1. Plurisubharmonic defining functions- Part I

We give an overview of known results concerning the existence of plurisubhar

monic defining functions for pseudoconvex domains.

2. Plurisubharmonic defining functions- Part II

We prove that the existence of a plurisubharmonic defining function for a 

smoothly bounded pseudoconvex domain D in C^n is a CR-invariant property of 

the boundary bD.

3. The core of a complex manifold- Part I We introduce the core of a complex

manifold, and we discuss questions and results related to analytic structure

and Liouville type properties of the core.

4. The core of a complex manifold- Part II We formulate sufficient conditions

for the existence of the Bergman metric on unbounded pseudoconvex domains, 

and we dicuss core sets with respect to families of plurisubharmonic functions

of different smoothness classes.