# An introduction to the Oka-Grauert principle

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Jasna Prezelj

created by daniele on 16 Jan 2018

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.