Rank one connections on abelian varieties

Abstract

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 $H^0(A,{\Omega }_A^1)$. Take any line bundle $L$ on $A^{\vee }$; let $\mathcal{C}(L)$ be the algebraic principal $H^0(A^{\vee },{\Omega }_{A^{\vee }}^1)$-bundle over $A^{\vee }$ given by the sheaf of connections on $L$. The line bundle $L$ produces a homomorphism $H^0(A,{\Omega }_A^1)\to H^0(A^{\vee },{\Omega }_{A^{\vee }}^1)$. We prove that $\mathcal{C}(L)$ is isomorphic to the principal $H^0(A^{\vee },{\Omega }_{A^{\vee }}^1)$-bundle obtained by extending the structure group of the principal $H^0(A,{\Omega }_A^1)$-bundle ${\mathcal{M}}_C$ using this homomorphism given by $L$. We compute the ring of algebraic functions on $\mathcal{C}(L)$.

Final version