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The complexity of the classification of Riemann surfaces and complex manifolds

Hjorth, G. and Kechris, A. S. (2000) The complexity of the classification of Riemann surfaces and complex manifolds. Illinois Journal of Mathematics, 44 (1). pp. 104-137. ISSN 0019-2082.

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In answer to a question by Becker, Rubel, and Henson, we show that countable subsets of 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 modern 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 ≥ 2.

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Additional Information:© 2000 by the Board of Trustees of the University of Illinois. Received May 20, 1998; received in final form December 14, 1998. Research partially supported by the National Science Foundation, the first-named author by grant DMS 96-22977 and the second-named author by grant DMS 96-19880.
Funding AgencyGrant Number
NSFDMS 96-22977
NSFDMS 96-19880
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Other Numbering System NameOther Numbering System ID
Mathematical Reviews number (MathSciNet)MR1731384
Zentralblatt MATH identifier0954.03052
Issue or Number:1
Classification Code:1991 Mathematics Subject Classification: Primary 04A15, 03A15, 32C10
Record Number:CaltechAUTHORS:20130521-075724313
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38586
Deposited By: Tony Diaz
Deposited On:21 May 2013 16:58
Last Modified:03 Oct 2019 04:58

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