How manifold-like is Aut(C^n) in the holomorphic category? - MathOverflow

How manifold-like is Aut(C^n) in the holomorphic category?

2010-12-01T01:01:12Z

This question is similar to, but not the same as this one. Take the space of automorphisms of $\mathbb{C}^n$ in the holomorphic category, with the compact-open topology. For $n=1$ this is just $\mathbb{C} \times \mathbb{C}^*$, but for larger $n$ is very complicated. It clearly contains $GL(n,\mathbb{C})$, and translations.

To give a taste of how big $Aut(\mathbb{C}^n)$ is, there is a theorem that given any two countable dense subsets $X,Y \subset \mathbb{C}^n$, $n >1$, there is a volume preserving automorphism taking $X$ to $Y$. But I have no idea about what this space is like. Is it some sort of infinite dimensional manifold? Analytic space? Does it contain an infinite dimensional Lie group as a (closed) subgroup?

Answer by zroslav for How manifold-like is Aut(C^n) in the holomorphic category?

2010-12-02T12:18:46Z

It is a group generated by flows of holomorphic vector fields. Its Lie algebra is a set of all holomorphic vector fields. As I can for now remember there were a computation of its group of cohomologies (by Feigin and Fuchs: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=3301&option_lang=eng

http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intf&paperid=93&option_lang=eng )

Maybe this book can help if you have no translation of these articles from Russian: Fuks D.B. Cohomology of infinite-dimensional Lie algebras, Consultants Bureau, 1986

I think that you have much to learn from this book.

Also there is an article: http://arxiv.org/abs/0708.3398 - it may be rather useful