| Résume | We will introduce the concept of Stein space and of weakly complete space, both in terms of the algebra of holomorphic functions and in terms of the cone of plurisubharmonic functions. In dimension 2, some typical examples of weakly complete spaces which are not Stein are
- modifications of Stein spaces of dimension 2
- spaces admitting a proper holomorphic map onto an open complex curve
- Grauert-type surfaces (foliated in complex curves whose closures are Levi-flat 3-dimensional compact hypersurfaces).
With Slodkowski and Tomassini we proved that, if the space admits an exhaustion function which is plurisubharmonic and real analytic, then these are the only three possibilities.
We will give some details of the proof and we will highlight the difficulties in extending this kind of result both in terms of regularity (smooth instead of real analytic) and in terms of dimension. |
