent hyperplane Tx X. A singular point of x of X
is singular if dfx = 0.
Therefore, the definition of a singular point is rather easy in the case of hypersurfaces, i.e. algebraic subsets of Cn defined by one algebraic equation.
In the case of algebraic sets X defined by several equations in Cn , one considers
the ideal I(X) of all equations of X in Cn . The finiteness Theorem of Hilbert shows
that this ideal I(X) is finitely generated in the polynomial ring C[X1 , . . . , Xn ]. Let
g1 , . . . , gk be a set of generators of I(X). One considers the matrix of partial
derivatives of this set of generators:


∂g1 /∂X1 , . . . , ∂g1 /∂Xn
.
...
J =
∂gk /∂X1 , . . . , ∂gk /∂Xn
Let ρ(x) be the rank of J at x ∈ X. Then, a point x of X is non-singular if
ρ(x) = ρ := maxx∈X ρ(x).
Notice that, if X has several irreducible components, with this definition, a nonsingular point x belongs to the component of smallest dimension. In fact, we have
the following result of H. Whitney [24]:
Proposition Let X be an irreducible complex algebraic set. The non-singular
points of X is an open dense subset of X which is a complex analytic sub-manifold
of X of dimension n − ρ.
One can define a local ring at each point x of X. If we assume X to be an
irreducible algebraic subset of CN , the ideal I(X) of polynomials in C[X1 , . . . , XN ]
which vanish on X is a prime ideal. The restrictions to X of the polynomials
in C[X1 , . . . , XN ] form a C-algebra A[X] which is isomorphic to the quotient of
C[X1 , . . . , XN ] by the ideal I(X):
A[X] = C[X1 , . . . , XN ]/I(X).
The regular functions of X at x form the localization OX,x of A[X] at x.
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LÊ DŨNG TRÁNG
Then, one can prove that:
Proposition Let X be an irreducible complex algebraic set. A point x of X is
non-singular if and only if its local ring OX,x is regular.
See [1] for the definition of a regular local ring.
If the complex algebraic set X has several components, one says that a point x
is non-singular if it has a neighborhood Ux in X which is analytically isomorphic
to an open set of an affine space Cd . Then, it is equivalent to say that the local
ring OX,x of X at x is regular of Krull dimension equal to d.
A point x of an algebraic set X is non-singular if it does not belong to two
distinct irreducible components of X and it is a non-singular point of the irreducible
component in which it is contained.
It is easier to study complex hypersurfaces, i.e. algebraic subsets of CN defined
by one reduced equation. In fact, it was the study of an isolated singularity of
hypersurface which lead E. Brieskorn to show that exotic spheres is given by an
algebraic equation ([3]). This result of E. Brieskorn triggered J. Milnor to make a
seminar in Princeton which lead to his famous book “Singular points of complex
hypersurfaces.” ([14]).
1. Basic results
A result that we shall often need is the following:
Proposition 1.1. Let K be the field of real numbers or the field of complex numbers.
Let E be a K-algebraic set and F be an algebraic subset of E, such that E − F is
non-singular. The restriction of a K-polynomial function to E − F has a finite
number of critical values.
As consequence, one has in particular:
Corollary 1.2. Let K be the field of real numbers or the field of complex numbers.
The number of singular fibers of a K-polynomial is finite.
The proof of the Proposition 1.1 uses a simple partition of a difference of real
algebraic sets:
Proposition 1.3. Let E and F be real algebraic sets. The difference E − F is the
disjoint union of a finite number of connected smooth manifolds.
Proof. We proceed by induction on the dimension of E. Let us denote ΣE the
algebraic subset of singular points of E.
We notice that E − F is the disjoint union of E − (ΣE ∪ F ) and ΣE − F . Since
dim ΣE < dim X, by induction it is enough to prove the Proposition 1.3 in the case
E − F is non-singular.
Now, let us assume that E − F is non-singular. We have the following partition
of E:
(1) Σ0 := E;
(2) Let i ≥ 0, Σi+1 = Σ(Σi ).
By Hilbert finiteness Theorem this sequence is stationary, so there is ` such that
Σ` 6= ∅ and Σ`+1 = Σ(Σ` ) = ∅. Therefore, we have:
a a
a a
E = (Σ0 \ Σ1 )
. . . (Σi \ Σi+1 )
...
Σ` .
COMPLEX SINGULARITIES
3
In this way, we have defined a partition of E defined by the strata Σi \ Σi+1 , for
0 ≤ i ≤ `.
This partition gives a partition of E−F with the strata Σi −(Σi+1 ∪F ), 0 ≤ i ≤ `.
Each stratum Σi − (Σi+1 ∪ F ) is smooth manifold. It remains to prove that it has
a finite number of connected components.
This results from the following:
Lemma 1.4. Let E − F be a difference of real algebraic sets. Assume that this
difference is non-singular, then it has a finite number of connected components.
Proof of the Lemma. Let f1 , . . . , fk be real polynomials which define F in E.
Pk
Then, g := 1 fi2 = 0 also define F in E.
The graph of 1/g : E − F → R is diffeomorphic to E − F . It is also an algebraic
subset of E × R defined by ug = 1 where u is the coordinate of the target of 1/g.
We have the general result:
Lemma 1.5. A non-singular real algebraic subset of RN has the homotopy type of
a finite complex.
The proof of this lemma is done using Morse theory (see [15]), since the square
of the distance function restricted to a non-singular real algebraic set E to almost
any point of RN is a Morse function (see [15] Theorem 6.6 p. 36). Then, a critical
point of such a function being non-degenerate, it is isolated, furthermore the critical
points form a real algebraic set.
Corollary A.2 of Appendix A of [14] shows that the number of critical points of
the square of the distance function to almost every point of RN must be finite. By
Morse theory, the real algebraic set E has the homotopy type of a finite complex.
In particular it has a finite number of connected components. This ends the proof
of the Lemma above and of the Proposition 1.3.
Then, the critical set of the restriction of a K-polynomial function G to a nonsingular difference E − F of K-algebraic sets is a difference of K-algebraic sets
ΣG − F , where the algebraic set ΣG is defined in the following way: let I(E) be the
ideal of E in Kn generated by the polynomials f1 , . . . , fk in K[X1 , . . . , Xn ]; consider
on each irreducible component Ei of E the maximal rank ρi of the jacobian matrix:


∂f1 /∂X1 , . . . , ∂f1 /∂Xn
.
...
J =
∂fk /∂X1 , . . . , ∂fk /∂Xn
Now, consider the algebraic subset Σi of Ei defined by the minors (ρi + 1) × (ρi + 1)
of:


∂G/∂X1 , . . . , ∂G/∂Xn
 ∂f1 /∂X1 , . . . , ∂f1 /∂Xn 
.
J(G) = 


...
∂fk /∂X1 , . . . , ∂fk /∂Xn
The algebraic subset ΣG is the union of the subsets Σi .
