Hopf manifold

Example

Let $n \ge 2$ and let $\lambda \in \C$ with $0 < \abs{\lambda} < 1$. Let $W = \C^n \setminus \set{0}$ and let $\Gamma$ be the group of maps $g_k(z) = \lambda^k z$, $k \in \Z$. The quotient $X = W / \Gamma$ is a (classical) Hopf manifold, a compact complex manifold of dimension $n$.

Check that $X$ is a complex manifold. The action is free: if $\lambda^k z = z$ with $z \ne 0$, then $\lambda^k = 1$, so $k = 0$. It is properly discontinuous: a compact $K \subset W$ lies in a shell $\set{r \le \abs{z} \le R}$ with $r > 0$. The set $g_k(K)$ lies in $\set{\abs{\lambda}^k r \le \abs{z} \le \abs{\lambda}^k R}$, which meets $K$ for only finitely many $k$. The maps $g_k$ are holomorphic, so the quotient inherits a complex structure. It is compact: it is the image of the compact shell $\set{\abs{\lambda} \le \abs{z} \le 1}$.

For $\lambda$ real, the map $z \mapsto (\abs{z}, z/\abs{z})$ identifies $W$ with $(0, \infty) \times S^{2n-1}$, and $\Gamma$ acts only on the first factor. So $X$ is diffeomorphic to $S^1 \times S^{2n-1}$; the same holds for all $\lambda$.