Hopf surface

Example

A Hopf surface is a compact complex surface whose universal cover is biholomorphic to $\C^2 \setminus \set{0}$. It is primary if its fundamental group is infinite cyclic.

Example: let $0 < \abs{\lambda} < 1$ and $X = (\C^2 \setminus \set{0}) / \langle g \rangle$ with $g(z, w) = (\lambda z, \lambda w)$, the Hopf manifold of dimension $2$. Since $\C^2 \setminus \set{0}$ is homotopy equivalent to $S^3$, it is simply connected, so it is the universal cover of $X$, and $\pi_1(X) \cong \Z$.

Check that $X$ is a class VII surface. Its first Betti number is $1$, since $X$ is diffeomorphic to $S^1 \times S^3$. Its Kodaira dimension is $-\infty$. A section of $K_X^{\otimes m}$ with $m \ge 1$ pulls back to $h(z, w)\,(dz \wedge dw)^{\otimes m}$ with $h$ holomorphic on $\C^2 \setminus \set{0}$, hence, by Hartogs's extension theorem, on $\C^2$. Invariance under $g$ gives $h(\lambda z, \lambda w)\, \lambda^{2m} = h(z, w)$, so each Taylor coefficient $c_{jk}$ of $h$ satisfies $c_{jk} \lambda^{j + k + 2m} = c_{jk}$; since $\abs{\lambda} < 1$, all $c_{jk}$ vanish and $h = 0$.