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?

  • $\begingroup$ I don't know what counts as an infinite-dimensional Lie group, but you can make lots of closed subgroups, for example the stabilizer of a point or a line or a function $\mathbb C^n\to \mathbb C$. $\endgroup$ – Tom Goodwillie Dec 1 '10 at 3:27
  • $\begingroup$ Say a Frechet Lie group. A Banach Lie group might be too much to ask. $\endgroup$ – David Roberts Dec 1 '10 at 4:23
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    $\begingroup$ In the algebraic case it's horrifying: not representable for any $n > 1$. Indeed, suppose rep'td by some $G$. Then Grothendieck's functorial criterion for being locally of finite presentation forces $G$ to be loc. of finite type, then smooth (by Cartier), so $G^0$ is a finite type connected smooth group variety. This forces the points of $G^0$ to act by automorphisms represented by polynomials of bounded degree, which is absurd for $n > 1$ since for two distinct variables $(x,y)$ can use $(x,y) \mapsto (x + t y^n, y)$ to get a connected family of auts joining id to arb. big degree. $\endgroup$ – BCnrd Dec 1 '10 at 5:06

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

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    $\begingroup$ But you can easily write down two polynomial vector fields that have global flows, and whose sum does not. $\endgroup$ – Tom Goodwillie Dec 2 '10 at 17:33
  • $\begingroup$ @Tom: Maybe. I really don't know if it so. $\endgroup$ – zroslav Dec 2 '10 at 20:04

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