  We explore multiband effects on d-wave superconducting instabilities driven
by electron-electron interactions. Our models on the two-dimensional square
lattice consist of a main band with an extended Fermi surface and predominant
weight from $d_{x^2-y^2}$ orbitals, whose orbital character is influenced by
the admixture of other energetically neighbored orbitals. Using a functional
renormalization group description of the superconducting instabilities of the
system and different levels of approximations, we study how the energy scale
for pairing and hence the critical temperature is affected by the band
structure. We find that a reduction of orbital admixture as a function of the
orbital energies can cause a $T_c$ enhancement although the Fermi surface
becomes more curved and hence less favorable for antiferromagnetic spin
fluctuations. While our study does not allow a quantitative understanding of
the $T_c$ differences in realistic high-$T_c$ cuprate systems, it may reveal an
underlying mechanism contributing to the actual material trends.
