  We study quantum vortex states of strongly interacting bosons in a
two-dimensional rotating optical lattice. The system is modeled by Bose-Hubbard
Hamiltonian with rotation. We consider lattices of different geometries, such
as square, rectangular and triangular. Using numerical exact diagonalization
method we show how the rotation introduces vortex states of different
ground-state symmetries and the transition between these states at discrete
rotation frequencies. We show how the geometry of the lattice plays crucial
role in determining the maximum number of vortex states as well as the general
characteristics of these states such as, the average angular momentum $<L_z>$,
the current at the perimeter of the lattice, phase winding, the relation
between the maximum phase difference, the maximum current and also the
saturation of the current between the two neighboring lattice points. The
effect of the two- and three-body interactions between the particles, both
attractive and repulsive, also depends on the geometry of the lattice as the
current flow or the lattice current depends on the interactions. We also
consider the effect of the spatial inhomogeneity introduced by the presence of
an additional confining harmonic trap potential. It is shown that the curvature
of the trap potential and the position of the minimum of the trap potential
with respect to the axis of rotation or the center of the lattice have a
significant effect on the general characteristics these vortex states.
