I am curious about following similar statements in algebraic geometry and complex geometry:
Algebraic Geometry Version: If $X$ is an integral scheme, the map from Cartier divisor group to Picard group (i.e. group of invertible sheaves) $CaCl(X) \to Pic(X)$ is surjective.
Complex Geometry Version: If $X$ is a complex manifold, then the image of $CaCl(X) \to Pic(X)$ is generated by those line bundles $L \in Pic(X)$ with $H^0(X,L) \neq 0$.
More interesingly, there is a remark for the complex version:
Remark: $CaCl(X) \to Pic(X)$ may not be surjective even for very easy manifolds, e.g. a generic complex torus of dimension two, this is no longer the case.
My questions is twofold:
Explicit one: How to justify the above remark, i.e. $CaCl(X) \to Pic(X)$ may not be surjective for a generic complex torus of dimension two.
Inexplicit one: Though I don't know the exact meaning of "integral manifold" (or the proper prototype of integral scheme ), it seems for me that "integrality" is natrual endowed by a complex manifold(suppose it is connected). And thus, the difference is quite unexpected. How could this come out, and what are reasonable category to guarantee subjectivity.