TOPOLOGY AND BILIPSCHITZ GEOMETRY OF COMPLEX
SINGULARITIES
ANNE PICHON
These are notes of the four lectures given in december 2013 at the CIMPA School
in Hanoi. They are an introduction to topology of complex singularities whose
objective is to make possible the next step : understanding bilipschitz geometry of
complex curves and surfaces, even for students beginning in singularity theory.
1. The conical structure theorem
Let U be an open subset of Cn which contains the origin. Let f1 , . . . , fr : U → C
be r holomorphic functions. Consider
X := {z ∈ U | fi (z) = 0, i = 1, . . . , r}.
Goal : understand the topology and geometry of X in small neighborhoods of X
around 0. So we are interesting in the germ (X, 0) defined by germs of functions
fi : (Cn , 0) → (C, 0) with fi ∈ C{z1 , . . . , zn }.
Example 1.1.
(1) n = 2 and r = 1. (plane curves)
a) X = {(x, y) ∈ C2 | y = 0}
b) X = {xy = 0}
c) X = {x3 − y 2 = 0}
(2) n = 3 and r = 1. (complex surfaces in C3 )
a) X = {(x, y, z) ∈ C3 | x2 + y 3 + z 5 = 0}
b) X = {(x, y, z) ∈ C3 | x2 + z 2 y 2 + y 3 = 0}
(3) n = 3 and r = 2
a) X = {x2 + y 3 + z 5 = 0, x = 0} ⊂ C3 (a complex curve in C3 ).
In the sequel, we restrict ourselves to the case r = 1 (however most of the results
can be stated in general case). So we consider a single germ of holomorphic function
f : (Cn , 0) → (C, 0) and X = f −1 (0).
Definition 1.2. The singular locus of X (or f ) is the set
∂f
Sing(X) = {z ∈ Cn |
(z) = 0, i = 1, . . . , n}.
∂zi
Remark 1.3. As the critical values of f are isolated, then locally, in a neighborhood
U of 0, one has either Sing(X) = ∅ or {0} ⊂ Sing(X) ⊂ X. In the former case,
(X, 0) is locally holomorphic to an open set of Cn−1 , so the local geometry of X is
fully understood.
Definition 1.4. X (or f ) has an isolated singularity at 0 if there exists a neighborhood U of 0 such that Sing(X) ∩ U = {0}
Example 1.5. Example (1.a) below has Sing(X) = ∅. The other examples have
isolated singularity unless (2.b) which has Sing(X) = {x = y = 0}.
In the sequel, we restrict ourselves to the case f has an isolated singularity at 0.
1
2
ANNE PICHON
1.1. The conical structure theorem. References: [3], [5] (and lecture 1 of D.T.
Lê).
Let > 0. Denote
n
X
B = {(z1 , . . . , zn ) ∈ Cn |
|zi |2 ≤ 2 }
i=1
and
S = ∂B = {(z1 , . . . , zn ) ∈ Cn |
n
X
|zi |2 = 2 }
i=1
Definition 1.6. The link of (X, 0) (at radius ) is the intersection:
X () = X ∩ S
√
Example 1.7. X = {x3 − y 2 =
0}. Exercise : Show that the link X ( 2) is the
√
trefoil. [Hint: remark that X ( 2) ⊂ {|x| = 1} × {|x| = 1} ⊂ S√2 and that a
√
parametrization of X (
2)
3
is given by: θ ∈ [0, 4π] 7→ (eiθ , ei 2 θ )].
Theorem 1.8 (Conical structure theorem, Milnor 1968). Let f : (Cn , 0) → (C, 0)
with isolated singularity and X = f −1 (0). There exists 0 > 0 such that for each
, 0 < ≤ 0 ,
(1) The intersection X ∩ S is transverse,
(2) The homeomorphism class of the pair (S , X () ) does not depend on ,
(3) There is a homeomorphism of pairs:
(B , X ∩ B ) ∼
= Cone(S , X () ).
2. The link of a plane curve germ
2.1. The Puiseux expansion theorem. References: [1], [2] or lecture 1 of D.T.
Lê.
Let C{x, y} be the ring of convergent power series in two variables.
Theorem 2.1. C{x, y} is a unique factorization domain. So each f ∈ C{x, y} is
expressed in a unique way (up to multiplications by non zero constants) as a product
of irreducible elements. When f ∈ C{x, y} is irreducible, the link of X = f −1 (0) is
connected.
