Supermassive stars ( SMSs ; \ga 10 ^ { 5 } ~ { } M _ { \odot } ) formed in the first protogalaxies with virial temperature T _ { vir } \ga 10 ^ { 4 } K are expected to collapse into seeds of supermassive black hole in the high-redshift Universe ( z \ga 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 disc , and the fate of the clumps formed in the disc by gravitational instability .
Interior to a characteristic radius R _ { f } = { few } \times 10 ^ { -2 } pc , the disc fragments into massive clumps with M _ { c } \sim 30 ~ { } M _ { \odot } .
The clumps grow via accretion and migrate inward rapidly on a time-scale of \sim 10 ^ { 4 } yr , which is comparable or shorter than the Kelvin-Helmholtz 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 \ga 3 \times 10 ^ { -4 } ~ { } Z _ { \odot } , because clump migration below this metallicity remains as rapid as in the primordial case .
Our results suggest that fragmentation of a compact , metal-poor disc can not prevent the formation of an SMS .