geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
physics, mathematical physics, philosophy of physics
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geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
A Kähler polarization of a symplectic manifold is a polarization by a compatible Kähler manifold structure.
Given a prequantization of a Kähler polarized symplectic manifold by a holomorphic line bundle, then the polarized sections are the holomorphic sections.
Hence the concept of Kähler polarization is that special case of polarization which connects most intimately the symplectic geometry to complex analytic geometry. The generalization of this from complex analytic geometry to more general algebraic geometry is the concept of a polarized algebraic variety.
For more see at
In terms of G-structures this means that it is a lift from an integrable $Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})$-G-structure (integrable almost symplectic structure) to a first-order integrable $U(n) \hookrightarrow Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})$-structure (first-order integrable almost Hermitian structure).
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Discussion of the functoriality of Kähler polarization quantization with respect to the choice of metaplectically corrected Kähler structure is in section 3 of