Grauert's theorem for infinite dimensional Frechet Lie groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T05:37:23Z http://mathoverflow.net/feeds/question/116786 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116786/grauerts-theorem-for-infinite-dimensional-frechet-lie-groups Grauert's theorem for infinite dimensional Frechet Lie groups Sebastian 2012-12-19T13:01:32Z 2012-12-19T13:01:32Z <p>Stein manifolds are complex analytic submanifolds of some $\mathbb{ C}^N.$</p> <p>(A version of) Grauert's theorem states that on a Stein manifold $X$ every continuous map $g\colon X\to G$ to a complex Lie group $G$ is homotopic to a holomorphic map, see Gromov "Oka's principle for holomorphic sections of elliptic bundles".</p> <p>This theorem was generalized to the case where $G$ is an infinite dimensional complex Banach Lie group, see Bungart "On analytic fiber bundles. I. Holomorphic fiber bundles with infinite dimensional fibers".</p> <p>I would like to know if the theorem still holds when $G$ is replaced by an infinite dimensional complex Frechet Lie group or at least in the case of $G=C^\infty(M,\mathbb{S}L(2,\mathbb{C}))$ for a compact mannifold $M.$</p>