Fields of CR meromorphic functions

TL;DR

This paper studies the structure of the field of CR meromorphic functions on certain compact CR manifolds, establishing bounds on transcendence degree, algebraic relations, and connections to algebraic varieties under specific extension conditions.

Contribution

It characterizes the field of CR meromorphic functions on manifolds with property E, relating it to algebraic extensions and projective varieties, extending classical results like Chow's theorem.

Findings

Transcendence degree of the field is at most n+k.

Field of CR meromorphic functions is a finite algebraic extension of rational functions.

When embedded in projective space, the field is isomorphic to rational functions on an algebraic variety.

Abstract

Let $M$ be a smooth compact $CR$ manifold of $CR$ dimension $n$ and $CR$ codimension $k$, which has a certain local extension property $E$. In particular, if $M$ is pseudoconcave, it has property $E$. Then the field $\Cal K(M)$ of $CR$ meromorphic functions on $M$ has transcendence degree $d$, with $d\leq n+k$. If $f_1, f_2, \hdots , f_d$ is a maximal set of algebraically independent $CR$ meromorphic functions on $M$, then $\Cal K(M)$ is a simple finite algebraic extension of the field $\Bbb C(f_1, f_2, \hdots, f_d)$ of rational functions of the $f_1, f_2, \hdots , f_d$. When $M$ has a projective embedding, there is an analogue of Chow's theorem, and $\Cal K(M)$ is isomorphic to the field $\Cal R(Y)$ of rational functions on an irreducible projective algebraic variety $Y$, and $M$ has a $CR$ embedding in $\roman{reg} Y$. The equivalence between algebraic dependence and analytic dependence fails when condition $E$ is dropped.