  Supermassive stars (SMSs) collapsing via the general-relativistic (GR)
instability are invoked as the possible progenitors of supermassive black
holes. Their mass and angular momentum at the onset of the instability are key
in many respects, in particular regarding the possibility for observational
signatures of direct collapse. Here, we study the stability of rotating,
rapidly accreting SMSs against GR and derive the properties of these stars at
death. On the basis of hylotropic structures, relevant for rapidly accreting
SMSs, we define rotation profiles under the assumption of local angular
momentum conservation in radiative regions, which allows for differential
rotation. We find that rotation favours the stability of rapidly accreting SMSs
as soon as the accreted angular momentum represents a fraction f > 0.1% of the
Keplerian angular momentum. For f = 0.3%-0.5% the maximum masses consistent
with GR stability are increased by an order of magnitude compared to the
non-rotating case. For f = 1%, the GR instability cannot be reached if the
stellar mass does not exceed 10^7-10^8 Msun. These results imply that, like in
the non-rotating case, the final masses of the progenitors of direct collapse
black holes range in distinct intervals depending on the scenario considered:
10^5 Msun < M < 10^6 Msun for primordial atomically cooled haloes; 10^6 Msun <
M < 10^9 Msun for metal-rich galaxy mergers. The models suggest that the
centrifugal barrier is inefficient to prevent the direct formation of a
supermassive black hole at the collapse of a SMS. Moreover, the conditions of
galaxy mergers appear as more favorable than those of atomically cooled haloes
for detectable gravitational wave emission and ultra-long gamma-ray bursts at
black hole formation.
