  The growth of the first super massive black holes (SMBHs) at z > 6 is still a
major challenge for theoretical models. If it starts from black hole (BH)
remnants of Population III stars (light seeds with mass ~ 100 Msun) it requires
super-Eddington accretion. An alternative route is to start from heavy seeds
formed by the direct collapse of gas onto a ~ 10^5 Msun BH. Here we investigate
the relative role of light and heavy seeds as BH progenitors of the first
SMBHs. We use the cosmological, data constrained semi-analytic model
GAMETE/QSOdust to simulate several independent merger histories of z > 6
quasars. Using physically motivated prescriptions to form light and heavy seeds
in the progenitor galaxies, we find that the formation of a few heavy seeds
(between 3 and 30 in our reference model) enables the Eddington-limited growth
of SMBHs at z > 6. This conclusion depends sensitively on the interplay between
chemical, radiative and mechanical feedback effects, which easily erase the
conditions that allow the suppression of gas cooling in the low metallicity gas
(Z < Zcr and JLW > Jcr). We find that heavy seeds can not form if dust cooling
triggers gas fragmentation above a critical dust-to-gas mass ratio (D > Dcr).
In addition, the relative importance of light and heavy seeds depends on the
adopted mass range for light seeds, as this dramatically affects the history of
cold gas along the merger tree, by both SN and AGN-driven winds.
