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.