A complex surface admitting a strongly plurisubharmonic function but no   holomorphic functions
  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. 
Franc Forstneri\v{c}
math.CV