PSEUCONVEX DOMAINS · Pseudoconvex Domains in Stein Spaces and Compact Kahler manifols
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2005-04-01 → 2007-03-31
- EU contribution
- €80,000
- Participants
- 1
- Scheme
- IRG
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Results in brief
Final Activity Report Summary - PSEUCONVEX DOMAINS (Pseudoconvex Domains in Stein Spaces and Compact Kahler manifols)
I proved that in the affine n-dimensional, n>1, complex space there exists a bounded connected Stein domain, D, which is not Runge and for every complex line l the intersection of l and D is simply connected (hence Runge in l). In this way I answered a question asked by H. Bremermenn in 1957 (Math. Annalen) and raised again by T. Ohsawa in his 2002 book. Moreover the domain can be chosen to be strictly pseudoconvex and with real analytic smooth boundary. I proved that there exists a sequence of disjoint polydiscs, P_j which is locally finite, their union is Runge but one cannot uniformly approximate holomorphic functions on sequences of compact subsets of P_j with global holomorphic functions. In a joint paper with Terrence Napier and Mohan Ramachandran we consider several notions of q-convexity on reduced complex spaces. The classes of functions that we define satisfy extension and approximation properties and we obtain an unified approach of results of Grauert-Riemenschneider, Greene-Wu, Ohasawa, Coltoiu, Demailly. In a joint paper with Daniela Joita we defined a notion of minor for weighted graphs. We prove that with this minor relation, the set of weighted graphs is directed. We also gave an algorithmic procedure such that - for any two given weights on a connected graph with the same total weight, we can transform one into the other using a sequence of edge subdivisions and edge contractions.
Data: CORDIS, © European Union
Project objective
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.
Original text from CORDIS.
Participants
- INSTITUTE OF MATHEMATICS OF THE ROMANIAN ACADEMY · BUCHARESTCoordinatorCity levelRomania
Links
Data: CORDIS, © European Union