  We explore the formation process of a black hole (BH) through the
pair-instability collapse of a rotating Population III very massive star in
axisymmetric numerical relativity. As the initial condition, we employ a
progenitor star which is obtained by evolving a rapidly rotating zero-age main
sequence (ZAMS) star with mass $320M_\odot$ until it reaches a pair instability
region. We find that for such rapidly rotating model, a fraction of the mass,
$\sim 10M_\odot$, forms a torus surrounding the remnant BH of mass $\sim
130M_\odot$ and an outflow is driven by a hydrodynamical effect. We also
perform simulations, artificially reducing the initial angular velocity of the
progenitor star, and find that only a small or no torus is formed and no
outflow is driven. We discuss the possible evolution scenario of the remnant
torus for the rapidly rotating model by considering the viscous and
recombination effects and show that if the energy of $\sim 10^{52}$ erg is
injected from the torus to the envelope, the luminosity and timescale of the
explosion could be of the orders of $10^{43}$ erg/s and yrs, respectively. We
also point out the possibility for observing gravitational waves associated
with the BH formation for the rapidly rotating model by ground-based
gravitational-wave detectors.
