In this proposal we plan to study several problems, most of them concerning pseudoconvex domains in analytic spaces. We plan to use Gromov-Hausdorff limits to study the Cauchy-Riemann equation on singular Stein spaces. We will try to find counter examples to the hyper-intersection problem in dimensions greater than three. A counterexample of dimension three was given by M. Coltoiu and K. Diederich.
We want to prove that in a complex Kahler manifold with positive bisectional curvature there is no relatively compact pseudoconvex domain with smooth real analytic boundary such that the set of points of the boundary at which the domain is not strongly pseudoconvex contains a sub-manifold of positive dimension. This is a special case of a conjecture of Diede rich and Ohsawa. We would like to prove that every Stein domain with smooth boundary of order one in the projective space is hyperconvex.
The same result holds for domains in the affine space and for domains in the projective space if the boundary is smooth of order two. This problem is motivated by the efforts to decrease the smoothness required in the non-existence of Levi-flat domains. We want to decide if the Russel cubic is biholomorphic to the complex affine space of dimension three. It is known that they are diffeomorphic and that they are not algebraically isomorphic. We want to give a counterexample to a problem of Bremermann that states that if a Stein domain in the affine complex space has Runge intersection with every line then it is Runge. Finally, we plan to study the four ball problem asking whether the union of four disjoint closed balls is polynomially convex.
Call for proposal
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