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 $U (1)$-bundle $L$ on a symplectic manifold $M$, we introduce an infinite-dimensional Kähler manifold $\mathscr P^{\mathrm{hw}}$ of half-weighted Planck cycles. With every Kähler polarization on $M$ we canonically associate a map $\mathscr P^{\mathrm{hw}}\overset{\gamma}{\to}H^{0}(M,L)$ to the space of holomorphic sections of the prequantization bundle. We show that this map has a constant Kähler 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 $S^3$.