  We derive effective Hamiltonians for lattice bosons with strong geometrical
frustration of the kinetic energy by projecting the interactions on the flat
lowest Bloch band. Specifically, we consider the Bose Hubbard model on the one
dimensional sawtooth lattice and the two dimensional kagome lattice. Starting
from a strictly local interaction the projection gives rise to effective
long-range terms stabilizing a supersolid phase at densities above nu_c=1/9 of
the kagome lattice. In the sawtooth lattice on the other hand we show that the
solid order, which exists at the magic filling nu_c=1/4, is unstable to further
doping. The universal low-energy properties at filling 1/4+delta nu are
described by the well known commensurate-incommensurate transition. We support
the analytic results by detailed numerical calculations using the Density
Matrix Renormalization Group and exact diagonalization. Finally, we discuss
possible realizations of the models using ultracold atoms as well as frustrated
quantum magnets in high magnetic fields. We compute the momentum distribution
and the noise correlations, that can be extracted from time of flight
experiments or neutron scattering, and point to signatures of the unique
supersolid phase of the kagome lattice.

Strongly Correlated Electrons;Quantum Gases