A Lecture on Filling Pseudoconcave Holes , Hayama , Japan , 1999

Let X− be a compact complex manifold with a strictly pseudoconcave boundary. We consider the following questions: Question 1: When does there exist a compact complex manifold V such that X− embeds into V as an open set? If such a V can be found then we say that X− is fillable. If dimX− ≥ 3 then using a result of Rossi we know that this can always be done, see [7]. However if dimX− = 2 then there are many examples which show that this is not always possible, see [3]. This suggests: Question 2: If dimX− = 2 and X− ↪→ V as above then describe the set of small deformations of the complex structures on X− which can be similarly filled. (a) Is the set of fillable structures closed in the C∞-topology? (b) Is the set of fillable structures path connected? (c) Does this set have the structure of a fiber space over a finite dimensional base? Question 3: Can V be taken to be a projective variety?