Why $S^1\times S^{2m-1}$ carries a complex structure.

Question

Let $S^n$ denotes $n$-sphere, then why $S^1\times S^{2m-1}$ carries a complex structure.

Answer 1

$\newcommand{\C}{\mathbf{C}}$Scalar multiplication by a complex number $\lambda$ of modulus strictly larger than $1$ defines a biholomorphism from $\C^{m}\setminus\{0\}$ to itself. The automorphism group generated by this mapping acts properly discontinuously, so the quotient admits the structure of a holomorphic manifold.

To identify the quotient space, note that $$ \{z \in \C^{m} : 1 \leq \|z\| \leq |\lambda|\} \simeq S^{2m-1} \times \bigl[1, |\lambda|\bigr] $$ is a fundamental domain, and the unit sphere (the "inner boundary") is glued to the sphere of radius $|\lambda|$ (the "outer boundary") by the action.

For simplicity, you may as well choose $\lambda$ to be real if you're mainly interested in seeing intuitively why the quotient is a product of spheres. You may also be interested in reading about Calabi-Eckmann manifolds.

(Note that, aside from elliptic curves arising when $m = 1$, none of these manifolds admits a Kähler metric, since $H^{2} = 0$.)