  The density-of-states at the Fermi energy, $N(E_F)$, is low in doped
superconducting semiconductors and high-$T_C$ cuprates. This contrasts with the
common view that superconductivity requires a large electron-boson coupling
$\lambda$ and therefore also a large $N(E_F)$. However, the generic Fermi
surfaces (FS) of these systems are relatively simple.
  Here is presented arguments showing that going from a 3-dimensional
multi-band FS to a 2-dimensional and simple FS is energetically favorable to
superconductivity. Nesting and few excitations of bosons compensate for a low
$N(E_F)$. The typical behavior of the 2-dimensional FS for cuprates, and small
3-dimensional FS pockets in doped semiconductors and diamond, leads to $T_C$
variations as a function of doping in line with what has been observed. Diamond
is predicted to attain higher $T_C$ from electron doping than from hole doping,
while conditions for superconductivity in Si and Ge are less favorable. A
high-$T_C$ material should ideally have few flat and parallel FS sheets with a
reasonably large $N(E_F)$.
