Fields of CR meromorphic functions

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$CR$meromorphicfunctionson$M$hastranscendencedegree$d$,with$d\leq n+k$\mathrm{.}If$f_1, f_2, \hdots , f_d$ is a maximal set of algebraically independent $CR$ meromorphic functions on $M$, then $\text{Cal}K(M)$ is a simple finite algebraic extension of the field $\mathbb{C}(f_1,f_2,\text{hdots},f_d)$ of rational functions of the $f_1,f_2,\text{hdots},f_d$. When $M$ has a projective embedding, there is an analogue of Chow’s theorem, and $\text{Cal}K(M)$ is isomorphic to the field $\text{Cal}R(Y)$ 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.