Let $V^n$ a be a $\mathbb{C}$-vector space. For $U\subset V$ a complete lattice, the holomorphic line bundles over $V/U$ are classified (see e.g. $\textit{Abelian varieties}$, D. Mumford) by data $(H,\alpha)$ consisting of 

$$
H(\cdot,\cdot)=E_\mathbb{R}(i\cdot,\cdot)+iE_\mathbb{R}(\cdot,\cdot), \; E\in \mathrm{Hom}(\Lambda^2U,\mathbb{Z})\\
\alpha: U\rightarrow S^1, \ \alpha(u_1+u_2)=(-1)^{E(u_1,u_2)}\alpha(u_1)\alpha(u_2)
$$

This is the content of Appell-Humbert theorem. For such a line bundle $L(H,\alpha)$, much of its hermitian and holomorphic geometry can be described completely explicitly by pull-back to $V$, e.g. its sections are determined through the computation of $\theta$-functions, etc.

Now let $n=2$. Standard complex geometry techniques allow to deduce $h^1(L(H,\alpha))>0$ for a wealth of line bundles. However I am not aware of an explicit description of smooth $L(H,\alpha)$-valued $(0,1)$-forms as Dolbeault representatives of classes in $H^1(V/U,L(H,\alpha))$ in the literature, in the spirit of Appell-Humbert or otherwise.