Algebraic dimension

Summary

The algebraic dimension $a(X)$ of a compact connected complex manifold $X$ is the transcendence degree over $\C$ of its field of meromorphic functions. It measures how many independent meromorphic functions $X$ carries. By Siegel's theorem, $0 \le a(X) \le \dim X$. The extreme case $a(X) = \dim X$ defines Moishezon manifolds, which are close to projective varieties, while $a(X) = 0$ occurs for generic complex tori.