A complex manifold isn't a sympletic manifold

Question

I want to think about an example of a complex manifold which isn't a sympletic manifold. I consider it in this way: $X=\mathbb{C}^2-\{0\}$, a group $\mathbb{Z}$ acts on X by $(n,z)=2^nz$, then I think that $X/\mathbb{Z}$ is a complex manifold which isn't a sympletic manifold, but I can't prove it.

Answer 1

Your manifold $(\Bbb{C}^2 - \{0\})/\Bbb Z$ is known as the Hopf surface and it is indeed diffeomorphic to $S^3 \times S^1$. Now $H^2(S^3 \times S^1) = 0$ so that every closed $2$-form on the Hopf surface is exact. Since the Hopf surface is a closed manifold, Stokes' theorem implies that it admits no symplectic form (symplectic forms on closed manifolds cannot be exact).