Let $T=\mathbb{C}^3/\Lambda$ be a complex torus of our interest and $L$ be a holomorphic line bundle on $T$, I am interested in $H^{0,2}_{\bar\partial_L}(T,L)$, i.e., the $(0,2)$ harmonic forms taking values in $L$. Now $H^{0,2}_{\bar\partial_L}(T,L)\cong H^2(T,\mathcal{O}(L))$. I want to know if there's any simple description of these forms using local coordinates in terms of the "multipliers" of $L$. My inspiration is that we can pull back the line bundle $L$ to $\mathbb{C}^3$ and we get a trivial line bundle on $\mathbb{C}^3$. Now for $H^0(T,\mathcal{O}(L))$ we get all the sections from Theta functions using the "multipliers" conditions. Using $H^{0,2}_{\bar\partial}(\mathbb{C}^3,\mathbb{C})=0$, do we get anything? Can we comment something on the dimension of $H^{0,2}_{\bar\partial_L}(T,L)$?
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