25 jan 2018 -- 11:30
Aula Tricerri, DiMaI, Firenze
Abstract.
We present a survey on Oka-Grauert Principle.
The Oka-Grauert or homotopy principle states that cohomologically formulated problems on Stein manifolds are holomorphically solvable if they are continuously solvable. After the results of Oka in 1940' for sheaves on Stein manifolds the next main result was Grauert?sclassification of principal holomorphic fiber bundles over Stein spaces. In particular, this problem was reduced to finding a holomorphic section of a certain principal bundle homotopic to a given continuous one. Gromov extended the principle to sections of elliptic submersions, where elliptic refers to manifolds with 'plenty' of images of Euclidean spaces. This gave rise to Oka manifolds, a concept introduced by Forstneri\v c around 2010. Modern Oka theory studies holomorphic maps from Stein manifolds and Stein spaces to Oka manifolds. We will also describe the present state of the art on Oka theory.