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, 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
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