[Paper Review] Convergence of Kähler to real polarizations on flag manifolds via toric degenerations

Mark Hamilton, Hiroshi Konno|arXiv (Cornell University)|May 4, 2011

Geometry and complex manifolds6 references3 citations

TL;DR

This paper constructs a one-parameter family of Kähler polarizations on complex flag manifolds that converge to the real polarization defined by the Gelfand-Cetlin integrable system. Using toric degenerations and symplectic potential deformations, the authors show that holomorphic sections of the prequantum line bundle converge in the limit to delta-function sections supported on Bohr-Sommerfeld fibers, establishing a direct geometric link between Kähler and real quantizations.

ABSTRACT

In this paper we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand-Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle converge to delta-function sections supported on the Bohr-Sommerfeld fibers. Our construction is based on a toric degeneration of flag varieties and a deformation of Kähler structure on toric varieties by symplectic potentials.

Motivation & Objective

• To establish a direct geometric correspondence between Kähler and real polarizations in geometric quantization on flag manifolds.
• To resolve the long-standing question of whether Kähler and real polarizations yield isomorphic quantum Hilbert spaces via a continuous family of complex structures.
• To extend the method of symplectic potential deformation—previously used in toric varieties—to non-toric flag manifolds using toric degenerations.
• To provide a constructive, analytic framework for the convergence of holomorphic sections to delta-function states on Bohr-Sommerfeld fibers.

Proposed method

• Utilizes the toric degeneration of flag varieties constructed by Kogan and Miller, which deforms the flag manifold into a toric variety while preserving diffeomorphism type.
• Applies symplectic potential deformation techniques from Guillemin and Abreu to construct a one-parameter family of Kähler structures on the flag manifold.
• Constructs a family of complex structures $ J_s $ on the flag manifold $ Fl_n $ via a deformation of the symplectic potential, leading to a family of holomorphic sections $ \sigma^m_s $ of the prequantum line bundle.
• Analyzes the $ L^1 $-norm and pointwise convergence of normalized sections $ \sigma^m_s / \|\sigma^m_s\|_{L^1} $, showing convergence to delta-function states on the fibers $ \mu_{GC}^{-1}(m) \cap V_{0,symp} $.
• Uses the degeneration map $ \tilde{\Psi}_0 $ to pull back delta-functions from the toric limit to the original flag manifold, ensuring covariant constancy under the connection.
• Employs compactness and uniform convergence estimates on tubular neighborhoods of the degeneration locus to control the $ C^0 $-norms of sections and their limits.

Experimental results

Research questions

• RQ1 Can a continuous family of Kähler polarizations be constructed on a flag manifold that converges to the real polarization from the Gelfand-Cetlin system?
• RQ2 Do holomorphic sections of the prequantum line bundle under this family converge to delta-function sections supported on Bohr-Sommerfeld fibers?
• RQ3 Is there a geometric mechanism—via symplectic potential deformation and toric degeneration—that realizes the physical principle of polarization independence in geometric quantization?
• RQ4 Can the convergence of quantum states be quantified in terms of $ C^0 $-norms and $ L^1 $-norms of sections on the degeneration limit?
• RQ5 How does the Gelfand-Cetlin system on the flag manifold relate to the integrable system on the limiting toric variety in the context of geometric quantization?

Key findings

• The paper constructs a continuous family of Kähler polarizations $ \{P_{J_s}\}_{s \in [0,\infty)} $ on the flag manifold $ Fl_n $, with $ P_{J_0} $ being the standard Kähler polarization and $ P_{J_s} \to P_\mu $ as $ s \to \infty $, where $ \mu $ is the Gelfand-Cetlin system.
• For each $ m \in \mathrm{Int}\Delta_{GC} \cap (\mathfrak{t}_{GC})^*_{\mathbb{Z}} $, the normalized holomorphic section $ \sigma^m_s / \|\sigma^m_s\|_{L^1} $ converges in the $ C^0 $-norm on compact subsets away from the degeneration locus to a delta-function state supported on the Bohr-Sommerfeld fiber $ \mu_{GC}^{-1}(m) \cap V_{0,symp} $.
• The convergence is established via uniform bounds on the $ C^0 $-norms of the normalized sections and their pullbacks under the degeneration map, ensuring the limit is a well-defined covariantly constant section on the fiber.
• The limit state on $ \mu_{GC}^{-1}(m) \cap V_{0,symp} $ is pulled back via $ \tilde{\Psi}_0 $ to a covariantly constant section $ \delta_m^\mathbb{F} $ on $ \mu_{GC}^{-1}(m) \subset Fl_n $, realizing the real polarization quantization.
• The convergence is quantified via estimates (7.13)–(7.17), showing that the normalized sections remain uniformly bounded and converge in the weak-$*$ topology to the delta-function state.
• The construction provides a direct, geometric realization of the principle of independence of polarization in geometric quantization, extending the known equality of dimensions to a strong, continuous convergence of states.