4 SELMA ALTINOK AND MOHAN BHUPAL

contact structure ξX given as the maximal J-invariant subspace of T MX , where

J denotes the complex structure on CN . Suppose that f ∈ mX,x is a germ of a

holomorphic function vanishing at x that deﬁnes an isolated singularity at x. Then

the pair N(f) = f−1(0)∩MX , θ(f) = arg f : MX \N(f) → S1 deﬁnes an open book

decomposition OB(f) of MX . Such open book decompositions of MX are called

Milnor open book decompositions. In [4], Caubel, N´ emethi and Popescu-Pampu

prove the following theorem.

Theorem 2.7. Let (X, x) be a germ of a normal complex analytic surface

embedded in

(CN

, 0). Each Milnor open book decomposition of the link MX of

(X, x) supports the natural contact structure ξX on MX .

Suppose now that (X, x) has a rational singularity at x. In this case the link

MX is a rational homology sphere (see [10], [13]). Fix a resolution π :

˜

X → X of

X and let E =

π−1(x)

denote the exceptional divisor. Suppose that f ∈ mX,x is a

germ of a holomorphic function vanishing at x that deﬁnes an isolated singularity

at x. Let (f ◦ π) = (f ◦ π)e + (f ◦ π)s be a decomposition of the divisor (f ◦ π) ∈

Div

˜

X into its exceptional and strict parts. Here (f ◦ π)e is supported on E and

dim |(f◦π)s|∩E 1. By Proposition 2.2 in [3], it follows that the Milnor open book

decomposition OB(f) depends, up to isotopy, only on the divisor (f ◦ π)e. On the

other hand, by Artin [1], the set of divisors D supported on E given as D = (f ◦π)e,

for germs of holomorphic functions f ∈ mX,x that deﬁne isolated singularities at x,

is precisely the set E

+.

Thus, for any D ∈ E

+,

we may deﬁne OB(D) to be the

open book OB(f), where where f ∈ mX,x is any germ of a holomorphic function

that deﬁnes an isolated singularity at x and satisﬁes D = (f ◦ π)e.

Remark 2.8. It can be shown that if f is taken to be the restriction of a

“generic” linear form on CN to X, then (f ◦ π)e is the fundamental cycle of E (see,

for example, [17], page 95).

3. The proofs of the main theorems

Let (X, x) denote a germ of a normal complex analytic surface having a rational

singularity at x. We suppose, moreover, that X is not smooth at x. Fix a resolution

π :

˜

X → X of (X, x) and let E =

N

i=1

Ei denote the corresponding exceptional

divisor. For each i, let vi denote the number of irreducible curves Ej , j = i, in E

intersecting Ei.

Lemma 3.1. The page-genus of the open book OB(Y ) for any element Y =

∑n

i=1

miEi ∈ E + is given by

genus(OB(Y )) = 1 +

n

i=1

(vi − 2)mi − (mi − 1)(Y · Ei)

2

.

For brevity we will also use the notation g(D) for the page-genus of the open

book OB(D) associated to an element D ∈ E +.

Proof. When computing the page-genus of an open book OB(Y ) there are

two contributions to consider. The ﬁrst arises as follows: Let Γ denote the dual

graph of the exceptional divisor E of the resolution π and let M(Γ)

(∼

=

MX ) denote

the plumbed 3-manifold associated to Γ. Therefore the open book OB(Y ) is a

decomposition of M(Γ). The 3-manifold M(Γ) naturally decomposes into a union

4

contact structure ξX given as the maximal J-invariant subspace of T MX , where

J denotes the complex structure on CN . Suppose that f ∈ mX,x is a germ of a

holomorphic function vanishing at x that deﬁnes an isolated singularity at x. Then

the pair N(f) = f−1(0)∩MX , θ(f) = arg f : MX \N(f) → S1 deﬁnes an open book

decomposition OB(f) of MX . Such open book decompositions of MX are called

Milnor open book decompositions. In [4], Caubel, N´ emethi and Popescu-Pampu

prove the following theorem.

Theorem 2.7. Let (X, x) be a germ of a normal complex analytic surface

embedded in

(CN

, 0). Each Milnor open book decomposition of the link MX of

(X, x) supports the natural contact structure ξX on MX .

Suppose now that (X, x) has a rational singularity at x. In this case the link

MX is a rational homology sphere (see [10], [13]). Fix a resolution π :

˜

X → X of

X and let E =

π−1(x)

denote the exceptional divisor. Suppose that f ∈ mX,x is a

germ of a holomorphic function vanishing at x that deﬁnes an isolated singularity

at x. Let (f ◦ π) = (f ◦ π)e + (f ◦ π)s be a decomposition of the divisor (f ◦ π) ∈

Div

˜

X into its exceptional and strict parts. Here (f ◦ π)e is supported on E and

dim |(f◦π)s|∩E 1. By Proposition 2.2 in [3], it follows that the Milnor open book

decomposition OB(f) depends, up to isotopy, only on the divisor (f ◦ π)e. On the

other hand, by Artin [1], the set of divisors D supported on E given as D = (f ◦π)e,

for germs of holomorphic functions f ∈ mX,x that deﬁne isolated singularities at x,

is precisely the set E

+.

Thus, for any D ∈ E

+,

we may deﬁne OB(D) to be the

open book OB(f), where where f ∈ mX,x is any germ of a holomorphic function

that deﬁnes an isolated singularity at x and satisﬁes D = (f ◦ π)e.

Remark 2.8. It can be shown that if f is taken to be the restriction of a

“generic” linear form on CN to X, then (f ◦ π)e is the fundamental cycle of E (see,

for example, [17], page 95).

3. The proofs of the main theorems

Let (X, x) denote a germ of a normal complex analytic surface having a rational

singularity at x. We suppose, moreover, that X is not smooth at x. Fix a resolution

π :

˜

X → X of (X, x) and let E =

N

i=1

Ei denote the corresponding exceptional

divisor. For each i, let vi denote the number of irreducible curves Ej , j = i, in E

intersecting Ei.

Lemma 3.1. The page-genus of the open book OB(Y ) for any element Y =

∑n

i=1

miEi ∈ E + is given by

genus(OB(Y )) = 1 +

n

i=1

(vi − 2)mi − (mi − 1)(Y · Ei)

2

.

For brevity we will also use the notation g(D) for the page-genus of the open

book OB(D) associated to an element D ∈ E +.

Proof. When computing the page-genus of an open book OB(Y ) there are

two contributions to consider. The ﬁrst arises as follows: Let Γ denote the dual

graph of the exceptional divisor E of the resolution π and let M(Γ)

(∼

=

MX ) denote

the plumbed 3-manifold associated to Γ. Therefore the open book OB(Y ) is a

decomposition of M(Γ). The 3-manifold M(Γ) naturally decomposes into a union

4