Does disk fragmentation prevent the formation of supermassive stars in
  protogalaxies?
  Supermassive stars (SMSs; >10^5 Msun) formed in the first protogalaxies with
virial temperature T_vir>10^4 K are expected to collapse into seeds of
supermassive black hole (SMBHs) in the high-redshift universe (z>7).
Fragmentation of the primordial gas is, however, a possible obstacle to SMS
formation. We discuss the expected properties of a compact, metal-free,
marginally unstable nuclear protogalactic disk, and the fate of the clumps
formed in the disk by gravitational instability. Interior to a characteristic
radius R_f=few*10^{-2} pc, the disk fragments into massive clumps with M_c~30
Msun. The clumps grow via accretion and migrate inward rapidly on a timescale
of ~10^4 yr, which is comparable or shorter than the Kelvin-Helmholz time >10^4
yr. Some clumps may evolve to zero-age main sequence stars and halt gas
accretion by radiative feedback, but most of the clumps can migrate inward and
merge with the central protostar before forming massive stars. Moreover, we
found that dust-induced-fragmentation in metal-enriched gas does not modify
these conclusions unless Z> 3*10^{-4} Zsun, because clump migration below this
metallicity remains as rapid as in the primordial case. Our results suggest
that fragmentation of a compact, metal--poor disk can not prevent the formation
of a SMS.
