How manifold-like is Aut(C^n) in the holomorphic category? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:00:03Z http://mathoverflow.net/feeds/question/47847 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47847/how-manifold-like-is-autcn-in-the-holomorphic-category How manifold-like is Aut(C^n) in the holomorphic category? David Roberts 2010-12-01T01:01:12Z 2012-10-19T09:22:01Z <p>This question is similar to, but not the same as <a href="http://mathoverflow.net/questions/26555/are-spaces-of-holomorphic-maps-manifolds" rel="nofollow">this one</a>. 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. </p> <p>To give a taste of how big $Aut(\mathbb{C}^n)$ is, there is a <a href="http://www.jstor.org/pss/2001110" rel="nofollow">theorem</a> 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?</p> http://mathoverflow.net/questions/47847/how-manifold-like-is-autcn-in-the-holomorphic-category/48040#48040 Answer by zroslav for How manifold-like is Aut(C^n) in the holomorphic category? zroslav 2010-12-02T12:18:46Z 2010-12-02T12:18:46Z <p>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: <a href="http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;jrnid=rm&amp;paperid=3301&amp;option_lang=eng" rel="nofollow">http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;jrnid=rm&amp;paperid=3301&amp;option_lang=eng</a></p> <p><a href="http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;jrnid=intf&amp;paperid=93&amp;option_lang=eng" rel="nofollow">http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;jrnid=intf&amp;paperid=93&amp;option_lang=eng</a> )</p> <p>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</p> <p>I think that you have much to learn from this book.</p> <p>Also there is an article: <a href="http://arxiv.org/abs/0708.3398" rel="nofollow">http://arxiv.org/abs/0708.3398</a> - it may be rather useful</p>