Seminar Algebra and Geometry: Frank Kutzschebauch (Universität Bern)

The density property for complex manifolds - a strong form of holomorphic flexibility

Compared to the real differentiable case complex manifolds in general are more rigid, their groups of holomorphic diffeomorphisms are rather small (in general trivial). A long known exception to this behavior is affine n-space Cⁿ for n at least 2, its group of holomorphic diffeomorphisms is infinite dimensional. In the late 1980’s Andersen - Lempert proved are markable theorem which stated in its generalized version due to Forstneric and Rosay that any local holomorphic phase flow given on a Runge subset ofCncan be locally uniformly approximated by a global holomorphic diffeomorphism. The main ingredient in the proof was formalized by Varolin to be called the density property: The Lie algebra generated by complete holomorphic vector fields is dense in the Lie algebra of all holomorphic vector fields. In these manifolds a similar local to global approximation of Andersen-Lempert type holds, It is a precise way of saying that the group of holomorphic diffeomorphisms is large. In the talk we will explain how this notion is related to other more recent flexibility notions in Complex Geometry, in particular to the notion of Oka-Forstneric manifold. We will give examples of manifolds with the density property and sketch applications of the density property. If time permits we will explain criteria for the density property developed by Kaliman and the speaker or sketch some future plans.

Veranstaltung übernehmen als iCal