Franc Forstneric

Abstract:   In this talk we shall discuss the basic dichotomy "flexibility versus rigidity" for complex manifolds. By "holomorphic flexibility" we mean the abundance of holomorphic maps from complex Euclidean spaces and, more generally, from Stein manifolds (closed complex submanifolds of Euclidean spaces) to the given manifold, while "rigidity" means limitations on such maps; for example, Kobayashi hyperbolicity of Y excludes nonconstant holomorphic maps from C to Y.

Our intention is to explain the hierarchy of several flexibility properties of a complex manifold Y such as the Oka property (which means the possibility of deforming any continuous map from a Stein manifold to Y into a holomorphic map), a Runge approximation property for maps from complex Euclidean spaces, the jet transversality theorem for holomorphic maps of Stein manifolds to Y, the existence of dominating holomorphic maps from Euclidean spaces, and the existence of holomorphic submersions from Stein manifolds to Y. We shall illustrate these concepts with many concrete examples with the emphasis on complex surfaces.