  Failed supernovae (FSN) are a possible channel for the formation of heavy
stellar-mass black holes ($M_{ BH}>\sim 30$ M$_\odot$). However, the effects of
metallicity, rotation and magnetic field on the islands of explodabilty of
massive stars are not clear. Here, we simulate the stellar structure and
evolution in the mass range between 6 and 55 $M_{\odot}$ with different initial
rotational velocities, metallicities, and magnetic fields from zero-age main
sequence (ZAMS) to pre-collapse. We find that the rapid rotating stars can
remain lower $\rm ^{12}C$ mass fraction at the time of C ignition, which allows
the transition, from convective carbon burning to radiative burning, to occur
at lower $M_{\rm ZAMS}$ than those from stars without rotation. However, the
rapid rotation is unfavorable for FSN occurring but is conducive to long
gamma-ray bursts (lGRBs) because it results in the specific angular momentum in
the CO core is greater than the last stable orbit at core collapse. The
increasing metallicity does not affect FSN islands, but high metallicity
inhibits rotational mixing and is unfavorable for producing lGRBs. A magnetic
field can constrain the mass-loss rate even for rapid rotating stars, resulting
in higher mass at pre-collapse. The magnetic braking triggered by the magnetic
field can reduce the rotation velocity for high-metallicity models, which
decreases the specific angular momentum in the CO core and is favorable for FSN
occurring. We suggest that the heavy-mass black holes detected by LIGO may
originate from rapidly rotating massive stars with strong magnetic fields,
rather than those with very low metallicity.
