Space of Kähler metrics (V) – Kähler quantization

Xiuxiong Chen; Song Sun

doi:10.1007/978-3-0348-0257-4_2

Space of Kähler metrics (V) – Kähler quantization

Xiuxiong Chen
, Song Sun

Mathematics

University of Wisconsin-Madison
Imperial College London

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

23 Scopus citations

Abstract

Given a polarized Kähler manifold (X,L). The space H of Kähler metrics in 2πc₁(L) is an infinite-dimensional Riemannian symmetric space. As a metric space, it has non-positive curvature. There is associated to H a sequence of finite-dimensional symmetric spaces B_k(k ∈ ℕ) of non-compact type. We prove that H is the limit of B_k as metric spaces in certain sense. As applications, this provides more geometric proofs of certain known geometric properties of the space H.

Original language	English
Title of host publication	Progress in Mathematics
Publisher	Springer Basel
Pages	19-41
Number of pages	23
DOIs	https://doi.org/10.1007/978-3-0348-0257-4_2
State	Published - 2012

Publication series

Name	Progress in Mathematics
Volume	297
ISSN (Print)	0743-1643
ISSN (Electronic)	2296-505X

Keywords

Geodesic distance
Kähler metrics
Quantization

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10.1007/978-3-0348-0257-4_2

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abstract = "Given a polarized K{\"a}hler manifold (X,L). The space H of K{\"a}hler metrics in 2πc1(L) is an infinite-dimensional Riemannian symmetric space. As a metric space, it has non-positive curvature. There is associated to H a sequence of finite-dimensional symmetric spaces Bk(k ∈ ℕ) of non-compact type. We prove that H is the limit of Bk as metric spaces in certain sense. As applications, this provides more geometric proofs of certain known geometric properties of the space H.",

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AB - Given a polarized Kähler manifold (X,L). The space H of Kähler metrics in 2πc1(L) is an infinite-dimensional Riemannian symmetric space. As a metric space, it has non-positive curvature. There is associated to H a sequence of finite-dimensional symmetric spaces Bk(k ∈ ℕ) of non-compact type. We prove that H is the limit of Bk as metric spaces in certain sense. As applications, this provides more geometric proofs of certain known geometric properties of the space H.

KW - Geodesic distance

KW - Kähler metrics

KW - Quantization

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Space of Kähler metrics (V) – Kähler quantization

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