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Understanding the link between massive (30 M_) stellar black holes (BHs) and their progenitor stars is a crucial step to interpret observations of gravitational-wave events. In this paper, we discuss the final fate of very massive stars (VMSs), with zero-age main sequence (ZAMS) mass >150 M_, accounting for pulsational pair-instability supernovae (PPISNe) and for pair-instability supernovae (PISNe). We describe an updated version of our population synthesis code SEVN, in which we added stellar evolution tracks for VMSs with ZAMS mass up to 350 M_ and we included analytical prescriptions for PPISNe and PISNe. We use the new version of SEVN to study the BH mass spectrum at different metallicity Z, ranging from Z=2. 0 10^-4 to Z=2. 0 10^-2. The main effect of PPISNe and PISNe is to favour the formation of BHs in the mass range of the first gravitational-wave event (GW150914), while they prevent the formation of remnants with mass 60 - 120 M_. In particular, we find that PPISNe significantly enhance mass loss of metal-poor (Z 2. 0 10^-3) stars with ZAMS mass 60 Mₙ₀₌ₒ/M_ 125. In contrast, PISNe become effective only for moderately metal-poor (Z<8. 0 10^-3) VMSs. VMSs with M ₙ₀₌ₒ220 M_ and Z<10^-3 do not undergo PISNe and form intermediate-mass BHs (IMBHs, with mass 200 M_) via direct collapse.
Spera et al. (Wed,) studied this question.