Geodesic rays in the space of Kähler metrics with T-symmetry

2022/11/10 by Naichung Conan Leung, Dan Wang, Leung, Naichung Conan +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2211.05324

openalex publication_date 2022/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (M, ω, J) be a Kähler manifold, equipped with an effective Hamiltonian torus action ρ: T → Diff(M, ω, J) by isometries with moment map μ: M → \mathfrakt^*. We first construct a singular mixed polarization P_mix on M. Second, we construct a one-parameter family of complex structures J_t on M which are compatible with ω. Furthermore, the path of corresponding Kähler metrics g_t is a complete geodesic ray in the space of Kähler metrics of M, when M is compact. Finally, we show that the corresponding family of Kähler polarizations P_t associated to J_t converges to P_mix as t → ∞.

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• Quantization in mixed polarization via transverse Poincaré-Birkhoff-Witt theorem

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