Milnor open books and Milnor fillable contact 3-manifolds

We say that an oriented contact manifold (M, xi) is Milnor fillable if it is contactomorphic to the contact boundary of an isolated complex-analytic singularity (X, x). In this article we prove that any three-dimensional oriented manifold admits at most one Milnor fillable contact structure up to contactomorphism. The proof is based on Milnor open books: we associate an open book decomposition of M with any holomorphic function f : (K, x) -). (C, 0), with isolated singularity at x and we verify that all these open books carry the contact structure of (M, xi)-generalizing results of Milnor and Giroux. (C) 2006 Elsevier Ltd. All rights reserved.