Let $X$ be a smooth projective variety over the complex numbers.
Let $$c_1 : \text{Pic}(X)\to H^2(X,\mathbf{Z}(1))$$
Its kernel is the subgroup of homologically trivial divisor classes on $X$. How does the proof that $$\text{im}(c_1) = \text{NS}(X)$$ go?
I can't track a source, and there is some content in it.
One way to define the Neron-Severi group $NS(X)$ is to fit it into the short exact sequence $$0 \rightarrow \mathrm{Pic}^0(X) \rightarrow \mathrm{Pic}(X) \rightarrow NS(X) \rightarrow 0$$ ($0$ being the appropriate identity). We also have the exponential sheaf sequence $$0 \rightarrow \mathbb{Z} \rightarrow \mathcal O \rightarrow \mathcal O^* \rightarrow 0$$ and, using the fact that $\mathrm{Pic}(X) \simeq H^1(X, \mathcal O^*)$ (Hartshorne, Ex. III.4.5), the long exact sequence of cohomology associated to this sheaf sequence gives us this map $c_1: \mathrm{Pic}(X) \to H^2(X, \mathbb Z)$. Now use the fact that $\mathrm{Pic}^0(X) = \ker c_1$ and an isomorphism theorem.
An interesting case of this is for K3 surfaces $X$. By definition, $H^1(X, \mathcal O) = 0$, so $c_1: \mathrm{Pic}(X) \hookrightarrow H^1(X, \mathbb Z)$ is an injection, and $NS(X)$ is identified with $\mathrm{Pic}(X)$.