  We present a quantitative study on the properties at death of fast-rotating
massive stars evolved at low-metallicity, objects that are proposed as likely
progenitors of long-duration gamma-ray bursts (LGRBs). We perform 1D+rotation
stellar-collapse simulations on the progenitor models of Woosley & Heger (2006)
and critically assess their potential for the formation of a black hole and a
Keplerian disk (namely a collapsar) or a proto-magnetar. We note that
theoretical uncertainties in the treatment of magnetic fields and the
approximate handling of rotation compromises the accuracy of stellar-evolution
models. We find that only the fastest rotating progenitors achieve sufficient
compactness for black-hole formation while the bulk of models possess a core
density structure typical of garden-variety core-collapse supernova (SN)
progenitors evolved without rotation and at solar metallicity. Of the models
that do have sufficient compactness for black-hole formation, most of them also
retain a large amount of angular momentum in the core, making them prone to a
magneto-rotational explosion, therefore preferentially leaving behind a
proto-magnetar. A large progenitor angular-momentum budget is often the sole
criterion invoked in the community today to assess the suitability for
producing a collapsar. This simplification ignores equally important
considerations such as the core compactness, which conditions black-hole
formation, the core angular momentum, which may foster a magneto-rotational
explosion preventing black-hole formation, or the metallicity and the residual
envelope mass which must be compatible with inferences from observed LGRB/SNe.
Our study suggests that black-hole formation is non trivial, that there is room
for accommodating both collapsars and proto-magnetars as LGRB progenitors,
although proto-magnetars seem much more easily produced by current
stellar-evolutionary models.
