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Rank one connections on abelian varieties

Let A be a complex abelian variety. The moduli space ${\mathcal M}_C$ of rank one algebraic connections on $A$ is a principal bundle over the dual abelian variety $A^\vee=\text{Pic}^0(A)$ for the group $H^0(A, Ω^1_A)$. Take any line bundle $L$ on $A^\vee$; let ${\mathcal C}(L)$ be the algebraic principal $H^0(A^\vee, Ω^1_{A^\vee})$-bundle over $A^\vee$ given by the sheaf of connections on $L$. The line bundle $L$ produces a homomorphism $H^0(A, Ω^1_A) \rightarrow H^0(A^\vee,\, Ω^1_{A^\vee})$. We prove that ${\mathcal C}(L)$ is isomorphic to the principal $H^0(A^\vee, Ω^1_{A^\vee})$-bundle obtained by extending the structure group of the principal $H^0(A,\, Ω^1_A)$-bundle ${\mathcal M}_C$ using this homomorphism given by $L$. We compute the ring of algebraic functions on ${\mathcal C}(L)$.