  The polar Kerr effect and the closely related anomalous charge Hall effect
are among the most distinguishing signatures of the superconducting state in
Sr$_2$RuO$_4$, as well as in several other compounds. These effects are often
thought to be derived from chiral superconducting pairing, and different
mechanisms have been invoked for the explanation. However, the intrinsic
mechanisms proposed previously often involve unrealistically strong interband
Cooper pairing. In this study we show that, even without interband pairing,
nonunitary superconducting states can support intrinsic anomalous charge Hall
effect, thanks to the quantum geometric properties of the Bloch electrons. The
key here is to have a normal-state spin Hall effect, for which a nonzero
spin-orbit coupling is essential. A finite charge Hall effect then naturally
arises at the onset of a spin-polarized nonunitary superconducting pairing. It
depends on both the spin polarization and the normal-state electron Berry
curvature, the latter of which is the imaginary part of the quantum geometric
tensor of the Bloch states. Applying our results to the weakly-paired
Sr$_2$RuO$_4$ we conclude that, if the reported Kerr effect is of intrinsic
origin, the superconducting state is most likely nonunitary and has odd-parity.
Our theory may be generalized to other superconductors that exhibit polar Kerr
effect.
