  Global symmetries are expected to be violated by gravity, which may cause a
serious problem to models based on these symmetries. A famous example is the
Peccei-Quinn solution to the strong CP problem in QCD axion models; it requires
a global U(1) symmetry to be respected with high quality, and even the
Planck-scale suppressed operators that violate the U(1) symmetry would spoil
this solution. Indeed, it is known that gravitational instantons do break the
U(1) Peccei-Quinn symmetry and induce the U(1)-violating operators, bringing
back the strong CP problem to the axion models. This conclusion is, however,
dependent highly on the structure of the theory around the Planck scale and,
therefore, may be evaded if we go beyond the minimal setup. In this paper, we
study the effect of a non-minimal coupling of the Peccei-Quinn field to
gravity, $\xi$, on the gravitational instanton effect. This setup is frequently
considered in cosmology as it can realize a successful inflation if $\xi
\lesssim 10^5$. We find that the U(1)-breaking effect of the gravitational
instantons can sufficiently be suppressed for $\xi \gtrsim 2 \times 10^3$,
which suggests that the U(1) symmetry may be maintained with high quality even
in the presence of gravity. Our result thus points to a new way to avoid the
quality problem of global symmetries.
