  We investigate the thermodynamic geometry of the quark-meson model at finite
temperature, $T$, and quark number chemical potential, $\mu$. We extend
previous works by the inclusion of fluctuations exploiting the functional
renormalization group approach. We use recent developments to recast the flow
equation into the form of an advection-diffusion equation. We adopt the local
potential approximation for the effective average action. We focus on the
thermodynamic curvature, $R$, in the $(\mu,T)$ plane, in proximity of the
chiral crossover, up to the critical point of the phase diagram. We find that
the inclusion of fluctuations results in a smoother behavior of $R$ near the
chiral crossover. Moreover, for small $\mu$, $R$ remains negative, signaling
the fact that bosonic fluctuations reduce the capability of the system to
completely overcome the fermionic statistical repulsion of the quarks. We
investigate in more detail the small $\mu$ region by analyzing a system in
which we artificially lower the pion mass, thus approaching the chiral limit in
which the crossover is actually a second order phase transition. On the other
hand, as $\mu$ is increased and the critical point is approached, we find that
$R$ is enhanced and a sign change occurs, in agreement with mean field studies.
Hence, we completely support the picture that $R$ is sensitive to a crossover
and a phase transition, and provides information about the effective behavior
of the system at the phase transition.
