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Convergence of the Kähler–Ricci Flow on a Kähler–Einstein Fano Manifold

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An Introduction to the Kähler-Ricci Flow

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2086))

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Abstract

The goal of these notes is to sketch the proof of the following result, due to Perelman and Tian–Zhu: on a Kähler–Einstein Fano manifold with discrete automorphism group, the normalized Kähler–Ricci flow converges smoothly to the unique Kähler–Einstein metric. We also explain an alternative approach due to Berman–Boucksom–Eyssidieux–Guedj–Zeriahi, which only yields weak convergence but also applies to Fano varieties with log terminal singularities.

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Notes

  1. 1.

    This functional seems to have been first explicitly considered by W.Y. Ding in [Ding88, p. 465], hence the chosen terminology.

  2. 2.

    In his seminar talk, Perelman apparently focused on his key estimates and did not say much about the remaining details.

References

  1. T. Aubin, Equation de type Monge-Ampère sur les variétés kählériennes compactes. Bull. Sci. Math. 102, 63–95 (1978)

    MathSciNet  MATH  Google Scholar 

  2. S. Bando, T. Mabuchi, Uniqueness of Einstein Kähler metrics modulo connected group actions, in Algebraic Geometry (Sendai, 1985), ed. by T. Oda. Advanced Studies in Pure Mathematics, vol. 10 (Kinokuniya, 1987), pp. 11–40 (North-Holland, Amsterdam, 1987)

    Google Scholar 

  3. R. Berman, S. Boucksom, V. Guedj, A. Zeriahi, A variational approach to complex Monge-Ampère equations. Publ. Math. I.H.E.S. 117, 179–245 (2013)

    Google Scholar 

  4. R. Berman, S. Boucksom, P.Eyssidieux, V. Guedj, A. Zeriahi, Kähler–Ricci flow and Ricci iteration on log-Fano varieties (2011). Preprint [arXiv]

    Google Scholar 

  5. B. Berndtsson, A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem (2011). Preprint [arXiv:1103.0923]

    Google Scholar 

  6. S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, Monge-Ampère equations in big cohomology classes. Acta Math. 205, 199–262 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. H.D. Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds. Invent. Math. 81(2), 359–372 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. X.X. Chen, H. Li, B. Wang, Kähler–Ricci flow with small initial energy. Geom. Funct. Anal. 18 (5), 1525–1563 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. X.X. Chen, G. Tian, Ricci flow on Kähler-Einstein surfaces. Invent. Math. 147(3), 487–544 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Chow, The Ricci flow on the 2-sphere. J. Differ. Geom. 33(2), 325–334 (1991)

    MATH  Google Scholar 

  11. T. Collins, G. Székelyhidi, The twisted Kähler–Ricci flow (2012). Preprint [arXiv:1207.5441]

    Google Scholar 

  12. W.-Y. Ding, Remarks on the existence problem of positive Kähler-Einstein metrics. Math. Ann. 282, 463–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. W.-Y. Ding, G. Tian, Kähler-Einstein metrics and the generalized Futaki invariant. Invent. Math. 110(2), 315–335 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. S.K. Donaldson, Kähler geometry on toric manifolds, and some other manifolds with large symmetry, in Handbook of Geometric Analysis, No. 1. Advanced Lectures in Mathematics (ALM), vol. 7 (International Press, Somerville, 2008), pp. 29–75

    Google Scholar 

  15. M. Gill, Convergence of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds. Comm. Anal. Geom. 19(2), 277–303 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. V. Guedj, A. Zeriahi, Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15(4), 607–639 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. V. Guedj, A. Zeriahi, The weighted Monge-Ampère energy of quasiplurisubharmonic functions. J. Funct. Anal. 250, 442–482 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Hamilton, The Ricci flow on surfaces, in Mathematics and General Relativity (Santa Cruz, CA, 1986). Contemporary Mathematics, vol. 71 (American Mathematical Society, Providence, 1988), pp. 237–262

    Google Scholar 

  19. S. Kołodziej, The complex Monge-Ampère equation. Acta Math. 180(1), 69–117 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. H. Li, On the lower bound of the K-energy and F-functional. Osaka J. Math. 45(1), 253–264 (2008)

    MathSciNet  MATH  Google Scholar 

  21. D.H. Phong, N. Sesum, J. Sturm, Multiplier ideal sheaves and the Kähler–Ricci flow. Comm. Anal. Geom. 15(3), 613–632 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. D.H. Phong, J. Song, J. Sturm, B. Weinkove, The Moser-Trudinger inequality on Kähler-Einstein manifolds. Am. J. Math. 130(4), 1067–1085 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. D.H. Phong, J. Sturm, On stability and the convergence of the Kähler–Ricci flow. J. Differ. Geom. 72(1), 149–168 (2006)

    MathSciNet  MATH  Google Scholar 

  24. D.H. Phong, J. Sturm, Lectures on stability and constant scalar curvature, in Handbook of Geometric Analysis, No. 3. Advanced Lectures in Mathematics (ALM), vol. 14 (International Press, Somerville, 2010), pp. 357–436

    Google Scholar 

  25. N. Sesum, G. Tian, Bounding scalar curvature and diameter along the Kähler–Ricci flow (after Perelman). J. Inst. Math. Jussieu 7(3), 575–587 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Sherman, B. Weinkove, Interior derivative estimates for the Kähler–Ricci flow (2011). Preprint [arXiv]

    Google Scholar 

  27. Y.T. Siu, in Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics. DMV Seminar, vol. 8 (Birkhäuser, Basel, 1987)

    Google Scholar 

  28. G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class. Invent. Math. 101(1), 101–172 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  29. G. Tian, Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130, 239–265 (1997)

    Article  Google Scholar 

  30. G. Tian, X. Zhu, Convergence of Kähler–Ricci flow. J. Am. Math. Soc. 20(3), 675–699 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. V. Tosatti, Kähler-Einstein metrics on Fano surfaces Expo. Math. 30(1), 11–31 (2012)

    MathSciNet  MATH  Google Scholar 

  32. C. Villani, Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften, vol. 338 (Springer-Verlag, Berlin, 2009), xxii + 973 pp.

    Google Scholar 

  33. X.J. Wang, X. Zhu, Kähler–Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188(1), 87–103 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. S.T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Comm. Pure Appl. Math. 31(3), 339–411 (1978)

    Article  MATH  Google Scholar 

  35. A. Zeriahi, Volume and capacity of sublevel sets of a Lelong class of psh functions. Indiana Univ. Math. J. 50(1), 671–703 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

It is a pleasure to thank D.H.Phong for patiently explaining several aspects of the proof of this result.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse and Institut Universitaire de France, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

    Vincent Guedj

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  1. Vincent Guedj
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Correspondence to Vincent Guedj .

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Editors and Affiliations

  1. Institut de Mathématiques de Jussieu, CNRS-Université Pierre et Marie Curie, Paris, France

    Sebastien Boucksom

  2. Institut Fourier and Institut Universitaire de France, Université Joseph Fourier, Saint-Martin d'Hères, France

    Philippe Eyssidieux

  3. Institut de Mathématiques de Toulouse and Institut Universitaire de France, Université Paul Sabatier, Toulouse, France

    Vincent Guedj

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Guedj, V. (2013). Convergence of the Kähler–Ricci Flow on a Kähler–Einstein Fano Manifold. In: Boucksom, S., Eyssidieux, P., Guedj, V. (eds) An Introduction to the Kähler-Ricci Flow. Lecture Notes in Mathematics, vol 2086. Springer, Cham. https://doi.org/10.1007/978-3-319-00819-6_6

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