Milnor open books and Milnor fillable contact 3-manifolds

Abstract

We say that a contact manifold (M, ξ) 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 3-dimensional oriented manifold admits at most one Milnor fillable contact structure up to contactomorphism. The proof is based on Milnor open books: we associate with any holomorphic function f : (X , x) → (C, 0), with isolated singularity at x (and any euclidian rug function ρ), an open book decomposition of M , and we verify that all these open books carry the contact structure ξ of (M, ξ) -generalizing results of Milnor and Giroux.

Smoothings of singularities and symplectic surgery

“…Let M denote the blow-up of X in a fiber. The fiber passing throgh the blownup point, together with a section now provides the configuration of two curves (A, B) with intersection patterns as in the resolution graph of the singularity given by Equation (3).…”

Symplectic surgeries and normal surface singularities

“…Notice that the link L of the singularity .V; 0/ admits a contact structure by considering the complex tangents along L. According to [2] this contact structure is unique up to contactomorphism. It is called the Milnor fillable contact structure on L. By a famous result of Bogomolov [1] the complex structure on a resolution z V can be deformed to a (possible blow-up of a) Stein filling, hence Milnor fillable contact structures are necessarily Stein fillable.…”

“…According to Caubel, Némethi and Popescu-Pampu [2], the link Y D @Z 1 of the singularity .S ; 0/ given by the (negative definite) plumbing graph admits a unique (up to contactomorphism) Milnor fillable contact structure M , for which Z 1 (with its Stein structure originating from the deformation) provides a Stein filling. In fact, our proof will not use the fact that Z 1 is a smoothing of .S ; 0/.…”

Surgery diagrams for horizontal contact structures

“…On the other hand, a recent discovery (cf. [1]) is that any 3-manifold admits at most one Milnor fillable contact structure, up to isomorphism. Note that any Milnor fillable contact structure is horizontal (cf.…”