Lift-Off Fellowship report: A strong Oka principle for circular domains

摘要

In complex geometry, Stein manifolds are of fundamental importance as those com- plex manifolds with a rich supply of holomorphic (that is, complex differentiable) functions from them to the complex numbers C. When studying holomorphically defined problems on Stein manifolds, an interesting phenomenon can arise in which the existence of a continuous solution is enough to give a holomorphic solution. This is somewhat surprising as holomorphic maps are much more rigid than contin- uous maps, so we would not necessarily expect that the only obstruction to solving a holomorphic problem is topological in nature. In such instances we say that the Oka principle holds, named for Kiyoshi Oka who gave one of the first results of this kind in 1939, showing that the holomorphic and topological classifications of line. bundles over a Stein manifold are the same.