TY - GEN
T1 - Tame complex analysis and o-minimality
AU - Peterzil, Ya'acov
AU - Starchenko, Sergei
PY - 2010
Y1 - 2010
N2 - 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.
AB - 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.
KW - Abelian varieties
KW - Complex analytic sets
KW - Non-Archimedean analysis
KW - O-minimality
KW - Real closed fields
KW - Theta functions
KW - Weierstrass function
UR - http://www.scopus.com/inward/record.url?scp=84877905894&partnerID=8YFLogxK
M3 - Conference contribution
AN - SCOPUS:84877905894
SN - 9814324302
SN - 9789814324304
T3 - Proceedings of the International Congress of Mathematicians 2010, ICM 2010
SP - 58
EP - 81
BT - Proceedings of the International Congress of Mathematicians 2010, ICM 2010
T2 - International Congress of Mathematicians 2010, ICM 2010
Y2 - 19 August 2010 through 27 August 2010
ER -