We know that the difference ΣG −F of algebraic sets is the finite disjoint union of
real connected smooth manifolds and the restriction of G to each of these connected
manifolds is critical, so it must be constant. So the restriction of G to the critical
set of the restriction of the K-polynomial function G to the non-singular difference
E − F has a finite number of values. This ends the proof of Proposition 1.1.
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LÊ DŨNG TRÁNG
Now, we have the following consequence of Proposition 1.1
Corollary 1.6. Let x be an isolated singular point of an algebraic set X. There
exists εx > 0, such that the sphere Sε (x), centered at x with radius ε > 0, ε ≤ εx ,
is transverse to X.
This corollary is consequence of Proposition 1.1 by considering the real polynomial which is the square of the distance to x. Since 0 is a critical value, there are
no critical values in the interval (0, εx ].
2. Plane curves
The complex hypersurfaces of dimension 1 are the complex plane curves. Singularities of complex plane curves are isolated. According to Corollary 1.6 a real
sphere Sε (x) centered at a singular point x of a plane curve with radius ε small
enough intersects the curve transversally.
As a real sphere of the complex plane C2 has dimension 3 and the non-singular
part of a complex curve is a real manifold of real dimension 2 the transverse intersection in C2 ' R4 has real dimension 1. Therefore it is a link. Since it exists
εx > 0, such that the spheres Sε (x), centered at x with radius ε > 0, ε ≤ εx , are
transverse to the curve C, for ε >, ε ≤ εx , the links:
Sε (x) ∩ C ⊂ Sε (x)
are diffeomorphic.
In fact, the type of these links are the same and are given by algebraic invariants
computed from the equation f (X, Y ) = 0 of C. These links are called the algebraic
link of the curve C at x (see [12]).
First, we have the existence of a local parametrization.
A method due to Newton shows that, if f (0, Y ) 6= 0, one can prove that at the
point (0, 0), there is a formal power series ϕ(X 1/n ) ∈ C[[X 1/n ]] in X 1/n , such that:
f (X, ϕ(X 1/n )) = 0.
This is known as the Theorem of Puiseux. The series ϕ(X 1/n ) are called a Puiseux
expansion or a Puiseux series of the curve {f = 0} at the point (0, 0)..
If (0, 0) is a non-singular point, n = 1 and the formal series ϕ(X) is given by the
implicit function Theorem. In the implicit function Theorem, one knows that this
series is convergent.
The Puiseux series gives a parametrization of the curve at (0, 0):
X = tn and Y = ϕ(t).
Precisely we have (see [19]):
Theorem 2.1 (Puiseux Theorem). Let K be a algebraically closed field of characteristic 0. Let f ∈ K[[X, Y ]]. We assume that f (0, 0) = 0 or equivalently f is in the
maximal ideal (X, Y ) of K[[X, Y ]]. Suppose that f (0, Y ) 6= 0, there is an integer
N and a power series Φ ∈ K[[X]], such that f (X, Φ(X 1/N )) = 0, where X 1/N is a
N -th root of X.
Remark that the choice of X 1/N is not unique. If ζ is a N -th root of 1, ζX 1/N
is another N -th root of X. We fix some N -th root of X.
The series ϕ(X 1/N ) are called a Puiseux expansion and one can prove that it is
convergent if the field K = C and f is a polynomial (see [6]).
COMPLEX SINGULARITIES
5
Assume that n is the smallest integer, such that ϕ(X 1/N ) belongs to K[[X 1/n ]].
Let K((X)) be the field of fractions of K[[X]]. One can prove that the field
extension K((X)) ⊂ K((X 1/n )) is a Galois extension and the Galois group G is
given by n-th roots of unity. In fact we have: {σ ∈ G} 7→ {X 1/n 7→ ζX 1/n }. The
extension being Galois we have:
Y
(Y − ϕ(ζX 1/n )) ∈ K[[X]][Y ] ⊂ K[[X, Y ]]
{ζ,ζ n =1}
and {ζ,ζ n =1} (Y − ϕ(ζX 1/n )) is an irreducible element of K[[X, Y ]].
Q
The element {ζ,ζ n =1} (Y − ϕ(ζX 1/n )) is called a branch of the curve:
Q
{f (X, Y ) = 0}
at the singular point (0, 0).
When K = C, each component of the link Sε ((0, 0)) ∩ {f (X, Y ) = 0} corresponds
to a branch of {f (X, Y ) = 0} at (0, 0).
Let us suppose f (X, Y ) to be irreducible in K[[X, Y ]]:
Y
f (X, Y ) = α
(Y − ϕ(ζX 1/n )) = 0
{ζ,ζ n =1}
where α ∈ K and 6= 0.
We define a finite sequence of exponents, called Puiseux exponents relatively to
the coordinates (X, Y ), which is important in the description of the local topology
of the singularity, when the field K is the field of complex numbers.
If n = 1, the Puiseux expansion is a formal series with coefficients in K. In this
case, there are no Puiseux exponent.
If n > 1, the set E1 = {k/n 6∈ N, ak 6= 0} is not empty, since n is the smallest
integer, such that ϕ(X 1/n ) ∈ K[[X 1/n ]].
Define the first Puiseux exponent relatively to the coordinates (X, Y ):
k1
= inf{k/n 6∈ N, ak 6= 0}.
n
Then, either (k1 , n) are relatively prime and there is only one Puiseux exponent,
or
k1
m1
=
n
n1
and n1 < n. The set E1 = {k/n 6∈ (1/n1 )N, ak 6= 0, k > k1 } is not empty, otherwise
ϕ(X 1/n ) belongs to K[[X 1/n1 ]].
Define the second Puiseux exponent by:
k2
k
1
:= inf{ 6∈
N, ak 6= 0, k > k1 }.
n
n n1
There is a unique way to write:
k2
m2
=
n
n1 n2
in such a way that (m2 , n2 ) are relatively prime.
Then, either n1 n2 = n and there are only two Puiseux exponents, or n1 n2 < n
and the set:
1
k
E2 = { 6∈
N, ak 6= 0, k > k2 }
n n1 n2
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LÊ DŨNG TRÁNG
is not empty.
By induction, one defines mh /n1 . . . nh , where (mh , nh ) are relatively prime.
Either, n1 . . . nh = n and there are h Puiseux exponents, or n1 . . . nh < n and the
set:
1
k
N, ak 6= 0, k > kh }
Eh = { 6∈
n n 1 . . . nh
is not empty, in which case inf Eh = kh+1 /n = mh+1 /n1 . . . nh+1 , where the pair of
integers (mh+1 , nh+1 ) are relatively prime and unique.
The process has to end, since n has a finite number of divisors.
Therefore the Puiseux expansion ϕ(X 1/n ), where n is the smallest integer such
that ϕ(X 1/n ) ∈ K[[X 1/n ]], has a finite number of Puiseux exponents relatively to
the coordinates (X, Y ):
k1
m1
kg
mg
=
=
,...,
,
n
n1
n
n1 . . . ng
n1 . . . ng = n.