Theorem 2.2 (Puiseux expansion theorem). Let f ∈ C{x, y}, f irreducible. Assume f (0, 0) = 0 and f (0, y) is not identically zero. There then exists an integer
n ≥ 1 and y(t) ∈ tC{t} such that if ξ is an nth primitive root of unity, then
f (x, y) =
n
Y
(y − y(ξ i t)),
i=1
n
where x = t .
Therefore, y(t) is a root of f in the Galois extension C{x, y} ⊂ C{x, t} with
x = tn , and the Galois group consists of the nth roots of unity.
We can then write a parametrization of X = f −1 (0) either under the form
X
x(t) = tn , y(t) =
ai ti ∈ C{t},
i≥1
TOPOLOGY AND BILIPSCHITZ GEOMETRY OF NORMAL SURFACE SINGULARITIES
3
where ai ∈ C, or alternatively under the form
X
y=
ai xi/n ∈ C{x1/n }.
i≥1
Definition 2.3. These are called Puiseux expansions of f (or of X).
The proof of Theorem 2.1 is based on Newton-Puiseux algorithm, which enables
one to built explicitly a Puiseux parametrization for each branch of any f ∈ C{x, y}.
We again refer to [1] for this algorithm.
Exercise. Find an equation for the plane curve germ with Puiseux expansion
7
3
y = x 2 + x 4 [Solution: f (x, y) = (x3 − y 2 )3 + 4x5 y + x7 .]
2.2. Construction of the link of a plane curve.
Example 2.4. Take the plane curve germ X with Puiseux expansion:
3
7
y = x2 + x4 .
The homeomorphism class of (B , X ∩ B ) remains unchanged if one considers a
“ball with corners” B0 = {|x| ≤ } × {|y| ≤ } instead of the round ball B 1. We
will construct the link X () as the intersection X ∩ ∂B0 .
When is sufficiently small, the link X ∩ B0 is included in the interior of the
solid torus {|x| = , |y| ≤ }. We proceed by successive approximations :
3
Consider the curve X1 parametrized by y = x 2 , i.e., obtained by truncation of
the Puiseux expansion of X after the first term. It is parametrized by
3
3
θ ∈ [0, 4π] 7→ (eθ , 2 e 2 θ )
Its link is then a (3, 2)-torus knot (same as in Example 1.7).
Consider now the curve X2 = X. It is parametrized by:
3
3
7
7
θ ∈ [0, 8π] 7→ (eθ , 2 e 2 θ + 4 e 4 θ )
7
3
As >> 1 and 23 < 74 , then 4 << 2 and X (2) is inside a small tubular
neighborhood of X (1) . It is a torus knot cabled around the (3, 2)-torus knot X1 .
P
General case f irreducible. y = i≤1 ai xqi ∈ C{x1/n }, with n minimal. X () is
constructed by successive approximations by the curves Xk whose Puiseux expansions are obtained by truncating the Puiseux expansion of X after exponent qk . (In
Example 2.4, there were only one approximating X1 i.e., X2 = X.) The process is
finite in virtue of the following result:
Proposition 2.5. Let s be the least exponent such that denom(qs ) = n. Then
Xs+1 is isotopic to Xs .
()
Proof. As denom(qs+1 ) ≤ n and Xs ∩ {x = } consists of n points, then Xs+1 is
an (1, a)-torus knot cabled on Xs , (for some a) so it is isotopic to Xs . (Intuitively,
Xs+1 is a little “spring” around Xs .)
1see e.g., A. Durfee, Neighborhoods of algebraic sets, Trans. Amer. Math. Soc. 276 (1983),
517-530.
4
ANNE PICHON
As a consequence, one obtains that X is isotopic to the link Xs .
Exercise. Show that there exists a unique minimal sequence i1 < . . . < ir (i.e.,
with r minimal) such that X () is isotopic to the link of
y = xqi1 + xqi2 + . . . + xqir ,
and describe explicitly the exponents qi1 . [hint: ir = s and qi1 is the least exponent
which is not an integer.]
Definition 2.6. The exponents qi1 , . . . , qir are called the essential exponents (or
characteristic exponents) of X.
General case f non irreducible. Let f1 , . . . , fr be the irreducible components
of f . One takes a Puiseux expansion for each of them and argue as before, taking
into account common leading parts between the different expansions (i.e., the so
called coincidence exponents between branches). The description of the link is then
more complicated but similar to that of the previous example. We refer to [1] (see
also [2]).
