Within ten years, flat band (FB) superconductivity has gained a huge interest
for its remarkable features and connection to quantum geometry. We investigate
the superconductivity in a FB system whose orbitals are inequivalent and in
which the gap and the quantum metric are tunable. The key feature of the
present theoretical study is to show a unique and simple constrain relation
that pairings obey. Furthermore, pairings and superfluid weight in partially
filled FB are shown to be controlled by those of the half-filled lattice. We
argue that the geometry of the lattice or the complexity of the hopping terms
have no impact on the features revealed in this work as far as the system is
bipartite.Comment: 9 pages, 6 figure