In the case K = C and the curve {f (X, Y ) = 0} is a branch at the point (0, 0),
the Puiseux exponents relatively to the coordinates (X, Y ) determine the knot
Sε ((0, 0)) ∩ {f (X, Y ) = 0}.
With the notations above, if the number of Puiseux pairs is one, it is a torus
knot (m1 , n1 ).
If the number of Puiseux pairs is g ≥ 2, it is a torus knot around the knot
determined by g − 1 pairs with meridian the transverse meridian and with parallel
a translate of the knot with g − 1 Puiseux pairs.
In short the local knot of a plane branch is an iterated torus knot determined
by the Puiseux exponents relatively to the coordinates (X, Y ).
In fact, the Puiseux exponents relatively to coordinates (X, Y ) do not depend on
the coordinates (X, Y ) if (X, Y ) are general coordinates. These Puiseux exponents
are called the local Puiseux exponents of the curve at the singular point 0.
The reader will find details in [6].
In the case of a curve having k branches {f1 = 0}, . . . , {fk = 0} at the singular
point (0, 0), a Theorem of M. Lejeune-Jalabert and O. Zariski shows that the link
is uniquely determined by the knots of the branches and the pairwise intersection
numbers:
({fi = 0}, {fj = 0})(0,0) = dimC C[[X, Y ]]/(fi , fj ).
3. Hypersurfaces with an isolated singularity
In the case of dimension ≥ 2 complex hypersurfaces might have a singular locus
of dimension ≥ 1. When the singular locus ΣX of a complex hypersurface X has
dimension ≥ 1 at a singular point x ∈ X any sphere Sε (x) intersects ΣX . This fact
makes the study of the topology of X near x difficult. We first assume that x is an
isolated singular point.
We have the following fibration Theorem which does not assume that the singular
point is isolated, but which is rather easy to prove when the singular point is
isolated:
Theorem 3.1. Let f : Cn+1 → C be a polynomial function (not necessarily reduced). For any point x ∈ f −1 (0), there is ε > 0 small enough and ηε > 0, such
COMPLEX SINGULARITIES
7
that, for any η, ηε ≥ η > 0 the map:
ϕε,η : Bε (x) ∩ f −1 (Sη ) → Sη
induced by f , where Bε (x) is the ball of Cn+1 of radius ε centered at the point x and
Sη is the circle of C centered at 0 of radius η, is a locally trivial smooth fibration.
Proof in the isolated singular case. We have seen in Corollary 1.6 that there
is εx > 0 such that, for any ε, εx ≥ ε > 0, the sphere Sε (x) intersects f −1 (0)
transversally. Using Proposition 1.1 with E = Sε (x),F = ∅ and the restriction of f
to Sε (x), it yields that there is η1 > 0 such that, for any t, |t| ≤ η1 , the hypersurface
f −1 (t) intersects the sphere Sε (x) in C2 .
The Corollary 1.2 implies that the restriction of f to the interior B̊ε (x) ∩ f −1 (Sη )
of Bε (x) ∩ f −1 (Sη ) has maximal rank if η, 0 < η < η1 , is small enough.
The transversality of the sphere Sε (x) and f −1 (t) for any t, |t| ≤ η1 implies that
the restriction of the polynomial f to the boundary Sε (x) ∩ f −1 (Sη ) of Bε (x) ∩
f −1 (Sη ) has maximal rank.
Ehresmann Lemma (that we recall below) implies that the restriction of f to
Bε (x) ∩ f −1 (Sη ) induces a locally trivial smooth fibration:
ϕε,η : Bε (x) ∩ f −1 (Sη ) → Sη .
Now, let us recall Ehresmann Lemma (see e.g. [7] (20.8) problème 4, or see also
[4] Theorem 8.12 p. 84 when the boundary is empty):
Lemma 3.2 (Ehresmann Lemma). Let (M, ∂M ) be a smooth manifold M with
boundary ∂M . Let ϕ : M → N be a proper smooth map onto a connected manifold
N . Assume:
(1) The map ϕ is proper;
(2) The restriction of ϕ to ∂M is submersive and surjective onto N ;
(3) The restriction of ϕ to M is submersive and surjective onto N .
Then, the map ϕ is a locally trivial smooth fibration.
Therefore, the Theorem 3.1 is quite easy in the case of an isolated singularity
and only depends on Ehresmann Lemma.
The local fibration given by Theorem 3.1 is called the Milnor fibration. By lifting
on the total space the unit vector field on the circle, one obtains a diffeomorphism
of a fiber of the fibration over itself by integrating the lifted vector field. This diffeomorphism is not unique, but its class of isotopy is well-defined. This isotopy class
is called the geometric monodromy. It induces an automorphism on the homology
or the cohomology of the fiber, called the monodromy of the fibration.
Let ϕ : E → S1 be a locally trivial smooth fibration on a circle and let h : F → F
be a diffeomorphism which represents the geometrical monodromy of ϕ. Then, the
map:
F × [0, 1]/{(x, 1) ∼ (h(x), 0), x ∈ F } → [0, 1]/(0 ∼ 1) ' S1
is isomorphic to ϕ.
In his book [14], J. Milnor proved:
Theorem 3.3. Let f : Cn+1 → C be a complex polynomial. Let x ∈ {f = 0}.
There is ε0 such that, for any ε, 0 < ε < ε0 , the quotient f /kf k induces a map:
Ψε : Sε − {f = 0} → S1
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LÊ DŨNG TRÁNG
which is a locally trivial smooth fibration.
Using Theorem 5.11 of [14], one can prove that if Theorem 3.1 is true, the
fibrations of 3.1 and 3.3 are isomorphic.
In the case of complex plane curve C, we have seen that to each singularity
x ∈ C one associates a local link Sε (x) ∩ C ⊂ Sε (x).
In the case that C has only one branch at x we know that this link is a knot.
J. Milnor proved that for knots whose complement fibers over S1 the Alexander
Polynomial equals the characteristic polynomial of the monodromy of the fibration.
In the case of the algebraic knot associated to a complex branch, O. Zariski in
[27] proved that the Puiseux exponents determine the fundamental group of the
complement of the knot, hence the Alexander Polynomial. W. Burau in [5] showed
that the Alexander polynomial of an algebraic knot determines the Puiseux pairs of
the branch defining the knot. This implies that in the case of one complex branch
the local topology of a plane curve singularity is equivalent to the knowledge of the
local Puiseux exponents.
The fiber of the fibration ϕε,η is called the Milnor fiber of the isolated singularity
of the polynomial f at x.
We have the following remarkable Theorem:
Theorem 3.4. The Milnor fiber of a hypersurface {f = 0} of dimension n at an
isolated singularity x = (x0 , . . . , xn ) has the homotopy type of a bouquet of spheres.
The number of spheres is:
µ(f, x) = dimC C{X0 − x0 , . . . , Xn − xn }/(∂f /∂X1 , . . . , ∂f /∂Xn ).