3. Graph manifolds in singularity theory
Let M be a C ∞ 3-dimensional compacte connected oriented manifold (∂M is
not necessarily empty).
Definition 3.1. M is a Seifert manifold if there exists a foliation of M by differentiable circles such that each leave has a compact tubular neighborhood (so a solid
torus) which is a union of leaves. The leaves are called Seifert fibers.
Example 3.2. Let p, q ∈ N with (p, q) = 1. Consider the S1 -action on the solid
torus S1 × D2 defined for t ∈ S1 and (x, y) ∈ S1 × D2 by:
t.(x, y) = (tp x, tq y).
The orbits define a Seifert fibration on D2 × S1 , and the orbits space is an orbifold
homeomorphic to D2 .
Example 3.3. Similarly, the thickened torus S1 ×S1 ×[0, 1] admits Seifert fibrations
obtained from that of D2 × S1 by removing from D2 × S1 the interior of a concentric
solid torus saturated with above Seifert fibres.
Example 3.4. Let X = {(x2 + y 3 + z 5 = 0)} ⊂ C3 . Consider the C∗ -action on X
defined by:
t.(x, y, z) = (t15 x, t10 y, t6 z).
It restricts to an S1 -action which preserves the links X () and whose orbits define
a Seifert fibration on each X () .
One can show that the boundary of a Seifert manifold consists of a disjoint union
of tori. A nice and short survey on Seifert and graph manifolds and more general
classification of 3-dimensional manifolds can be find as a section in [6]. It gives
precise references for more detailed literature.
Definition 3.5. M is a graph manifold (or Waldhausen manifold) if there exists
Seifert manifolds M1 , . . . , Mn embedded in M such that
Sn
(1) M = i=1 Mi ,
TOPOLOGY AND BILIPSCHITZ GEOMETRY OF NORMAL SURFACE SINGULARITIES
5
(2) for all i, j with i 6= j, Mi ∩ Mj is either empty or a disjoint union of solid
tori (which are the common boundary components of Mi and Mj ).
Sn
Such`
a union M = i=1 Mi is called a graph decomposition of M and the family
T = i6=j Mi ∩ Mj is a separating family of tori.
Theorem 3.6. Let X be the germ of a plane curve and let N be a tubular neighborhood of its link X () in S3 . The manifold S3 r N is a graph manifold.
Theorem 3.7. The link of a normal surface singularity (for example an hyper
surface in (C3 , 0) with isolated singularity) is a graph manifold.
Remark 3.8. The classical proof of this two results uses resolution theory and
plumbing technics that we won’t develop here. We refer to [6] for Theorem 3.6
and to [4] for Theorem 3.7. We will give alternative proofs based on carrousel
decomposition of the complement of a plane curve. This construction which will
be useful to study bilipschitz geometry of complex surfaces.
4. Carrousel decomposition adapted to a plane curve germ
We describe such a decomposition on an example. General case is similar and
can be found in [7].
Consider again the germ of curve (X, 0) ⊂ (C2 , 0) with Puiseux expansion y =
7
3
a1 x 2 + a2 x 4 . Its tangent cone is the x-axis. Fix a small real η > 0 and consider
the cone V = {(x, y) | |y| ≤ η|x|}. When is sufficiently small, then X ∩ B ⊂ V .
We define a decomposition of B into semi-algebraic sets as follows :
• Set
B(1) := B r V .
The manifold B(1) is foliated (outside 0) by the complex lines y = λx where
|λ| ≥ η.
• Let β1 such that 0 < |a1 | < β1 .
3
The semi-algebraic set B 0 ( 32 ) := {|y| ≤ β1 |x| 2 } has the following properties :
3
– it contains X1 : y = a1 x 2 ,
3
– It is foliated by the complex curves y = λx 2 where |λ| ≤ β1
3
– its intersection with the slice {x = x0 } is a disc with diameter |x0 | 2 .
We set
3
3
A(1, ) := V r B 0 ( ).
2
2
• Let γ1 such that 0 < |a1 | − γ1 and |a1 | + γ1 < β1 .
3
3
We set B 00 ( 23 ) := {|y − a1 x 2 | ≥ γ1 |x| 2 } and
3
3
3
B( ) = B 0 ( ) r B 00 ( )
2
2
2
3
B( 23 ) is still foliated by complex curves y = λx 2 .
• Let β2 , such that |a2 | < β2 . We set
7
3
7
B 0 ( ) := {|y − a1 x 2 | ≤ β2 |x| 4 }
4
and
3 7
3
7
A( , ) = B 00 ( ) r B( ).
