nLab Kähler polarization

Idea

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

• polarization – Kähler polarizations

• geometric quantization – Quantum state space as space of polarized sections

Definition

In terms of $G$-structures

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).

In terms of distributions in the complexified tangent bundle

(…)

$$\mathcal{P} \subset T_{\mathbb{C}} T X$$

$$\mathcal{P} \cap \overline{\mathcal{P}} = 0$$

$$\mathcal{P} = \left\{J v - i v \;|\; v \in T X \right\}$$

Related concepts

• real polarization

• p-convex polarization

• almost Kähler quantization

• spin^c quantization

• metaplectic quantization

References

Discussion of the functoriality of Kähler polarization quantization with respect to the choice of metaplectically corrected Kähler structure is in section 3 of

• Lauridsen, Aspects of quantum mathematics – Hitchin connections and the AJ conjecture, PhD thesis Aarhus 2010 (pdf)