Background: By Chow's theorem, if a complex manifold can be embedded holomorphically into complex projective space, then this complex manifold must be algebraic.


Question: Suppose X is a compact complex manifold (not necessarily algebraic). Let $f:X--> {\mathbb{CP}}^n$ be a meromorphic map that is injective on its domain. Does this imply that X is algebraic? 



Can a non-algebraic complex manifold be embedded meromorphically into projective space?
If $X$ is not compact there are loads of problems so I follow you in adding compactness as a condition.
Under the assumption of the question $X$ is bimeromorphic to a projective variety and hence by a result of Artin it is an algebraic space. Hence if you accept algebraic spaces as being algebraic the answer is yes. As there are smooth proper algebraic spaces that are not schemes if you don't the answer is no.


