  The Fermi surface (FS) of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$
  (Bi2212) predicted by band theory displays Bi-related pockets around the
$(\pi,0)$ point, which have never been observed experimentally. We show that
when the effects of hole doping either by substituting Pb for Bi or by adding
excess O in Bi2212 are included, the Bi-O bands are lifted above the Fermi
energy ($E_F$) and the resulting first-principles FS is in remarkable accord
with measurements. With decreasing hole-doping the Bi-O bands drop below $E_F$
and the system self-dopes below a critical hole concentration. Computations on
other Bi- as well as Tl- and Hg-based compounds indicate that lifting of the
cation-derived band with hole doping is a general property of the electronic
structures of the cuprates.
