A note on characterization of Moishezon spaces (Q2639206)

Main theorem: An irreducible compact complex manifold X is Moishezon if and only if there exist a line bundle L and a point x on X such that \(H^ 1(X,L\otimes m_ x^ r)=0\) for \(r=1,2\), where \(m_ x\) is the maximal ideal sheaf of x. Proof of ``if'' part. The condition for \(r=1\) implies that x is not a base point of L, so the rational map \(\rho\) defined by \(| L|\) is holomorphic at x. The condition for \(r=2\) implies that is of maximal rank at x. Hence dim \(\rho\) (X)\(=\dim X\) and X is Moishezon. The ``only if'' part is equally easy. It is a wonder that such a simple criterion is not mentioned in literatures.