We are going to give the proof that I gave in [10].
Pi=n
Let f + i=0 ai xi a linear perturbation of the polynomial f . We have:
Lemma 3.5. Let U be an open neighborhood of x. There is an open neighborhood
V of x in Cn+1 such that for almost all (a0 , . . . , an ) in V, there are µ(f, x) nonPi=n
degenerate critical points of f + i=0 ai xi in U.
Proof. Consider the germ of complex analytic map:
Φ : (Cn+1 , 0) → (Cn+1 , 0).
Since the point x is an isolated singular point of {f = 0}, it is also an isolated
critical point of f , because of Proposition 1.1. Therefore, the quotient:
C{X0 = x0 , . . . , Xn − xn }/(∂f /∂X0 , . . . , ∂f /∂Xn )
of the ring of convergent power series C{X0 −x0 , . . . , Xn −xn } by the ideal generated
by the partial derivatives of f is a finite dimensional complex vector space. This
means that 0 is an isolated point in the fiber of Φ over the origin 0.
The geometrical Weierstrass preparation Theorem (see [9]) shows that there are
neighborhoods U0 of x and V0 of 0 in Cn+! , such that Φ induces a finite morphism
of U0 onto V0 , i.e. a proper morphism with finite fibers of U0 on V0 .
In fact, one can define U0 , V0 such that the set Γ of critical points of Φ is a
complex analytic subspace of U0 and the image ∆ of Γ by Φ is a complex analytic
hypersurface of V0 , such that Φ induces a locally trivial smooth covering of U0 −
Φ−1 (∆) over V0 − ∆ (see e.g. [22]) . By [18] the degree of this covering is:
dimC C{X0 − x0 , . . . , Xn − xn }/((∂f /∂X0 , . . . , ∂f /∂Xn ).
COMPLEX SINGULARITIES
9
Let us choose the point −(a0 , . . . , an ) ∈ V0 −∆. The points of Φ−1 (−(a0 , . . . , an ))
are non critical points of Φ, so they are non-degenerate critical points of the function
Pi=n
f + i=0 ai xi and their number is (cf e.g.[18]):
dimC C{X0 − x0 , . . . , Xn − xn }/((∂f /∂X0 , . . . , ∂f /∂Xn )
which is the Milnor number µ(f, x) of f at x.
This method which consists in deforming the function f to obtain only nondegenerate critical points is usually called the Morsification of f .
One can choose (a0 , . . . , an ) in such a way that the complex line from the point
(0, . . . , 0) to the point (a0 , . . . , an ) is intersecting ∆ locally at an isolated point
(0, . . . , 0).
Let us choose (a0 , . . . , an ) in V0 − ∆ to satisfy this previous condition. Then, for
Pi=n
t 6= 0 small enough the function f + i=0 tai xi has µ(f, 0) ordinary critical point
Pi=n
in U . Assume that the origin 0 is not a critical value of f + i=0 tai xi for t 6= 0
small enough..
Pi=n
Consider the restriction ϕ of the real function |f + i=0 tai xi | to Bε (x).
Pi=n
If |t| is small enough, the critical points of the restriction of ft := f + i=0 tai xi
to Bε (x) are lying in the interior of the closed disk Dη .
Lemma 3.6. For |t| small enough, the spaces Bε (x)∩f −1 (Dη ) and Bε (x)∩ft−1 (Dη )
are diffeomorphic.
Proof. Just notice that the space Bε (x) ∩ f −1 (Dη ) is a manifold with corners. the
corners are precisely Sε (x) ∩ f −1 (Dη ). One has to adapt Ehresmann Lemma to
this new situation: instead of a boundary, we have a boundary and corners and the
restriction to the corners must also be submersive.
Details are left to the reader.
Pi=n
Now, assume that the origin 0 is not a critical value of f + i=0 tai xi for t 6= 0
small enough. Choose t0 6= 0 small enough.
In a disk De centered at 0 with radius e small enough, there are no critical value
of ft0 .
Since, the critical points of ft0 are non-degenerate, one can prove that the critical
points of |ft0 | in Bε (x) ∩ ft−1
(Dη − De ) are Morse points of index equal to n + 1.
0
Consequence. Applying Morse theory to the real function ϕ restriction of the real
smooth function |ft0 | to Bε (x) ∩ ft−1 (Dη ), we obtain that the space Bε (x) ∩ ft−1 (Dη )
has the homotopy type of Bε (x) ∩ ft−1 (De ) to which we have attached µ(f, x) cells
of dimension n + 1.
The Theorem 7.1 will be a consequence of the two following Lemmas:
Lemma 3.7. The space Bε (x) ∩ f −1 (Dη ) is contractible.
Lemma 3.8. Assume that a contractible space B has the homotopy type of a space
A to which is attached cells e1 , . . ., ek of dimension n + 1, then A has the homotopy
type of a bouquet of k spheres of dimension n.
4. Stratifications
Let E be a complex algebraic subset of Cn+1 . Let S1 , . . . , Sk be a finite partition
of E:
a a
E = S1
...
Sk .
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LÊ DŨNG TRÁNG
We say that it is a complex algebraic partition if the closures S i of Si in Cn+1 and
differences S i − Si are complex algebraic sets, for 1 ≤ i ≤ k.
Similarly let E be a real semi-algebraic subset of RN . Let S1 , . . . , Sk a finite
partition of E:
a a
E = S1
...
Sk .
We say that this partition is a real algebraic partition of E if the closures S i of Si
in RN and the differences S i − Si are semi-algebraic sets, for all i, 1 ≤ i ≤ k.
Any set Si in a partition S of E, 1 ≤ i ≤ k, is called a stratum of the partition.
Let E be a complex algebraic subset of Cn+1 .
We say that (Si )1≤i≤k is a complex stratification, if it is a complex algebraic
partition and each stratum Si is a connected complex manifold, and the partition
satisfies the frontier condition, i.e. Si ∩ S j 6= ∅ implies Si ⊂ S j .
Analogously let E be a real semi-algebraic subset of RN and let S1 , . . . , Sk be
an algebraic partition of E.
We say that the partition (Si )1≤i≤k is a real stratification of E, if it is a real
algebraic partition and each stratum Si is a connected smooth manifold and the
partition satisfies the frontier condition, i.e. Si ∩ S j 6= ∅ implies Si ⊂ S j .
Whenever we shall consider the stratification of a real algebraic set, we shall
consider a stratification by semi-algebraic sets, considering a real algebraic set as a
particular semi-algebraic set. Similarly, one may consider a complex algebraic set
as a real algebraic set and it may be endowed with a real stratification.
We say that a map:
ϕ:E→F
of a stratified set (E, S) into a stratified set (F, T ) is a stratified map if for any
stratum Si ∈ S, there is a stratum Tj ∈ T such that ϕ induces a submersive and
surjective map of Si onto Tj .
An example.
