# Moishezon manifold with vanishing $b_2$

Does there exist a closed Moishezon manifold with zero second Betti number?

• $M = \mathrm{pt}$ works, though you're probably looking for a positive-dimensional example. – Arun Debray Dec 8 '18 at 20:19

## 1 Answer

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.

• Note: $a(X)$ is the transcendence degree of the field of meromorphic functions of $X$ over $\mathbf{C}$ — and $X$ is implicitly assumed connected. – YCor Dec 9 '18 at 13:56