  We present a theoretical survey of ferromagnetic transition temperatures in
cubic (III,Mn)V semiconductors based on a model with $S=5/2$ local moments
exchange-coupled to itinerant holes in the host semiconductor valence band.
Starting from the simplest mean-field theory of this model, we estimate the
$T_c$ enhancement due to exchange and correlation in the itinerant-hole system,
and the $T_c$ suppression due to collective fluctuations of the ordered
moments. We show that high critical temperatures in these ferromagnetic
semiconductors require both the large magnetic susceptibility contribution from
the valence band's heavy holes and the large spin stiffness that results from
strong spin-orbit coupling in the valence band. Our calculations demonstrate
that this model for the ferromagnetism of these systems is fully consistent
with the room-temperature ferromagnetism reported for Mn doped nitrides.
