On the compactification of hyperconcave ends and the theorems of Siu-Yau   and Nadel
  We show that the pseudoconcave holes of some naturally arising class of manifolds, called hyperconcave ends, can be filled in, including the case of complex dimension 2 . As a consequence we obtain a stronger version of the compactification theorem of Siu-Yau and extend Nadel's theorems to dimension 2. 
George Marinescu and Tien-Cuong Dinh