Let E be a complex or real algebraic set. Let Σ(E) be the subset of singular
points of E. We defined above a decreasing sequence of algebraic sets defined by
induction:
(1) Σ0 := E
(2) Let i ≥ 0, Σi+1 = Σ(Σi )
By Hilbert finiteness Theorem this sequence is stationary, so there is ` such that
Σ` 6= ∅ and Σ`+1 = Σ(Σ` ) = ∅. Therefore, we have:
a a
a a
E = (Σ0 \ Σ1 )
. . . (Σi \ Σi+1 )
...
Σ` .
In this way, we have defined a partition of E defined by the strata Σi \ Σi+1 , for
1 ≤ i ≤ ` + 1.
The strata are smooth manifolds. However, the partition might not satisfy the
frontier condition.
A complex or real algebraic partition T of a complex or real algebraic set is said
to be finer than the algebraic partition S if the closures of the strata of S are union
of strata of T . We also say that the partition T is a refinement of S.
In [25] (Theorem 18.11) H. Whitney proves that:
COMPLEX SINGULARITIES
11
Proposition 4.1. Any partition of a complex or real algebraic set has a refinement
which is a stratification with connected strata.
The condition (a) of Whitney
Let S = (Si )1≤i≤k be a stratification of a complex algebraic subset E of CN
(resp. a semi-algebraic set). Let (Si , Sj ) be a pair of strata such that Si ⊂ S j . Let
p ∈ Si . We say that the pair (Si , Sj ) satisfies the condition (a) of Whitney along Si
at the point p, if, for any sequence (pn ) of points of Sj which converges to p such
the sequence of tangent spaces (Tpn Sj ) has a limit T , then the tangent space Tp Si
is contained in T .
We say that, the pair (Si , Sj ) satisfies the condition (a) of Whitney along Si , if
it satisfies the condition (a) of Whitney along Si at any point p of Si .
We say that the stratification S satisfies the condition (a) of Whitney if, for any
pair (Si , Sj ) such that Si ⊂ S j , the pair satisfies the condition (a) of Whitney.
Example: Consider the complex algebraic set E defined by X 2 − Y 2 Z = 0, which
is known as “Whitney umbrella”.
The singular set of E is given by X = Y = 0. This is a line Σ contained in E.
A stratification of E is given by E \ Σ and Σ. The pair (Σ, E \ Σ) does not satisfy
the condition (a) of Whitney along Σ at the origin 0 of C3 .
One may consider a sequence (pn ) of points of E \ Σ given by pn = (0, yn , 0),
where limn→∞ yn = 0. Then, p = (0, 0, 0). We have Tpn (E\Σ) = (X, Y )−plane and
Tp (Σ) = Z − axis. The limit at the point (0, 0, 0) of the sequence (Tpn (E \ Σ))n∈N
of tangent spaces is the (X, Y ) − plane and it does not contain the Z − axis.
The condition (b) of Whitney
Let (Si , Sj ) be a pair of strata such that Si ⊂ S j . Let p ∈ Si . We say that the
pair (Si , Sj ) satisfies the condition (b) of Whitney along Si at the point p, if for any
sequence pn of points of Sj and any sequence (qn ) of points of Si , which converges
to p, such that the sequence of tangent spaces Tpn Sj has a limit T and the sequence
of lines qn pn has a limit `, we have ` ⊂ T .
We say that the pair (Si , Sj ) satisfies the condition (b) of Whitney along Si if it
satisfies the condition (b) of Whitney along Si at any point p of Si .
We say that the stratification S satisfies the condition (b) of Whitney if, for any
pair (Si , Sj ) such that Si ⊂ S j , the pair satisfies the condition (b) of Whitney.
Examples: In the example given above the pair (Σ, E \ Σ) does not satisfy the
condition (b) of Whitney along Σ at the origin 0 of C3 .
Now, consider the complex algebraic set F given by X 2 − Y 2 Z 2 − Z 3 = 0 in C3 .
The singular locus Σ is the line X = Z = 0. One can prove that this stratification
satisfies the condition (a) of Whitney, but it does not satisfies the condition (b) at
the origin 0.
The sequences pn = (0, 1/n, −1/n2 ) and qn = (0, 1/n, 0) give sequences of tangent planes (Tpn (F \ Σ)) and of lines (qn pn ) whose limits T and ` are such that:
` 6⊂ T .
In fact, the condition (b) of Whitney implies the condition (a) of Whitney (see
[13] Proposition 2.4):
12
LÊ DŨNG TRÁNG
Lemma 4.2. Let E be a complex analytic subset of CN (resp. a real semi-algebraic
subset of RN ). Let S = (S1 , . . . , Sk ) be a stratification of E. Suppose that Si ⊂ S j .
Let p ∈ Si . Suppose that the pair (Si , Sj ) satisfies the (b)-condition of Whitney at
p, then it satisfies the (a)-condition of Whitney at p.
Proof. The following proof was given to me orally by D. Cheniot.
Assume that the pair (Si , Sj ) satisfies the condition (b) of Whitney at p ∈ Si .
Consider a sequence (pn )n∈N a sequence of point of Sj which converges to p and
such that the sequence of tangent spaces Tpn (Sj ) converge to T . We have to show
that: Tp (Si ) ⊂ T . Suppose otherwise. Then, there is a line D in CN (resp. RN )
through the origin, such that D ⊂ Tp (Si ), but D 6⊂ T .
By definition of the tangent space Tp (Si ), there is a sequence of points (qn )n∈N of
points of Si which converges to p and such that the lines pqn converge to D. Since
the sequence (pn ) converges to p, endowing the projective space of line directions
with a metric, for any k ∈ N, we can find nk such that the distance between the line
directions pqk and qk pnk is bounded by 1/k. Therefore, qk pnk converges to D, and
this would contradict that (Si , Sj ) satisfies the condition (b) of Whitney at p ∈ Si .
One can extend the definition of stratifications to complex or real projective
varieties by considering the local affine covering. One can also define stratified
algebraic maps with these objects.
In [24] H. Whitney proved that:
Theorem 4.3. Let S be an algebraic partition of an algebraic variety V . There
is a stratification of V finer than S which satisfies the property (a) (resp. (b)) of
Whitney.
The importance of the Whitney condition comes from the fact that one may
assert a generalization of Ehresmann Lemma:
Theorem 4.4 (First isotopy Theorem of Thom-Mather [13]). Let ϕ : X → T be a
proper algebraic map. Assume that X is stratified by S = (S1 , . . . , Sk ) and that T
is connected and non-singular. If the restrictions of ϕ to each strata Si (1 ≤ i ≤ k)
are smooth maps which are submersive into T , then the map ϕ is a locally trivial
topological fibration.
Beware that the locally trivial fibration is not smooth, but topological, as the
following example shows.
Example. Let E be the real algebraic subset of R3 defined by:
XY (X − Y )(X − T Y ) = 0.
Consider the projection onto the T -axis. Consider the “tube” given by X 2 +Y 2 ≤ 1.
