  We theoretically investigate how the Berry curvature, which arises in
multi-band structures when the electrons can be described by an effective
single-band Hamiltonian, affects the superconducting properties of
two-dimensional electronic systems. Generically the Berry curvature is coupled
to electric fields beyond those created by the periodic crystal potential. A
potential source of such electric fields, which vary slowly on the lattice
scale, is the mutual interaction between the electrons. We show that the Berry
curvature provides additional terms in the Hamiltonian describing interacting
electrons within a single band. When these terms are taken into account in the
framework of the usual BCS weak-coupling treatment of a generic attractive
interaction that allows for the formation of Cooper pairs, the coupling
constant is modified. In pure singlet and triplet superconductors, we find that
the Berry curvature generally lowers the coupling constant and thus the
superconducting gap and the critical temperature as a function of doping. From
an experimental point of view, a measured deviation from the expected BCS
critical temperature upon doping, e.g. in doped two-dimensional
transition-metal dichalcogenides, may unveil the strength of the Berry
curvature.
