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Space of Kähler Metrics (V) – Kähler Quantization

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Metric and Differential Geometry

Part of the book series: Progress in Mathematics ((PM,volume 297))

Abstract

Given a polarized Kähler manifold (X,L). The space ℋ of Kähler metrics in\( 2_\Pi {c_1}{(L)}\)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\( \mathcal{B}_{k}{({k}\, \epsilon \,\mathbb{N})} \) of non-compact Type. We prove that ℋ is the limit of \( \mathcal{B}_{k} \)as metric spaces in certain sense. As applications, this provides more geometric proofs of certain known geometric properties of the space ℋ.

Mathematics Subject Classification (2000). 53C55.

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References

  1. M. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Springer, 1999.

    Google Scholar 

  2. E. Calabi, X.-X. Chen. Space of Kähler metrics and Calabi flow, J. Differential Geom. 61 (2002), no. 2, 173–193.

    MathSciNet  MATH  Google Scholar 

  3. X.-X. Chen. The space of Kähler metrics, J. Differential Geom. 56( 2000), no. 2, 189–234.

    Google Scholar 

  4. X.-X. Chen. Space of Kähler metrics III – The greatest lower bound of the Calabi energy, Invent. math. 175, 453–680 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  5. X.-X. Chen. Space of Kähler metrics IV – On the lower bound of the K-energy, arxiv: math/0809.4081.

    Google Scholar 

  6. X.-X. Chen. On the lower bound of the Mabuchi energy and its application, Internet. Math. Res. Notices 2000, no. 12, 607–623.

    Google Scholar 

  7. X.-X. Chen, G. Tian. Geometry of Kähler metrics and foliations by holomorphic discs, Publications Mathématiques de L’IHéS, Vol 107, 1–107.

    Google Scholar 

  8. S.K. Donaldson. Remarks on gauge theory, complex geometry and 4-manifold topology, Fields Medalists’ Lectures, World Sci. Publ., Singapore, 1997, 384–403.

    Book  Google Scholar 

  9. S.K. Donaldson. Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, 13–33, Amer. Math. Soc. Transl. Ser. 2, 196, Amer. Math. Soc., Providence, RI, 1999.

    Google Scholar 

  10. S.K. Donaldson. Scalar curvature and projective embeddings, I. J. Differential. Geom. 59 (2001), no. 3, 479–522.

    MathSciNet  MATH  Google Scholar 

  11. S.K. Donaldson. Holomorphic discs and the complex Monge-Amp`ere equation, J. Symplectic Geom. 1 (2002), no. 2, 171–196.

    MathSciNet  MATH  Google Scholar 

  12. S.K. Donaldson. Scalar curvature and projective embeddings, II. Q. J. Math. 56 (2005), no. 3, 345–356.

    Article  MathSciNet  MATH  Google Scholar 

  13. S.K. Donaldson. Lower bounds on the Calabi functional, J. Differential Geom. 70(2005), 453–472.

    MathSciNet  MATH  Google Scholar 

  14. J. Fine. Calabi flow and projective embeddings, arxiv: math/0811.0155.

    Google Scholar 

  15. Z. Lu. On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math, 122 (2000), 235–273.

    Article  MathSciNet  Google Scholar 

  16. Z. Lu, G. Tian. The log term of the Szegö kernel, Duke Math. J. 125 (2004), no. 2, 351–387.

    Article  MathSciNet  MATH  Google Scholar 

  17. T. Mabuchi. Some symplectic geometry on compact Kähler manifolds I, Osaka, J. Math, no. 24, 227–252, 1987.

    Google Scholar 

  18. D.H. Phong, J. Sturm. The Monge-Amp`ere operator and geodesics in the space of Kähler potentials, Invent. Math. 166 (2006), 125–149.

    Article  MathSciNet  MATH  Google Scholar 

  19. D.H. Phong, J. Sturn. Test Configurations for K-stability and Geodesic Rays, J. Symplectic Geom. 5 (2007), no. 2, 221–247.

    MathSciNet  MATH  Google Scholar 

  20. W.-D. Ruan. Canonical coordinates and Bergman metrics, Comm. Anal. Geom. 6 (1998), no. 3, 589–631.

    MathSciNet  MATH  Google Scholar 

  21. Y.A. Rubinstein, Geometric quantization and dynamical constructions on the space of Kähler metrics, Ph.D. Thesis, MIT, 2008.

    Google Scholar 

  22. Y.A. Rubinstein, S. Zelditch. Bergman approximations of harmonic maps into the space of Kähler metrics on toric varieties, arxiv:math/0803.1249.

    Google Scholar 

  23. J. Song, S. Zelditch. Bergman metrics and geodesics in the space of Kähler metrics on toric varieties, arxiv:math/0707.3082.

    Google Scholar 

  24. G. Tian. On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geometry. 32 (1990), 99–130.

    MATH  Google Scholar 

  25. G. Tian, X-H. Zhu. Convergence of Kähler-Ricci flow, J. Amer. Math. Soc. 20 (2007), no. 3, 675–699.

    Google Scholar 

  26. S. Zelditch. Szegö kernel and a theorem of Tian, Int. Math. Res. Notices 6(1998), 317–331.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI, 53706, USA

    Xiuxiong Chen & Song Sun

  2. Department of Mathematics, Stony Brook University, Stony Brook, 11794, NY, USA

    Xiuxiong Chen

  3. Department of Mathematics, Imperial College London, London, SW7 2AZ, UK

    Song Sun

Authors
  1. Xiuxiong Chen
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  2. Song Sun
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Corresponding author

Correspondence to Xiuxiong Chen .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of California, South Hall 6511, Santa Barbara, 93106, New Jersey, USA

    Xianzhe Dai

  2. , Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, 08854, New Jersey, USA

    Xiaochun Rong

Additional information

Dedicated to Jeff Cheeger for his 65th birthday

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Chen, X., Sun, S. (2012). Space of Kähler Metrics (V) – Kähler Quantization. In: Dai, X., Rong, X. (eds) Metric and Differential Geometry. Progress in Mathematics, vol 297. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0257-4_2

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