  We study the quasiparticle band gap of the hybrid organic-inorganic lead
halide perovskite CH$_3$NH$_3$PbI$_3$, using many-body perturbation theory
based on the $GW$ approximation. We perform a systematic analysis of the band
gap sensitivity to relativistic spin-orbit effects, to the description of
semicore Pb-5$d$ and I-4$d$ electrons, and to the starting Kohn-Sham
eigenvalues. We find that the inclusion of semicore states increases the
calculated band gap by 0.2 eV, and self-consistency on the quasiparticle
eigenvalues using a scissor correction increases the band gap by 0.4 eV with
respect to the $G_0W_0$ result. These findings allow us to resolve an
inconsistency between previously reported $GW$ calculations for
CH$_3$NH$_3$PbI$_3$. Our most accurate band gap is 1.65 eV, and is in good
agreement with the measured optical gap after considering a small excitonic
shift as determined in experiments.
