Let $\Sigma$ be a compact Riemann surface. Let $\mathcal{O},\mathcal{O}^*, \mathbb{Z}$ be, respectively, the sheaves of holomorphic functions, nonvanishing holomorphic functions, and integer-valued locally constant functions on $\Sigma$. Pointwise addition makes $\mathbb{Z},\mathcal{O}$ sheaves of abelian groups. Pointwise multiplication makes $\mathcal{O}^*$ a sheaf of abelian groups.
Consider the short exact sequence $$0\rightarrow \mathbb{Z} \rightarrow \mathcal{O}\rightarrow \mathcal{O}^* \rightarrow 0,$$ where the first map is the inclusion and the second is $f\mapsto \exp{(2\pi i f)}$. As part of the associated long exact sequence, we get a connecting homomorphism $$ H^1(\mathcal{O}^*)\rightarrow H^2(\mathbb{Z}).$$ We identify $H^1(\mathcal{O}^*)$ with the group of isomorphism classes of holomorphic line bundles on $\Sigma$. (Use a Čech cocycle representing an element of $H^1(\mathcal{O}^*)$ as transition data for a line bundle.)
How do you prove that the image of a line bundle under the boundary map is the same as the topologically defined Euler class of the bundle?
I just realized this question was discussed in the comments for this question: Algebraic versus topological line bundles As pointed out there, there's a proof using Chern-Weil theory in Chapter III, Section 4 of Differential Analysis on Complex Manifolds, by Wells.