The intersection:
V = E ∩ {X 2 + Y 2 ≤ 1} ∩ {0 < T < 1}
is semi-algebraic. The projection ϕ onto the T -axis induces a proper map of V onto
the open interval 0 < T < 1. Let us stratify V with S1 = {(0, 0)} × {0 < T < 1},
S2 = V ∩ {X 2 + Y 2 = 1} − S1 and S3 = V − (S1 ∪ S2 ). The restrictions of ϕ to the
connected components of each of Si , i = 1, 2, 3, induce submersive maps onto the
open interval {0 < T < 1}. By Theorem 4.4 above the map ϕ is a locally trivial
COMPLEX SINGULARITIES
13
topological fibration. It cannot be a locally trivial smooth fibration, because the
cross-ratio of the four lines varies continuously.
5. Hypersurfaces with non-isolated singularity
In this section we shall prove the Theorem 3.1 in the case x ∈ f −1 (0) is not an
isolated singularity.
In view of Ehresmann Lemma (Lemma 3.2) one has to prove that, for ε > 0
small enough, there is η > 0, such that, the space Bε (x) ∩ f −1 (Sη ) is a manifold
with boundary and the restrictions of f to the interior B̊ε (x) ∩ f −1 (Sη ) and to the
boundary Sε (x) ∩ f −1 (Sη ) of that manifold have maximal rank.
In fact, we are going to prove:
Lemma 5.1. if ε > 0 is small enough, for any point y ∈ Sε ∩ f −1 (0), for any
sequence yn ∈ Sε (x) − f −1 (0) which tends to y and for which the tangent spaces
Tyn (f −1 (f (yn ))) have a limit T , the limit T is transverse to Sε (x) in Cn+1 .
To prove this assertion we shall introduce a new coordinate T and consider the
hypersurface XA defined by:
FA (X0 , . . . , Xn , T ) = f (X0 , . . . , Xn ) + T A = 0.
Then:
Proposition 5.2. For A large enough and for ε > 0 small enough the limit of
tangent hyperplanes at sequences of non-singular points zn ∈ XA which tend to a
point:
y ∈ (Bε (x) × {0}) ∩ XA
is transverse to the hyperplane T = 0 in Cn+1 × C.
Proof Since the polynomial function f is, in particular a holomorphic function
having at x a critical point, there is an open neighborhood U of x in Cn+1 , a
constant C and a number α, 1 > α > 0 such that in U, one has the inequality
(Lojasiewicz inequality):
Ckgrad f (z)k ≥ |f (z)|α
for any z ∈ U.
A−1
The gradient of the polynomial function FA is the vector (grad f, AT
).
Let (yn ) := (zn , Tn ) be a sequence of non-singular points of {FA = 0} ∩ (U × C)
which tends to a point y of {f = 0} ∩ (U × {0}). Since FA (yn ) = f (zn ) + TnA = 0
we have:
|f (zn )| = |Tn |A .
Assume that the directions of gradient at un has a limit `. Since, for any z ∈ U,
Ckgrad f (z)k ≥ |f (z)|α , in particular we have Ckgrad f (zn )k ≥ |f (zn )|α . If A is
big enough:
A−1
1>
> α.
A
Therefore, the gradient of the polynomial function f + T A at yn is:
A−1
(grad f (zn ), AT n
)
and has the same direction as:
A−1
(
grad f (zn ) AT n
grad f (zn )
A−1−Aα
,
)=(
, AT n
)
Aα
Aα
Aα
Tn
Tn
Tn
14
LÊ DŨNG TRÁNG
Since A − 1 > Aα. and Ckgrad f (zn )k ≥ |f (zn )|α = |Tn |Aα , the limit ` of these
directions of gradient is contained in the hyperplane Cn+1 × {0} = {T = 0}.
Since the tangent space at a smooth point y to XA is orthogonal to the gradient
at y of FA , this proves our proposition.
Now, recall that a stratification S = (S1 , . . . , Sk ) of a real semi-algebraic set
satisfies the property (a) of Whitney, if for any pair of strata (Si , Sj ), such that
Si ⊂ S j , for any point y ∈ Si , for any sequence (yn ) of points of Sj which tends to
y such that the limit of tangents Tyn Sj exists and equals T , we have T ⊃ Ty Si .
Theorem 4.3 asserts any semi-algebraic set has a stratification of Whitney with
the property (a):
Proposition 5.3. Let E be a semi-algebraic set. Any partition of E have a refinement which is a stratification satisfying the property (a) of Whitney and any
subspace of non-singular points can be a stratum of a stratification with the property (a) of Whitney.
Let A be chosen big enough such that A − 1 > Aα and let S be a stratification
of the hypersurface {f + T A = 0} of Cn+2 such that it satisfies the property (a)
of Whitney, the space {f + T A = 0} − (Cn+1 × {0}), which is non-singular, is a
stratum and {f + T A = 0} ∩ (Cn+1 × {0}) is a union of strata.
n+2
Let ε1 > 0 such that the closed ball B̊n+2
is a subset of the open
ε1 (x) of C
n+2
set U × C of C
where U is the open neighborhood of x in Cn+1 where we have
Lojasiewicz inequality for f near x.
The intersections of the strata of S with the open ball B̊n+2
ε1 (x) give a stratificaA
n+2
tion of {f + T = 0} ∩ B̊ε1 (x).
The strata of {f + T A = 0} ∩ (Cn+1 × {0}) define strata T1 , . . . Tk of the hypersurface {f = 0} ∩ B̊ε1 (x) in the open ball B̊ε1 (x) centered at x with radius ε
small enough in Cn+1 . We can add the stratum T0 := B̊ε1 (x) − {f = 0} to obtain
a stratification of B̊ε1 (x).
Let (yn ) := (zn , tn ) be a sequence of points of the hypersurface {f + T A =
0} ∩ (U × {0}) such that tn 6= 0. Assume that it has a limit (y, 0) ∈ {f = 0} ∩
A
(Cn+1 × {0}) ∩ B̊n+2
= 0} at yn have a
ε1 (x) and the tangent spaces Tyn of {f + T
limit T .
Since we have a stratification with the property (a) of Whitney, we have:
T ⊃ Ty (Tj ),
if y ∈ Tj .
On the other hand, we saw above that the limits of tangents T to {FA = 0} at
any point of Bε1 (x) × {0} ⊂ Cn+1 × {0} is transverse to the hyperplane Cn+1 ×
{0}. One deduces that the limit of the sequence of spaces Tyn ∩ {T = tn } equals
T ∩ (Cn=1 × {0}) and it contains Ty (Tj ), if y ∈ Tj , since Ty (Tj ) is contained in T
because of Property (a) of Whitney and Ty (Tj ) is contained in Cn+1 because Tj is
in Cn+1 .
The affine spaces Tyn ∩ {T = tn } are tangent to {f + tA
n = 0}. Conversely we
observe that any sequence (zn ) of (Cn+1 − f −1 (0)) ∩ U defines a sequence of tangent
spaces Tzn (f −1 f (yn )), which are equal to Tyn ∩ {T = tn }, where yn = (zn , tn ) and
tn equals a A-th root of −f (yn ).
