A Hopf surface is a compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. I would like to know whether anyone has any of the following examples:


(i) A non-compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. 


(ii) A compact complex surface whose universal cover is smoothly isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$, but not complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. 



Fun examples relating to Hopf surfaces
**(ii)** I would like to prove that there are no complex surfaces that satisfy (ii).


Indeed, suppose that the universal cover $\widetilde X$ of a complex surface $X$ is diffemorphic to $\mathbb C^2\setminus 0$. Let's prove that $\widetilde X$ is biholomorphic to $\mathbb C^2\setminus 0$.


First, we note that $X$ has a finite cover that is diffeomorphic to $S^3\times S^1$. Indeed, take any 


$$S^3\subset \widetilde X=\mathbb C^2\setminus 0$$ 


that encircles $0$. Then, since the action of $\pi\_1(X)$ on $\mathbb C^2\setminus X$ is discreet, there exists only finite number of elements $g\in \pi\_1(X)$ such that $g\cdot S^3$ intersects $S^3$ in $\mathbb C^2\setminus 0$. Let's take such $g\_1\in\pi\_1(X)$ that $g\_1\cdot S^3$ is disjoint from $S^3$. Let $\mathbb Z=\langle g\_1\rangle $ be the group generated by $g\_1$. Then it is not hard to see that $(\mathbb C^2\setminus 0)/\langle g\_1\rangle$ is diffeomorphic to $S^1\times S^3$. Clearly $S^1\times S^3$ is a finite cover of $X$.


It remains now to apply the result of Bogomolov that complex surfaces with $b\_2=0$ are either Hopf surfaces or Inoue surfaces <https://en.wikipedia.org/wiki/Surface_of_class_VII> . Since $\pi\_1(S^1\times S^3)=\mathbb Z$, the complex structure on $S^1\times S^3$ is that of a Hopf surface. We conclude that $\widetilde X$ is byholomorphic to $\mathbb C^2\setminus 0$.


**(i)** As for (i), you can take $(\mathbb C^2\setminus 0)/\Gamma$, where $\Gamma$ is any finite subgroup of $SU(2)$.


