  We study secular stability against a quasi-radial oscillation for rigidly
rotating stars with soft equations of state in general relativity. The
polytropic equations of state with polytropic index $n$ between 3 and 3.05 are
adopted for modeling the rotating stars. The stability is determined in terms
of the turning-point method. It is found that (i) for $n \agt 3.04$, all the
rigidly rotating stars are unstable against the quasi-radial oscillation and
(ii) for $n \agt 3.01$, the nondimensional angular momentum parameter $q \equiv
cJ/GM^2$ (where $J$, $M$, $G$, and $c$ denote the angular momentum, the
gravitational mass, the gravitational constant, and the speed of light,
respectively) for all marginally stable rotating stars is larger than unity. A
semi-analytic calculation is also performed, and good agreement with the
numerical results is confirmed. The final outcome after axisymmetric
gravitational collapse of rigidly rotating and marginally stable massive stars
with $q > 1$ is predicted, assuming that the rest-mass distribution as a
function of the specific angular momentum is preserved and that the pressure
never halt the collapse. It is found that even for $1 < q \alt 2.5$, a black
hole may be formed as a result of the collapse, but for $q \agt 2.5$, the
significant angular momentum will prevent the direct formation of a black hole.
