A complex surface admitting a strongly plurisubharmonic function but no holomorphic functions

TL;DR

This paper constructs a domain in the complex projective plane that admits a strongly plurisubharmonic function yet admits no non-constant holomorphic functions, answering a longstanding question in complex analysis.

Contribution

It provides an explicit example of a complex surface with a strongly plurisubharmonic function but no non-constant holomorphic functions, solving an open problem.

Findings

Existence of such a domain in CP^2

Domain is diffeomorphic to a real 2-plane bundle over S^2

No non-constant holomorphic functions on the domain

Abstract

Answering an old question, we find a domain X in the complex projective plane CP^2 which admits a strongly plurisubharmonic function, but such that every holomorphic function on X is constant. The domain X can be chosen diffeomorphic to an oriented real 2-plane bundle over the 2-sphere.