  Context. The seeds of the supermassive black holes (SMBHs) with masses of
~10^9 M_Sun observed already at z ~ 6 may have formed through the direct
collapse of primordial gas in T_vir >~ 10^4 K halos, whereby the gas must stay
hot (~10^4 K) in order to avoid fragmentation.
  Aims. The interplay between magnetic fields, turbulence, and a UV radiation
background during the gravitational collapse of primordial gas in a halo is
explored; in particular, the possibilities for avoiding fragmentation are
examined.
  Methods. Using an analytical one-zone model, the evolution of a cloud of
primordial gas is followed from its initial cosmic expansion through
turnaround, virialization, and collapse up to a density of 10^7 cm-3.
  Results. It was found that in halos with no significant turbulence, the
critical UV background intensity (J_21^crit) for keeping the gas hot is lower
by a factor ~10 for an initial comoving magnetic field B_0 ~ 2 nG than for the
zero-field case, and even lower for stronger fields. In turbulent halos,
J_21^crit is found to be a factor ~10 lower than for the
zero-field-zero-turbulence case, and the stronger the turbulence (more massive
halo and/or stronger turbulent heating) the lower J_21^crit.
  Conclusions. The reduction in J_21^crit is particularly important, since it
exponentially increases the number of halos exposed to a supercritical
radiation background.
