  Observations of hyper-luminous quasars at $z>6$ reveal the rapid growth of
supermassive black holes (SMBHs $>10^9 \rm M_{\odot}$) whose origin is still
difficult to explain. Their progenitors may have formed as remnants of massive,
metal free stars (light seeds), via stellar collisions (medium-weight seeds)
and/or massive gas clouds direct collapse (heavy seeds). In this work we
investigate for the first time the relative role of these three seed
populations in the formation of $z>6$ SMBHs within an Eddington-limited gas
accretion scenario. To this aim, we implement in our semi-analytical
data-constrained model a statistical description of the spatial fluctuations of
Lyman-Werner (LW) photo-dissociating radiation and of metal/dust enrichment.
This allows us to set the physical conditions for BH seeds formation, exploring
their relative birth rate in a highly biased region of the Universe at $z>6$.
We find that the inclusion of medium-weight seeds does not qualitatively change
the growth history of the first SMBHs: although less massive seeds ($<10^3 \rm
M_\odot$) form at a higher rate, the mass growth of a $\sim 10^9 \rm M_\odot$
SMBH at $z<15$ is driven by efficient gas accretion (at a sub-Eddington rate)
onto its heavy progenitors ($10^5 \rm M_\odot$). This conclusion holds
independently of the critical level of LW radiation and even when medium-weight
seeds are allowed to form in higher metallicity galaxies, via the so-called
super-competitive accretion scenario. Our study suggests that the genealogy of
$z \sim 6$ SMBHs is characterized by a rich variety of BH progenitors, which
represent only a small fraction ($< 10 - 20\%$) of all the BHs that seed
galaxies at $z > 15$.
