Calabi–Eckmann manifold

Example

Let $p, q \ge 0$ be integers and let $\tau \in \C$ with $\operatorname{Im} \tau \ne 0$. Let $W = (\C^{p+1} \setminus \set{0}) \times (\C^{q+1} \setminus \set{0})$, and let the additive group $\C$ act on $W$ by $$t \cdot (u, v) = (e^{t} u, \; e^{\tau t} v).$$ The quotient $X_{p,q} = W / \C$ is a Calabi–Eckmann manifold, a compact complex manifold of dimension $p + q + 1$.

Check that $X_{p,q}$ is diffeomorphic to $S^{2p+1} \times S^{2q+1}$: each orbit meets $S^{2p+1} \times S^{2q+1}$ in exactly one point. Write $t = x + iy$. The conditions $\abs{e^t u} = 1$ and $\abs{e^{\tau t} v} = 1$ read $x = -\log\abs{u}$ and $x \operatorname{Re}\tau - y \operatorname{Im}\tau = -\log\abs{v}$. Since $\operatorname{Im}\tau \ne 0$, they have exactly one solution $(x, y)$. The action is free and proper and is by holomorphic maps, so the quotient is a complex manifold of dimension $(p + 1) + (q + 1) - 1$.