What topology has $Pic(X)$?

Question

Let $Pic(X):=\{\mbox{Holomorphic line bundles on } X\}/\sim$ be the group of isomorphism classes of line bundles on $X$. It is well-know and easy to prove that, working in Cech cohomology, $Pic(X) \simeq H^1(X, \mathcal{O}_X^*)$. Is there a good method in order to give a topology on $Pic(X)$? Is it possible to do this using the exact sequence in cohomology $$ \cdots \to H^1(X,\mathcal{O}_X^*) \to H^2(X,\mathbb{Z}) \to H^2(X, \mathcal{O}) \to \cdots \,\,\, ?$$ Thank you in advance!

Answer 1

For a compact Kähler manifold $X$ the exact sequence $$0 \to H^1(X,\mathbb{Z}) \to H^1(X, \mathcal{O}) \to H^1(X,\mathcal{O}_X^*) \stackrel {c_1}{\to} H^2(X,\mathbb{Z})$$ allows you to endow $Pic^0(X)$, the group of isomorphism classes of line bundles of first Chern class $c_1=0$ , with the structure of a compact complex torus .
Indeed $ H^1(X,\mathbb Z)\subset H^1(X, \mathcal O)$ is a lattice by Hodge theory and the quotient $H^1(X, \mathcal O)/H^1(X,\mathbb Z)$ is thus a compact (in general non-algebraic) torus of dimension $$g=\operatorname {dim}_\mathbb C H^1(X, \mathcal O)={dim}_\mathbb C H^{01}=h^{01}=b_1/2$$ (half the first Betti number of $X$).

In the case when $X$ is algebraic, the construction of an algebraic variety structure on $Pic(X)$ is much more complicated, as I mentioned in a comment to the question.
Kleiman's fantastic survey on the Picard scheme, whose history begins in the 17th century with James Bernoulli's study of a lemniscate linked to the bending of rods, can be found here.