Take the 2-minute tour ×  Consider a compact connected complex
manifold $X$ of dimension $n$. Siegel proved in 1955 that its
field of meromorphic functions $\mathcal M (X)$ has
transcendence degree over $\mathbb C$ at most $n$. Moishezon
studied those complex manifolds for which the degree is $n$, and
consequently these manifolds are now called Moishezon manifolds.
In dimension 2, every Moishezon surface is projective algebraic
according to a theorem of Chow-Kodaira proved in 1952, long
before Moishezon formally introduced his concept. However in
dimensions 3 and more, there exist nonprojective Moishezon
manifolds (you can see an example in Shafarevich's book Basic
Algebraic geometry).  Nevertheless a Moishezon manifold $X$ is
close to projective: Moishezon's main result is that after a
finite number of blow-ups with smooth centers, $X$ becomes
algebraic projective. So if a blow-up of $X$ is projective, you
can't deduce that $X$ was projective. However this main result
says nothing about the dimensions of th