  By using the dual vortex method (DVM), we develop systematically a simple and
effective scheme to use the vortex degree of freedoms on dual lattices to
characterize the symmetry breaking patterns of the boson insulating states in
the direct lattices. Then we apply our scheme to study quantum phases and phase
transitions in an extended boson Hubbard model slightly away from $ 1/3 $ ($
2/3 $) filling on frustrated lattices such as triangular and kagome lattice. In
a triangular lattice at $ 1/3 $, we find a X-CDW, a stripe CDW phase which was
found perviously by a density operator formalism (DOF). Most importantly, we
also find a new CDW-VB phase which has both local CDW and local VB orders, in
sharp contrast to a bubble CDW phase found previously by the DOF. In the Kagome
lattice at $ 1/3 $, we find a VBS phase and a 6 fold-CDW phase. Most
importantly, we also identify a CDW-VB phase which has both local CDW and local
VB orders which was found in previous QMC simulations. By analyzing carefully
the saddle point structures of the dual gauge fields in the translational
symmetry breaking sides and pushing the effective actions slightly away from
the commensurate filling $ f=1/3 $($ 2/3 $), we classified all the possible
types of supersolids and analyze their stability conditions. In a triangular
lattice, there are X-CDW supersolid, stripe CDW supersolid, but absence of any
valence bond supersolid (VB-SS). There are also a new kind of supersolid:
CDW-VB supersolid. In a Kagome lattice, there are 6 fold-CDW supersolid, stripe
CDW supersolid, but absence of any valence bond supersolid (VB-SS). There are
also a new kind of supersolid: CDW-VB supersolid. Implications on QMC
simulations with both nearest neighbor and next nearest neighbor interactions
in both lattices are given.
