Is there a compact complex manifold with trivial $H_2$?

Question

I don't believe that every complex manifold should have nontrivial $H_2$, otherwise we would easily prove the Chern's conjecture... But the problem is I don't have any counterexample. The Kähler manifold will have nontrivial $H_2$ and so do Riemann surface. Hence I guess there would be a complex surface with trivial $H_2$?

Answer 1

Yes, consider the Hopf torus, $({\mathbb C}^2-0)/{\mathbb Z}$, where the generator of ${\mathbb Z}$ acts by a dilation, say, $(x,y)\to 2(x,y)$.