Let $X$ be Hopf surface. (i.e. $X =\mathbb{C}^2\backslash\{(0,0)\}/∼$ , $(z_1,z_2)∼(2z_1,2z_2)$). $\mathcal{O}$ is the structure sheaf of $X$(i.e. $\mathcal{O}(X)=\{f\ |f$ is holomorphic on $X$$\}$). $\underline{\mathbb{Z}}$ is the constant sheaf. (When $X$ is a manifold, $H^k(X,\underline{\mathbb{Z}})$ is isomorphic to the singular cohomology or simplicial cohomology.) I want to compute $H^k(X,\mathcal{O})$.
From https://https://math.stackexchange.com/questions/539143/de-rham-coohomology-of-hopf-surface, I know $X=\mathbb C^2-\{0\}/\mathbb Z$ is diffeomorphic to $S^3\times S^1$ and $$H^k(X,\underline{\mathbb{Z}})\simeq H_{dR}^k(X)\simeq H_k(S^3\times S^1) \simeq \bigoplus_{i+j=k} H_i(S^3)\otimes H_j(S^1)$$ A very useful exact sequence is the following $$0\to \underline{\mathbb{Z}}\to\mathcal{O}\to\mathcal{O}^*\to 0$$ We get the exact sequence $$0\to H^0(X,\underline{\mathbb{Z}})\to H^0(X,\mathcal{O})\to H^0(X,\mathcal{O}^*)\to H^1(X,\underline{\mathbb{Z}})\to H^1(X,\mathcal{O})\to H^1(X,\mathcal{O}^*)\to H^2(X,\underline{\mathbb{Z}})\to \dots$$ I know $H^1(X,\mathcal{O}^*)\cong Pic(X)$. But I can't compute $H^k(X,\mathcal{O})$ and $H^k(X,\mathcal{O}^*)$ for $i\ge 1$.
Does this work or am I doing something wrong?
A good reference is [Barth-Peters-Hulek-Van de Ven; Compact Complex Surfaces], whose Chap. V.18 is devoted to (generalized) Hopf surfaces.
The general theory is that, for a compact complex surface $X$ the Froehlicher spectral sequence $$E_1^{p,q}=H^{p,q}(X) \Rightarrow H^{p+q}(X,\mathbb{C}) $$ degenerates at $E_1$ page. Also we know that the complex conjugate $H^{1,0}(X)\rightarrow H^{0,1}(X)$ is injective. All these are explained in the referred book. It follows, by using the de Rham cohomology description, that $$H^0(X,\mathcal{O})=\mathbb{C},\ H^1(X,\mathcal{O})=\mathbb{C},\ H^2(X, \mathcal{O})=0.$$ From the exponential exact sequence, it also follows $$H^0(X,\mathcal{O}^*)=\mathbb{C}^*,\ H^1(X,\mathcal{O}^*)=\mathbb{C}^*,\ H^2(X, \mathcal{O}^*)=\mathbb{Z}.$$