Tame complex analysis and o-minimality

Ya'acov Peterzil, Sergei Starchenko

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We describe here a theory of holomorphic functions and analytic manifolds, restricted to the category of definable objects in an o-minimal structure which expands a real closed field R. In this setting, the algebraic closure K of the field R, identified with R2, plays the role of the complex field. Although the ordered field R may be non-Archimedean, o-minimality allows to develop many of the basic results of complex analysis for definable K-holomorphic functions even in this non-standard setting. In addition, o-minimality implies strong theorems on removal of singularities for definable manifolds and definable analytic sets, even when the field R is ℝ. We survey some of these results and several examples. We also discuss the definability in o-minimal structures of several classical holomorphic maps, and some corollaries concerning definable families of abelian varieties.

Original languageEnglish
Title of host publicationProceedings of the International Congress of Mathematicians 2010, ICM 2010
Number of pages24
StatePublished - 2010
EventInternational Congress of Mathematicians 2010, ICM 2010 - Hyderabad, India
Duration: 19 Aug 201027 Aug 2010

Publication series

NameProceedings of the International Congress of Mathematicians 2010, ICM 2010


ConferenceInternational Congress of Mathematicians 2010, ICM 2010


  • Abelian varieties
  • Complex analytic sets
  • Non-Archimedean analysis
  • O-minimality
  • Real closed fields
  • Theta functions
  • Weierstrass function

ASJC Scopus subject areas

  • General Mathematics


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