  The inflationary origin of primordial black holes (PBHs) relies on a large
enhancement of the power spectrum $\Delta_\zeta$ of the curvature fluctuation
$\zeta$ at wavelengths much shorter than those of the cosmic microwave
background anisotropies. This is typically achieved in models where $\zeta$
evolves without interacting significantly with additional (isocurvature) scalar
degrees of freedom. However, quantum gravity inspired models are characterized
by moduli spaces with highly curved geometries and a large number of scalar
fields that could vigorously interact with $\zeta$ (as in the cosmological
collider picture). Here we show that isocurvature fluctuations can mix with
$\zeta$ inducing large enhancements of its amplitude. This occurs whenever the
inflationary trajectory experiences rapid turns in the field space of the model
leading to amplifications that are exponentially sensitive to the total angle
swept by the turn, which induce characteristic observable signatures on
$\Delta_\zeta$. We derive accurate analytical predictions and show that the
large enhancements required for PBHs demand non-canonical kinetic terms in the
action of the multifield system.
