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 -