Stein fillability and the realization of contact manifolds

TL;DR

This paper proves the equivalence of intrinsic and extrinsic notions of Stein fillability for contact manifolds, and explores their embeddings and cohomological properties.

Contribution

It establishes the equivalence of intrinsic and extrinsic Stein fillability notions and analyzes the uniqueness of the germ of the Dolbeault cohomology.

Findings

Intrinsic and extrinsic Stein fillability are equivalent.

The germ of the Dolbeault cohomology of any border is unique.

Stein fillable 3-manifolds can be embedded in C^4 or immersed in C^3.

Abstract

There is an intrinsic notion of what it means for a contact manifold to be the smooth boundary of a Stein manifold. The same concept has another more extrinsic formulation, which is often used as a convenient working hypothesis. We give a simple proof that the two are equivalent. Moreover it is shown that, even though a border always exists, it's germ is not unique; nevertheless the germ of the Dolbeault cohomology of any border is unique. We also point out that any Stein fillable compact contact 3- manifold has a geometric realization in C^4 via an embedding, or in C^3 via an immersion.