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In his paper "Complex Structures on $S^1\times S^3$", Kodaira defines the following quotient of $W=\mathbb{C}^2\setminus \lbrace(0,0)\rbrace$: $$ (z_1,z_2)\sim (pz_1+\lambda z_2^m,qz_2)\,,\quad 0<|p|\leq |q|<1\,,\quad \lambda(q^m-p)=0\,. $$ He shows that this surface is diffeomorphic to $S^3\times S^1$.

My question is about the importance of $0<|p|\leq |q|<1$ when $\lambda=0$. Clearly, for $\lambda\neq 0$, one needs a requirement like this. However, is this condition also necessary when we take $\lambda=0$? For example, when $\lambda=0$, can we define the surface for $|p|,|q|>0$ without any further conditions on $p,q$?

sam
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  • Not a complete answer, but for $\lambda = 0$, (i) if $|p| > 1$ and $|q| > 1$ we can replace $p$ and $q$ by their reciprocals; (ii) if $|p| = 1$ or $|q| = 1$ we either don't get a properly-discontinuous action (think $e^{2\pi i\theta}$ for irrational $\theta$) or we don't get a quotient with the stated topology (same, for rational $\theta$). – Andrew D. Hwang Jun 20 '22 at 13:08
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    If $\lambda=0$ and $|p|<1<|q|$ then the infinite cyclic group will not act properly on the complement to zero in the affine space. This is what Kodaira was aiming to avoid. – Moishe Kohan Jun 20 '22 at 13:26

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Suppose that $\lambda=0$. Then Kodaira considers the quotient space of $W={\mathbb C}^2 \setminus \{(0,0)\}$ by the (holomorphic) action of the cyclic group $\Gamma$ generated by the transformation $$ \gamma(z_1, z_2)= (p z_1, q z_2), p, q\in {\mathbb C}^\times. $$ In order for the quotient to be a complex surface, one needs the group action on $W$ to be proper. (Properness is also a sufficient condition.) This is where the conditions on $p, q$ are coming from. Up to inverting $\gamma$ (which does not change $\Gamma$), we can assume that $$ |p|\le |q|. $$ Now, it is a pleasant exercise in general topology, that the $\Gamma$-action on $W$ is not proper if $|p|<1<|q|$, or if $|p|=1<|q|$, or if $|p|<|q|=1$ or $|p|=|q|=1$ and one of $p, q$ is not a root of unity. If $p$ and $q$ are roots of unity, then the quotient space will not be compact and Kodaira aims to construct compact complex surfaces. This leaves us with: $$ |p|\le |q|<1, $$
exactly what Kodaira writes.

Moishe Kohan
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