Abelian Lagrangian algebraic geometry

This paper begins a detailed exposition of a geometric approach to quantization, which is presented in a series of preprints ([23], [24], ...) and which combines the methods of algebraic and Lagrangian geometry. Given a prequantization -bundle  on a symplectic manifold , we introduce an infinite-dimensional Kahler manifold of half-weighted Planck cycles. With every Kahler polarization on  we canonically associate a map to the space of holomorphic sections of the prequantization bundle. We show that this map has a constant Kahler 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 .