  The "direct collapse" scenario has emerged as a promising evolutionary track
for the formation of supermassive black holes early in the Universe. In an
idealized version of such a scenario, a uniformly rotating supermassive star
spinning at the mass-shedding (Keplerian) limit collapses gravitationally after
it reaches a critical configuration. Under the assumption that the gas is
dominated by radiation pressure, this critical configuration is characterized
by unique values of the dimensionless parameters $J/M^2$ and $R_p/M$, where $J$
is the angular momentum, $R_p$ the polar radius, and $M$ the mass. Motivated by
a previous perturbative treatment we adopt a fully nonlinear approach to
evaluate the effects of gas pressure on these dimensionless parameters for a
large range of masses. We find that gas pressure has a significant effect on
the critical configuration even for stellar masses as large as $M \simeq 10^6
M_{\odot}$. We also calibrate two approximate treatments of the gas pressure
perturbation in a comparison with the exact treatment, and find that one
commonly used approximation in particular results in increasing deviations from
the exact treatment as the mass decreases, and the effects of gas pressure
increase. The other approximation, however, proves to be quite robust for all
masses $M \gtrsim 10^4 M_{\odot}$.
