Sheaf cohomology: meaning of $H^1(X,2\pi i\Bbb{Z})$

Question

Let $X$ be a complex algebraic surface.

We have the exponential sequence $0\to 2\pi i\Bbb{Z}\to\mathcal{O}_X\to\mathcal{O}_X^*\to 0$.

This this article says that $H^1(X,2\pi i\Bbb{Z})$ is the singular cohomology group and that $H^1(X,\mathcal{O}_X^*)$ may be identified with $\text{Pic}(X)$.

I don't understand this. How did we get from sheaf cohomology to singular cohomology and the group of divisors modulo linear equivalence?

Answer 1

This is two different questions.

For the first, you need a comparison theorem between Cech and singular cohomology; you can find this in a variety of places. (EDIT: full references here https://mathoverflow.net/questions/310306/outline-of-the-proof-that-cech-cohomology-and-singular-cohomology-coincide-on-an).

For the second, you need that $H^1(X, \mathcal{O}_X^*)$ parametrizes line bundles; this is straightforward from Cech cohomology -- the transition functions for a line bundle give a class in $H^1$, check that two line bundles are isomorphic iff they differ (multiplicatively) by the trivial class. (EDIT: it looks like you've seen Pic defined in terms of divisors mod linear equivalence, so you also need that these are the same as line bundles!)