On the strongly pseudoconcave boundary of a compact complex surface

TL;DR

This paper develops a method for attaching holomorphic handles to strongly pseudoconcave boundaries of complex surfaces, enabling new results in contact topology and complex cobordism, including Kähler structures.

Contribution

It introduces a holomorphic handle attaching technique for strongly pseudoconcave boundaries, proving all contact 3-manifolds can be realized as such boundaries and are cobordant, with Kähler options.

Findings

Every closed contact 3-manifold can bound a complex surface.

Any two contact 3-manifolds are complex cobordant.

Such complex surfaces can be Kähler.

Abstract

We establish the method of holomorphic handle attaching to the strongly pseudoconcave boundary of a complex surface. We use this for proving the following statements: (1) every closed connected oriented contact 3-manifold can be filled as the strongly pseudoconcave boundary of a compact complex surface; (2) any two closed connected oriented contact 3-manifolds are complex cobordant. Moreover, we show that such complex surface (or complex cobordism) can be taken K\"ahler.