TITLE: Is a complex manifold projective just because its blow-up at a point is ?
QUESTION [21 upvotes]: 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 the manifolds you blow up. I've heard it claimed that if only one point  is blown-up, you can't get from a non-projective to a projective manifold, but I could obtain neither precise reference nor proof.
Hence my question :
If a compact complex manifold becomes projective algebraic after blowing-up a point, was it already projective algebraic?

REPLY [5 votes]: There is a sledgehammer available for this particular nut: Mori's results on extremal rays, in his paper (Annals 1979?) on smooth projective varieties $Y$ where $K_Y$ is not nef: if $Y=Bl_PX$, with exceptional divisor $E$, then any line in $E$ will span an extremal ray and then can be contracted in the category of projective varieties. Since all curves in $E$ lie in the same ray, $E$ must be contracted to a point, so the result of the contraction must be $Y$.