Let
M
g
,
n
{\mathcal M}_{g,n}
denote the moduli space of smooth, genus
g
≥
1
g\geq 1
curves with
n
≥
0
n\geq 0
marked points. Let
A
h
{\mathcal A}_h
denote the moduli space of
h
h
-dimensional, principally polarized abelian varieties. Let
g
≥
3
g\geq 3
and
h
≤
g
h\leq g
. If
F
:
M
g
,
n
→
A
H
F:{\mathcal M}_{g,n} \to {\mathcal A}_H
is a nonholomorphic map, then
h
=
g
h=g
and
F
F
is the classical period mapping, assigning to a Riemann surface
X
X
its Jacobian.