By Proposition 1.1 one can choose ε small enough such that 0 < ε < ε1 and
Sε (x) is transverse to all the strata T1 , . . . , Tk .
COMPLEX SINGULARITIES
15
From our reasoning above, for any i, 1 ≤ i ≤ k, and for any z ∈ Ti ∩ Sε (x) ⊂
Sε (x) ∩ {f = 0} and, for any sequence (zn ) of Sε (x) tending to z, such that the
tangent space Tzn (f −1 (f (zn ))) tends to T , we have that T contains Tz (Ti ) and is
transverse to Tz (Sε (x)) in Cn+1 .
Therefore, we can choose ηε > 0 such that, for any z ∈ Sε (x), 0 < |f (z)| ≤ ηε ,
the tangent space Tz (f −1 (f (z))) is transverse to Sε (x) in Cn+1 . Otherwise we can
choose a sequence of points zn such that |f (zn )| = 1/n, has a limit:
z ∈ Sε (x) ∩ {f = 0}
and the tangent space Tzn (f −1 (f (zn ))) is contained in Tzn (Sε (x)) which is a contradiction.
Summarizing, the intersection Sε (x) ∩ f −1 (f (x)) is a manifold, boundary of Bε ∩
−1
f (f (x)), the restriction of ϕε,η to the boundary Sε (x) ∩ f −1 (f (x)) is submersive.
The restriction to the manifold B̊ε ∩ f −1 (f (x)) is subversive, because the fibers of f
are non-singular. Finally the manifold with boundary Bε ∩ f −1 (f (x)) is compact,
so the map ϕε,η is proper.
Therefore, we can apply Ehresmann Lemma and we obtain that ϕε,η is a locally
trivial smooth fibration.
6. Thom Condition
When one considers a stratified map f , there is an important condition on the
stratification of the source called Thom condition or af condition (see [23] or [13]
p. 65).
Let ϕ : V → W be an algebraic map. Let S = (S1 , . . . , Sk ) be a stratification of
V.
We say that the stratification S of V satisfies the Thom condition or the aϕ
condition if, for any pair (Si , Sj ) of strata, such that Si ⊂ S j , for any point p ∈ Si
and any sequence qn of points of Sj converging to p for which the limit of the
tangent spaces at qn to the fibers ϕ−1 (ϕ(qn )) ∩ Sj exists and is equal to T , we have
T ⊃ Tp (ϕ−1 (ϕ(p)) ∩ Si ).
Example. Let e : E → C2 be the blowing-up of the point 0 in the complex plane
C2 . Let us stratify the map e by E \ D and D in the source, where D is the
exceptional divisor of the blowing-up e and {0} and C2 \ {0} in the target. This
stratified map does not satisfy the Thom condition, since the fibers of e outside 0
are points.
Note that, pulling back the map e by e itself, the new map that we obtain can
be stratified to satisfy the Thom condition (see [20]).
Notice that in the preceding section we obtain a stratification of f which satisfies
Thom condition for f in a neighborhood of x.
Hironaka Theorem
A theorem of H. Hironaka will allow us to generalize Milnor fibration Theorem
(see [9] Corollary 1 §5 p. 248):
Theorem 6.1. Let f : E → C be an algebraic map from a complex algebraic
set E into a non-singular complex curve. One can stratify the map f by Whitney
stratifications such that the stratification of E satisfies Thom condition for f .
16
LÊ DŨNG TRÁNG
An example
In general when the target of the map has not complex dimension one, there are
no reason to obtain stratifications with Thom conditions.
For instance, consider the polynomial map F : C3 → C2 given by:
F (X, Y, Z) = (Y 2 − X 2 Z, X).
The critical locus is Y = X 2 = 0. Its image by F is (0, 0).
One can check that at the origin one cannot find a Whitney stratification which
satisfies Thom condition.
In fact, in [20] it is proved that after a base change the original map gives a new
map which can be stratified with Thom condition.
This result has been little used in the litterature.
Second isotopy theorem of Thom-Mather
Now we can formulate the second isotopy Theorem of Thom-Mather (see [13]
Proposition 11,2).
Consider the following diagram:
g
/F
E@
@@ f ◦g
~
@@
~~
@@
~~f
~
~
T
where g is a stratified map, where E is stratified by S = (S1 , . . . , Sk ) and F is
stratified by S 0 = (S10 , . . . , S`0 ), and T is non-singular. We say that g is a Thom
map over T if:
(1) The maps g and f are proper;
(2) For each stratum Si0 of S 0 the restriction f |Si0 is submersive;
(3) For any stratum Sj of S there is a stratum Si0 of S 0 such that g(Sj ) ⊂ Si0
and the map induced by g from Sj into Si0 is submersive;
(4) The stratification S satisfies the Thom condition relatively to g.
Then:
Theorem 6.2 (Second isotopy theorem of Thom-Mather). If g is a Thom map
over T , the map g is topologically locally trivial over T .
The second isotopy Theorem gives a criterion to have that family of maps are
topologically trivial.
The general fibration theorem
Now, consider g the restriction of a polynomial function to a complex algebraic
subset X of CN :
g : X → C.
We have a theorem similar to the Theorem 3.1, but the locally trivial fibration is
topological.
Theorem 6.3. For any point x ∈ g −1 (0), there is ε > 0 small enough and ηε > 0,
such that, for any η, ηε ≥ η > 0 the map:
ϕε,η : Bε (x) ∩ X ∩ g −1 (Sη ) → Sη
COMPLEX SINGULARITIES
17
induced by g, where Bε (x) is the ball of CN of radius ε centered at the point x
and Sη is the circle of C centered at the origin 0 of radius η, is a locally trivial
topological fibration.
Proof. Hironaka Theorem says that there is Whitney stratifications of X and
C such that g is stratified and the Whitney stratification S = (S1 , . . . , Sk ) of X
satisfies the Thom condition.
Let x ∈ g −1 (0). According to Proposition 1.1 there is ε0 such that the spheres
Sε (x), centered at the point x with radius ε, 0 < ε < ε0 , intersect transversally the
strata Si , 1 ≤ i ≤ k.
Let us fix such a ε > 0. The stratification of E induces a stratification of B̊ε (x)∩X
and a stratification of Sε (x) ∩ X. Both these stratifications give a stratification of
Bε (x) ∩ X which satisfies the condition of Whitney.
Since the stratification S satisfies the Thom condition, a reasoning as above in
the proof of the Fibration Theorem in the section 5 shows that there is η such
that, for all i, 1 ≤ i ≤ k the restriction of g to Si ∩ Sε (x) has maximal rank in
Si ∩ Sε (x) ∩ g −1 (Dη − {0}).
