  We study collective modes and superfluidity of spin-1 bosons with
antiferromagnetic interactions in an optical lattice based on the
time-dependent Ginzburg-Landau (TDGL) equation derived from the spin-1
Bose-Hubbard model. Specifically, we examine the stability of supercurrents in
the polar phase in the vicinity of the Mott insulating phase with even filling
factors. Solving the linearized TDGL equation, we obtain gapless spin-nematic
modes and gapful spin-wave modes in the polar phase that arise due to the
breaking of $S^2$ symmetry in spin space. Supercurrents exhibit dynamical
instabilities induced by growing collective modes. In contrast to the
second-order phase transition, the critical momentum of mass currents is finite
at the phase boundary of the first-order superfluid-Mott insulator (SF-MI)
phase transition. Furthermore, the critical momentum remains finite throughout
the metastable SF phase and approaches zero towards the phase boundary, at
which the metastable SF state disappears. We also study the stability of spin
currents motivated by recent experiments for spinor gases. The critical
momentum of spin currents is found to be zero, where a spin-nematic mode causes
the dynamical instability. We investigate the origin of the zero critical
momentum of spin currents and find it attributed to the fact that the polar
state becomes energetically unstable even in the presence of an infinitesimal
spin current. We discuss implications of the zero critical momentum of spin
currents for the stability of the polar state.
