Illinois Journal of Mathematics

The complexity of the classification of Riemann surfaces and complex manifolds

G. Hjorth and A. S. Kechris

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In answer to a question by Becker, Rubel, and Henson, we show that countable subsets of $\mathbb{C}$ can be used as complete invariants for Riemann surfaces considered up to conformal equivalence, and that this equivalence relation is itself Borel in a natural Borel structure on the space of all such surfaces. We further proceed to precisely calculate the classification difficulty of this equivalence relation in terms of the modem theory of Borel equivalence relations.

On the other hand we show that the analog of Becker, Rubel, and Henson's question has a negative solution in (complex) dimension $n \geq 2$.

Article information

Illinois J. Math., Volume 44, Issue 1 (2000), 104-137.

First available in Project Euclid: 19 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E15: Descriptive set theory [See also 28A05, 54H05]
Secondary: 30F20: Classification theory of Riemann surfaces 32Q99: None of the above, but in this section


Hjorth, G.; Kechris, A. S. The complexity of the classification of Riemann surfaces and complex manifolds. Illinois J. Math. 44 (2000), no. 1, 104--137. doi:10.1215/ijm/1255984956.

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