  Observations of high redshift quasars at $z>6$ indicate that they harbor
supermassive black holes (SMBHs) of a billion solar masses. The direct collapse
scenario has emerged as the most plausible way to assemble SMBHs. The nurseries
for the direct collapse black holes are massive primordial halos illuminated
with an intense UV flux emitted by population II (Pop II) stars. In this study,
we compute the critical value of such a flux ($J_{21}^{\rm crit}$) for
realistic spectra of Pop II stars through three-dimensional cosmological
simulations. We derive the dependence of $J_{21}^{\rm crit}$ on the radiation
spectra, on variations from halo to halo, and on the impact of X-ray
ionization. Our findings show that the value of $J_{21}^{\rm crit}$ is a few
times $\rm 10^4$ and only weakly depends on the adopted radiation spectra in
the range between $T_{\rm rad}=2 \times 10^4-10^5$ K. For three simulated halos
of a few times $\rm 10^{7}$~M$_{\odot}$, $J_{21}^{\rm crit}$ varies from $\rm 2
\times 10^4 - 5 \times 10^4$. The impact of X-ray ionization is almost
negligible and within the expected scatter of $J_{21}^{\rm crit}$ for
background fluxes of $J_{\rm X,21} \leq 0.1$. The computed estimates of
$J_{21}^{\rm crit}$ have profound implications for the quasar abundance at
$z=10$ as it lowers the number density of black holes forming through an
isothermal direct collapse by a few orders of magnitude below the observed
black holes density. However, the sites with moderate amounts of $\rm H_2$
cooling may still form massive objects sufficient to be compatible with
observations.
