  Predicting the fate of an interacting system in the limit where the
electronic bandwidth is quenched is often highly non-trivial. The complex
interplay between interactions and quantum fluctuations driven by the band
geometry can drive competition between various ground states, such as charge
density wave order and superconductivity. In this work, we study an electronic
model of topologically-trivial flat bands with a continuously tunable
Fubini-Study metric in the presence of on-site attraction and nearest-neighbor
repulsion, using numerically exact quantum Monte Carlo simulations. By varying
the electron filling and the minimal spatial extent of the localized flat-band
Wannier wavefunctions, we obtain a number of intertwined orders. These include
a phase with coexisting charge density wave (CDW) order and superconductivity,
i.e., a supersolid. In spite of the non-perturbative nature of the problem, we
identify an analytically tractable limit associated with a 'small' spatial
extent of the Wannier functions and derive a low-energy effective Hamiltonian
that can well describe our numerical results. We also provide unambiguous
evidence for the violation of any putative lower bound on the zero-temperature
superfluid stiffness in geometrically non-trivial flat bands.
