Suppose $X$ is a compact complex surface, $$ X=\mathbb{C}^2\backslash\{(0,0)\}/\sim, $$ where the relation is $(z_1,z_2)\sim(2z_1,2z_2)$. Compute $H^0(X, \mathscr{O}), H^2(X, \mathscr{O}),H^1(X, \mathscr{O}), H^0(X, \mathscr{O}^*)$.
I just read the Appendix of Complex Geometry_An Introduction by Huybrechts. I wonder how to compute a concrete example of sheaf cohomology.