  Flat-band superconductivity has theoretically demonstrated the importance of
band topology to correlated phases. In two dimensions, the superfluid weight,
which determines the critical temperature through the
Berezinksii-Kosterlitz-Thouless criteria, is bounded by the Fubini-Study metric
at zero temperature. We show this bound is nonzero within flat bands whose
Wannier centers are obstructed from the atoms - even when they have identically
zero Berry curvature. Next, we derive general lower bounds for the superfluid
weight in terms of momentum space irreps in all 2D space groups, extending the
reach of topological quantum chemistry to superconducting states. We find that
the bounds can be naturally expressed using the formalism of real space
invariants (RSIs) that highlight the separation between electronic and atomic
degrees of freedom. Finally, using exact Monte Carlo simulations on a model
with perfectly flat bands and strictly local obstructed Wannier functions, we
find that an attractive Hubbard interaction results in superconductivity as
predicted by the RSI bound beyond mean-field. Hence, a nonzero superfluid
weight constitutes a nontrivial bulk property that distinguishes obstructed
bands from trivial bands in the presence of interactions.
