  We perform axisymmetric simulations for gravitational collapse of a massive
iron core to a black hole in full general relativity. The iron cores are
modeled by $\Gamma = 4/3$ equilibrium polytrope for simplicity and a parametric
equation of state is adopted during the collapse. Performing simulations for a
wide variety of initial conditions changing the mass, the angular momentum, the
rotational velocity profile of the core, and the parameters of the equations of
state (which are chosen so that the maximum mass of the cold spherical
polytrope is $\approx 1.6M_{\odot}$), a criterion for the prompt black hole
formation is clarified in terms of them. It is found that (i) with the increase
of the thermal energy generated by shocks, the threshold mass for the prompt
black hole formation is increased by 20--40%, (ii) the rotational centrifugal
force increases the threshold mass by $\alt 25%$, (iii) with the increase of
the degree of differential rotation, the threshold mass is also increased, and
(iv) the amplification factors shown in the results (i)--(iii) depend
sensitively on the equation of state. We also find that the collapse dynamics
and the structure of the shock formed at the bounce depend strongly on the
stiffness of the adopted equation of state. In particular, as a new feature, a
strong bipolar explosion is observed for the collapse of rapidly rotating iron
cores with an equation of state which is stiff in subnuclear density and soft
in supranuclear density. Gravitational waves are computed in terms of a
quadrupole formula. It is also found that the waveform depends sensitively on
the equations of state.
