Summary: UIC Model Theory Seminar, March 2, 2004
K-analytic geometry over o-minimal structures
(joint work with Sergei Starchenko)
Kobi Peterzil, University of Haifa
Let M be an o-minimal expansion of a real closed field R, and let K be the
algebraic closure of R.
The notion of a K-holomorphic function in one and several variables, (for
M-definable partial functions on Kn) is defined in analogy to classical com-
plex functions. K-manifolds and K-analytic subsets are defined similarly.
Interesting examples of such definable objects arise from algebraic geometry
and from the theory of compact complex manifolds, including its elementary
I will discuss some of the theorems which, when interpreted over the
complex numbers, yield seemingly stronger theorems than the classical ones
(under the assumptions that the objects are definable in some o-minimal
Theorem 1. (A finiteness theorem) If M is a K-manifold and A is a de-
finable closed subset of M which is locally K-analytic then there are finitely
many definable open sets covering A, on each of which A is the zero set of
finitely many K-holomorphic functions.