Now, let us fix ε > 0 and η > 0 as before. Let X ∩ Bε (x) ∩ g −1 (Dη − {0}) be
endowed with the intersections of the strata of S with X ∩ S̊ε (x) ∩ g −1 (Dη − {0})
and X ∩Sε (x)∩g −1 (Dη −{0}). With this choice, the stratification satisfies Whitney
condition, the map:
Φε.η : X ∩ Bε (x) ∩ g −1 (Dη − {0}) → Dη − {0}
is proper and the restriction of g to the strata of X ∩ Bε (x) ∩ g −1 (Dη − {0}) are
submersive.
The first isotopy Theorem of Thom - Mather implies that the map Φε.η is a
locally trivial topological fibration. This proves the Theorem.
7. General local fiber of a complete intersection
In this section we are going to give a proof of the result of [11].
Let X be an algebraic subset of CN which is a complete intersection, i.e. the
number of generators of the ideal I(X) of polynomials of CN which vanish on X
is equal to the codimension of X in CN . For instance a complex hypersurface is a
complete intersection.
Then, we have:
Theorem 7.1. Let x ∈ X. For almost all linear form ` of CN the Milnor fiber
at the point x of the restriction of ` to X is a bouquet of spheres of dimension
dim X − 1.
Proof. We shall first assume that X is a complex hypersurface define by the polynomial function f : Cn+1 → C. However f might have non-isolated singularities.
We have seen that we can stratify the map f with Whitney stratifications satisfying Thom condition (see above Theorem 6.1 by Hironaka).
Let ` be a general linear form of Cn+1 . It defines a map:
Φ = (`, f ) : Cn+1 → C2 .
Consider the germ of Φ at the point x ∈ X. The critical locus of the germ of Φ at x
depends linearly on `. A classical theorem of Bertini says that the singular points
18
LÊ DŨNG TRÁNG
of a linear system lie in the fixed points of the linear system or in the singular set
of the ambient variety. In this case, since the smbient space is non-singular the
singular points lie in the fixed points of the linear system which are precisely the
critical space of f at x.
Therefore in a sufficiently small neighborhood U of x in Cn+1 the critical space
of Φ is non-singular outside the hypersurface X. The closure in U of this critical
space is empty or a curve Γ that we call the relative polar curve of f relatively to
the linear form `..
We can prove as in Theorem 6.3 that there are 1 ε η > 0 such that Φ
induces:
Φε,η (x) : Bε (x) ∩ Φ−1 (B̊η (Φ(p))) → B̊η (Φ(x))
which is a stratified map with Thom condition. Let ∆ be the union of the image of
the complex curve Γ ∩ Bε (x) ∩ Φ−1 (B̊η (Φ(x)) by Φε,η (x) and the trace in B̊η (Φ(x))
of the line C × {f (x)}:
∆ := Φε,η (x)((Γ ∩ Bε (x) ∩ Φ−1 (B̊η (Φ(x)))) ∪ (B̊η (Φ(x)) ∩ C × {f (x)})).
As we have done in §6, we can prove that the map Φε,η (x) induces a locally trivial
smooth fibration on B̊η (Φ(x)) − ∆.
Notice that the germ of the critical locus of f at x which might be of dimension
≥ 1 has its image by the germ of Φ at x contained in C × {f (x)}.
Since the linear form ` is general, if not empty, the space Γ − {x} is non-singular,
which means that, for x0 ∈ Γ − {x} the fiber Φε,η (x)−1 (Φ(x0 )) has an ordinary
quadratic point at x’.
This shows that the fibers of Φε,η (x) over B̊η (Φ(x))−C×{f (x)} are transverse to
Sε (x) in Cn+1 , because the fibers over ∆ − (C × {f (x)}) have an isolated singularity
at the points of Γ − {x}.
For t sufficiently small, the fiber of Φε,η (x) above the point (`(x) + t, f (x)) has
dimension dim X − 1. Let us call (u, v) the coordinates of C2 such that;
u = ` and v = f.
Consider the line u = `(x) + t. One can prove that the space of complex dimension
n:
Φε,η (x)−1 (({u = `(x) + t}) ∩ (B̊η (Φ(x))))
is contractible and the space:
Φε,η (x)−1 (D)
where D is a small closed disc of radius r of the line {u = `(x) + t} centered at
(`(x) + t, f (x)) retracts by deformation on the fiber Φε,η (x)−1 (`(x) + t, f (x))).
The restriction of |f | to the space Φε,η (x)−1 (({u = `(x) + t}) ∩ (B̊η (Φ(x))))
defines a real function, Let us start with the value r. As the values of |f | meet the
values of the intersection points of the line ({u = `(x) + t}) with ∆, the restriction
of |f | to Φε,η (x)−1 ({u = `(x) + t}) acquires Morse points with index equal to n (see
[10] p. 30).
Lemma 3.8 can be formulated in a more general way:
Lemma 7.2. A topological space which becomes a space homotopically equivalent
to a bouquet of real spheres of dimension n after attaching cells of dimension n is
homotopically equivalent to a bouquet of real spheres of dimension n − 1.
COMPLEX SINGULARITIES
19
This implies Theorem 7.1 in the case of hypersurfaces.
Namely the space Φε,η (x)−1 (D) is homotopically equivalent to a bouquet of real
spheres of dimension n − 1. Since this space retracts by deformation to the fiber
Φε,η (x)−1 ((`(x) + t, f (x))),
this fiber is also homotopically equivalent to a bouquet of real spheres of dimension
n − 1.
In the case of complete intersections, suppose that X is defined by:
f1 = . . . = fs = 0.
We may replace the equations f1 , . . . , fs by general linear combinations of f1 , . . . , fs .
Then, X is defined by fs on the complete intersection:
X1 = {f1 = . . . = fs−1 = 0}.
By replacing the original equations by general linear combinations, the singular
points of X1 lie in the fixed points, i.e. in X = {f1 = . . . = fs = 0}.
Therefore X1 − X is non-singular and we can repeat a similar argument as the
one we have developed above for hypersurfaces. The Lemma 7.2 will show that by
attaching cells of dimension dim X1 − 1 the Milnor fiber of a general linear form
restricted to X at x gives the Milnor fiber of a general linear form restricted to X1
at x.
Since, by induction on the number of equations, we can suppose that the Milnor
fiber of a general linear form restricted to X1 at x has the homotopy type of a
bouquet of real spheres of dimension dim X1 − 1. By making an argument using
Morse theory as done in the case of hypersurfaces and by applying Lemma 7.2 the
Milnor fiber of a general linear form restricted to X at x has the homotopy type of
a bouquet of spheres of dimension dim X − 1 = dim X1 − 2.
Details are left to the reader.
8. Miscellaneous
Most of the results in these notes work for germ of complex analytic functions
of a germ of complex analytic spaces.
The reader might have to change some Lemmas or Propositions like Proposition
1.1.
We leave to the reader to make the necessary changes.
Note that stratifications are no more finite partitions, but locally finite partitions.
However in the proofs we deal with a finite number of strata since we are considering
germs near a given point.
20
LÊ DŨNG TRÁNG
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