There is a line bundle-divisor correspondence. Namely, given a divisor $D$ which is a formal sum of codimension $1$ subvarieties, we can write $\mathcal{O}_X(D)$ to be the line bundle whose sections have prescribed zeroes and poles.
Line bundles on complex tori $V/\Lambda$ can also be described by a hermitian form $H$ on $V$ with a corresponding semicharacter on $H$ by Appel-Humbert Theorem. The content of Appel-Humbert theorem is that every line bundle can be described up to isomorphism by such line bundles $L(H,\chi)$.
Suppose we have a complex abelian variety (i.e. a complex tori with an ample line bundle or a positive definite Riemann form) and we look at $\mathcal{O}_X(D)$ for a divisor $D$.
Is there a way to describe what $\mathcal{O}_X(D)$ is in the language of Appel-Humbert? For example, I would like to know how we can describe $H=c_1(\mathcal{O}_X(D))$. Maybe this is too ambitious.
I would also like to know if $\deg D$ (i.e. if $D=\sum n_i D_i$, $\deg D=\sum n_i$) is equal to the degree of the isogeny, $\lambda_{\mathcal{O}_X(D)}, a\mapsto t_a^*L\otimes L^{-1}$ where $L=\mathcal{O}_X(D)$. Of course knowing what $c_1(\mathcal{O}_X(D))$ would help since if we know $\lambda_L$ is of type $(d_1,...,d_g)$ its degree is simply $\prod_{i=1}^g d_i$.