  In the absence of H_2 molecules, the primordial gas in early dark matter
halos with virial temperatures just above T_vir >~ 10^4 K cools by collisional
excitation of atomic H. Although it cools efficiently, this gas remains
relatively hot, at a temperature near T ~ 8000 K, and consequently might be
able to avoid fragmentation and collapse directly into a supermassive black
hole (SMBH). In order for H_2--formation and cooling to be strongly suppressed,
the gas must be irradiated by a sufficiently intense ultraviolet (UV) flux. We
performed a suite of three--dimensional hydrodynamical adaptive mesh refinement
(AMR) simulations of gas collapse in three different protogalactic halos with
T_vir >~ 10^4 K, irradiated by a UV flux with various intensities and spectra.
We determined the critical specific intensity, Jcrit, required to suppress H_2
cooling in each of the three halos. For a hard spectrum representative of
metal--free stars, we find (in units of 10^{-21} erg s^{-1} Hz^{-1} sr^{-1}
cm^{-2}) 10^4<Jcrit<10^5, while for a softer spectrum, which is characteristic
of a normal stellar population, and for which H^{-} --dissociation is
important, we find 30<Jcrit<300. These values are a factor of 3--10 lower than
previous estimates. We attribute the difference to the higher, more accurate
H_2 collisional dissociation rate we adopted. The reduction in Jcrit
exponentially increases the number of rare halos exposed to super--critical
radiation. When H_2 cooling is suppressed, gas collapse starts with a delay,
but it ultimately proceeds more rapidly. The infall velocity is near the
increased sound speed, and an object as massive as M ~ 10^5 solar mass may form
at the center of these halos, compared to the M ~ 10^2 solar mass stars forming
when H_2--cooling is efficient.
