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Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a reasonable moduli space, is it finite or infinite dimensional?

Remark. For $n=1$ there exist exactly two complex structures on $\mathbb{R}^2$: one is isomorphic to $\mathbb{C}$, the other one to the unit disc. This is a special case of the uniformization theorem saying that any simply connected complex 1-dimensional manifold is isomorphic either to unit disk, or to $\mathbb{C}$, or to $\mathbb{C}\mathbb{P}^1$.

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  • $\begingroup$ This paper of Hjorth and Kechris may be of interest: projecteuclid.org/euclid.ijm/1255984956. It deals with using descriptive set theory to understand the classification problem for arbitrary Riemann surfaces (not just structures on $\mathbb{R}^{2n}$), and shows that even in complex dimension $1$, this is extremely complicated (the "moduli space" is "Borel equivalent" to the quotient of $\{0,1\}^{F_2}$ by the left shift action of the free group $F_2$). For higher dimensions, it is more complicated ("not classifiable by countable structure" in the subject's parlance). $\endgroup$ – Iian Smythe Apr 20 '15 at 14:35
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There exist infinitely many inequivalent complex structures in $\mathbb{R}^{2n}$ for all $n \geq 2$.

See for instance the paper

K. Diederich, N. Sibony: Strange complex structures on Euclidean space, Journal für die reine und angewandte Mathematik 311-312, page 397-407 (1979)

and the references given therein.

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There are even infinitely many inequivalent complex structures without non-constant holomorphic functions on $\Bbb R^4$ download.

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