# Representatives of line bundle cohomology over tori

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. `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.