  We study the superfluid state of atomic fermions in a tunable optical kagome
lattice motivated by recent experiments. We show that imposed superflow induces
spatial modulations in the density and order parameter of the pair condensate
and leads to a charge modulated superfluid state analogous to a supersolid
state. The spatial modulations in the superfluid emerge due to the geometric
effect of the kagome lattice that introduces anisotropy in hopping amplitudes
of fermion pairs in the presence of superflow. We also study superflow
instabilities and find that the critical current limited by the dynamical
instability is quite enhanced due to the large density of states associated
with the flatband. The charge modulated superfluid state can sustain high
temperatures close to the transition temperature that is also enhanced due to
the flatband, and is therefore realizable in experiments.
