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Pseudoconvex Domains in Stein Spaces and Compact Kahler manifols

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.