  We study the phase diagram of ultra-cold bosonic polar molecules loaded on a
two-dimensional optical lattice of hexagonal symmetry controlled by external
electric and microwave fields. Following a recent proposal in Nature Physics
\textbf{3}, 726 (2007), such a system is described by an extended Bose-Hubbard
model of hard-core bosons, that includes both extended two- and three-body
repulsions. Using quantum Monte-Carlo simulations, exact finite cluster
calculations and the tensor network renormalization group, we explore the rich
phase diagram of this system, resulting from the strongly competing nature of
the three-body repulsions on the honeycomb lattice. Already in the classical
limit, they induce complex solid states with large unit cells and macroscopic
ground state degeneracies at different fractional lattice fillings. For the
quantum regime, we obtain effective descriptions of the various phases in terms
of emerging valence bond crystal states and quantum dimer models. Furthermore,
we access the experimentally relevant parameter regime, and determine the
stability of the crystalline phases towards strong two-body interactions.
