Let $X=S^{2n-1} \times S^1$. I have to compute $H^{(1,0)}_{\bar{\partial}} (X)$ and

$H^{(0,1)}_{\bar{\partial}}(X)$ . I don't know how to do this but if we use Kunnet formula we

have that $dim(H^{(0,1)}_{\bar{\partial}}(X))$ $+$

$dim(H^{(1,0)}_{\bar{\partial}}(X))=1$, so we

have that $H^{(1,0)}_{\bar{\partial}}(X)\simeq \mathbb{C}$

and $H^{(0,1)}_{\bar{\partial}} (X)\simeq 0$

or vice versa. I find that $X \simeq (\mathbb{C}^{n}-\{0\})/\mathbb{Z} $ via the

action $d:(z_1, \cdots, z_n) \mapsto(\lambda^dz_1, \cdots, \lambda^dz_n)$ with $n>1$,

$\lambda \in \mathbb{C}$ and $|\lambda|>1$. Futhermore I read that

$H=\bar{\partial}log(|z_1|^2+ \cdots +|z_n|^2)$ represents a non trivial class. But how can

I prove that $H$ is a closed but non exact form?