Differential operators on a polarized abelian variety

Abstract

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