The metaplectic correction in geometric quantization

Let P be a polarization on a symplectic manifold for which there exists a metalinear frame bundle. We show that for any other compatible polarization P′ there exists a unique metalinear frame bundle such that the BKS-pairing is well defined. This means that we do not need the metaplectic frame bundle (nor a positivity condition on P) to achieve this goal, and thus the name “metalinear correction” would be more appropriate than the commonly used name “metaplectic correction”.