Abelian Lagrangian algebraic geometry

This paper begins a detailed exposition of a geometric approach to quantization, which combines the methods of algebraic and Lagrangian geometry. Given a prequantization U(1)-bundle L on a symplectic manifold M, we introduce an infinite-dimensional Kaehler manifold Phw of half-weighted Planck cycles. With every Kaehler polarization on M we canonically associate a map Phw→γH0(M,L) to the space of holomorphic sections of the prequantization bundle. We show that this map has a constant Kaehler angle and its 'twisting' to a holomorphic map is the Borthwick-Paul-Uribe map. The simplest non-trivial illustration of all these constructions is provided by the theory of Legendrian knots in S3