TITLE: Moishezon manifold with vanishing $b_2$
QUESTION [6 upvotes]: Does there exist a closed  Moishezon manifold  with zero second Betti number?

REPLY [5 votes]: If $X$ has positive dimension, the answer is no. In fact, the following holds:

Proposition. Let $X$ be a compact manifold such that $a(X)= n >0$. Then $b_2(X)>0$.

The proof is essentially based on the well-known fact that the assumption $a(X)=n$ implies that $X$ is a bimeromorphic modification of a projective manifold. Look at Lemma 1.4 of the paper 
Campana, Frédéric; Demailly, Jean-Pierre; Peternell, Thomas, The algebraic dimension of compact complex threefolds with vanishing second Betti number, Compos. Math. 112, No. 1, 77-91 (1998). ZBL0910.32032.