Phase diagram for the final fates of massive Pop III stars, in the plane of the mass and the fraction of the Keplerian value of the equatorial rotational velocity on the zero-age main sequence. The dotted line denotes the limit above which the rotational velocity at the surface would exceed the critical rotation velocity (v_crit in Eq. (2)). The thick solid line marks the boundary between the regimes for the chemically homogeneous evolution (CHE), and the non-CHE evolution (normal evolution or transitionary evolution; see Sect. 3). The regions for different final fates are divided by thin solid lines. Each acronym has the following meaning. SN IIP: Type IIP supernova, NS: neutron star, BH: black hole, SN II: type II supernova, PISN: pair-instability supernova, Puls. PISN: pulsational pair-instability supernova, GRB: gamma-ray bursts, HN: hypernova, SNIbc: type Ib or Ic supernova.