On the strongly pseudoconcave boundary of a compact complex surface

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.