On the pairing of polarizations

Abstract

IfF is a positive Lagrangian sub-bundle of a symplectic vector bundle (E, ω) we show by elementary means that the Chern classes ofF are determined by ω. The notions of metaplectic structure for (E, ω), metalinear structure forF and square root ofKF, the canonical bundle ofF are shown to be essentially the same. IfF andG are two positive Lagrangian sub-bundles with\(F \cap \bar G = D^\mathbb{C} \), we define a pairing ofKF andKG into the bundleD−2(D) of densities of order −2 onD. This is the square of Blattner's half-form pairing and so characterizes the latter up to a sign.