  We show that certain type II string amplitudes at genus $g$ are given by the
topological partition function $F_g$ discussed recently by Bershadsky, Cecotti,
Ooguri and Vafa. These amplitudes give rise to a term in the four-dimensional
effective action of the form $\sum_g F_g W^{2g}$, where $W$ is the chiral
superfield of $N=2$ supergravitational multiplet. The holomorphic anomaly of
$F_g$ is related to non-localities of the effective action due to the
propagation of massless states. This result generalizes the holomorphic anomaly
of the one loop case which is known to lead to non-harmonic gravitational
couplings.
