Differential operators on a polarized abelian variety

Abstract:

Let $L$ be an ample line bundle over a complex abelian variety $A$. We show that the space of all global sections over $A$ of $\operatorname {Diff}^{n}_A(L,L)$ and $S^n(\operatorname {Diff}^1_A(L,L))$ are both of dimension one. Using this it is shown that the moduli space $M_X$ of rank one holomorphic connections on a compact Riemann surface $X$ does not admit any nonconstant algebraic function. On the other hand, $M_X$ is biholomorphic to the moduli space of characters of $X$, which is an affine variety. So $M_X$ is algebraically distinct from the character variety if $X$ is of genus at least one.