  The effect of a periodic pinning array on the vortex state in a 2D
superconductor at low temperatures is studied within the framework of the
Ginzburg-Landau approach. It is shown that attractive interaction of vortex
cores to a commensurate pin lattice stabilizes vortex solid phases with long
range positional order against violent shear fluctuations. Exploiting a simple
analytical method, based on the Landau orbitals description, we derive a rather
detailed picture of the low temperatures vortex state phase diagram. It is
predicted that for sufficiently clean samples application of an artificial
periodic pinning array would enable one to directly detect the intrinsic shear
stiffness anisotropy characterizing the ideal vortex lattice.