2 4
2
4
6
ANNE PICHON
B 0 ( 74 ) has the following properties :
– it contains the curve X,
3
3
– It is foliated by complex curves y = a1 x 2 + λx 2 ,
– its intersection with the slice {x = x0 } consists of two discs with
diameter |x0 |7/4 .
• At last, we choose β < a2 and we set:
3
7
7
7
7
7
D( ) := {|y − (a1 x 2 + a2 x 4 )| < β} and B( ) = B 0 ( ) r D( )
4
4
4
4
Exercise. Draw the intersections of a slice V ∩ {x = x0 } where x0 is fixed, with
the sets B(q), A(p, q) and D(q).
Show that The intersections A(1, 32 ) ∩ B and A( 32 , 74 ) ∩ B are thickened tori and
that the intersection D( 74 ) ∩ S is a tubular neighborhood of the link X () in S .
Definition 4.1. We call the union
3
3 7
7
3
B = B(1) ∪ A(1, ) ∪ B( ) ∪ A( , ) ∪ B( )
2
2
2 4
4
a carrousel decomposition of the germ (C2 , 0) adapted to the curve X.
General case A carrousel decomposition of the germ (C2 , 0) adapted to a plane
curve X is construct similarly by first choosing truncations of the Puiseux expansions of the different branches. For details see [7].
proof of Theorem 3.6. The carrousel decomposition intersected with S defines a
graph decomposition of S . Indeed, each B(q) ∩ S is foliated but the links of the
complex curves foliating B(q) outside 0, and each A(p, q) ∩ S is a thickened torus
so it admits a Seifert fibration (see Example 3.3). By construction, the curve X is
contained in the interior of the innermost D(q) sets, and the intersections of these
sets with S form a tubular neighborhood N of X () in S .
proof of Theorem 3.7. For convenience of notations, let us write the proof for an
hypersurface X = f −1 (0) in C3 (the proof is the same in the general case). Choose
coordinates in such a way that the restriction ` |X : (X, 0) → (C2 , 0) of the linear
projection ` = (x, y) : C3 → C2 has finite fibers. Then ` |X is a finite cover which
is branched on the complex curve
∂f
Π = {x ∈ X| Ker` ⊂ Tx X} = {
= 0, f = 0}.
∂x
Definition 4.2. Π is called the polar curve of ` |X and the curve ∆ = `(Π) is the
discriminant curve.
Take a carrousel decomposition of (C2 , 0) adapted to (∆, 0):
n
[
B4 =
Wi ,
i=1
B4
where
is a Milnor ball for ∆.
Adapting the coordinates, we can also assume that the ball with corners B 6 =
B4 × {|z| ≤ 0 } is a Milnor ball for the surface X and that
X ∩ B 6 ⊂ S3 × {|z| < 0 }.
Then ` |X restricts to a cover
` |X () : X () → S3
TOPOLOGY AND BILIPSCHITZ GEOMETRY OF NORMAL SURFACE SINGULARITIES
7
branched over the link ∆() . Using the fact that the curve ∆() is contained in the
interior of the innermost B(q) sets, and the intersections of these sets with S is a
()
tubular
the graph decomposition
S neighborhood N of X , it is easy to conclude that
S = i (Wi ∩ S ) lifts to a graph decomposition of X () .
References in progressive order with respect to the lectures:
References
[1] E. Brieskorn, H. Knörrer: Plane Algebraic Curves. Birkhäuser, Basel, Boston, Stuttgart
(1986).
[2] D. Eisenbud, W. D Neuman, Three dimensional link theory and invariants of plane curves
singularities, Annals of Mathematics Studies 110, (Princeton University Press, 1985).
[3] John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, 61
(Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo (1968).
[4] Walter D. Neumann, A calculus for plumbing applied to the topology of complex surface
singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), 299–
343.
[5] J. seade, on the topology of isolated singularities in analytic spaces, Birkhäuser, Basel, Boston,
Berlin (2006).
[6] Lê Dũng Tráng, Françoise Michel, Claude Weber, Courbes polaires et topologie des courbes
planes. Ann. Sci. École Norm. Sup. 24 (1991), 141–169.
[7] Lev Birbrair, Walter D Neumann and Anne Pichon, The thick-thin decomposition and
the bilipschitz classification of normal surface singularities, to appear in Acta Math..
arXiv:1105.3327v2.