The Nash Conjecture for Nonprojective Threefolds

We prove that for every compact, connected, differentiable 3--manifold $M$ there is a compact complex manifold $X$ which can be obtained from projective 3--space by a sequence of smooth, real blow ups and downs such that $M$ is diffeomorphic to the set of real points of $X$. By earlier results, such an $X$ can almost never be projective.