Direct collapse black hole ( DCBH ) formation with mass \gtrsim 10 ^ { 5 } ~ { } M _ { \odot } is a promising scenario for the origin of high-redshift supermassive black holes .
It has usually been supposed that the DCBH can only form in the primordial gas since the metal enrichment enhances the cooling ability and causes the fragmentation into smaller pieces .
What actually happens in such an environment , however , has not been explored in detail .
Here , we study the impact of the metal enrichment on the clouds , conducting hydrodynamical simulations to follow the cloud evolution in cases with different degree of the metal enrichment Z / Z _ { \odot } = 10 ^ { -6 } -10 ^ { -3 } .
Below Z / Z _ { \odot } = 10 ^ { -6 } , metallicity has no effect and supermassive stars form along with a small number of low-mass stars .
With more metallicity Z / Z _ { \odot } \ga 5 \times 10 ^ { -6 } , although the dust cooling indeed promotes fragmentation of the cloud core and produces about a few thousand low-mass stars , the accreting flow preferentially feeds the gas to the central massive stars , which grows supermassive as in the primordial case .
We term this formation mode as the super competitive accretion , where only the central few stars grow supermassive while a large number of other stars are competing for the gas reservoir .
Once the metallicity exceeds 10 ^ { -3 } ~ { } Z _ { \odot } and metal-line cooling becomes operative , the central star can not grow supermassive due to lowered accretion rate .
Supermassive star formation by the super competitive accretion opens up a new window for seed BHs , which relaxes the condition on metallicity and enhances the seed BH abundance .