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Some Dynamical Systems of Extremal Measures

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Complex Analysis and Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 144))

Abstract

We consider the dynamical system of extremal measures on a compact Kähler manifold. And we show that the dynamical system converges to the canonical measure, if we assume the abundance of the canonical bundle.

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Notes

  1. 1.

    We have abused the notations \(\mid \!\!A\!\!\mid \), \(\mid \!\!K_{X}\!\!\mid \) here. These notations are similar to the notations of corresponding linear systems. But we shall use the notation if without fear of confusion.

References

  1. Aubin, T.: Equations du type Monge-Ampere sur les varietes kahleriennes compactes. C.R. Acad. Sci. Paris, Ser. A 283, 119 (1976)

    Google Scholar 

  2. Bergman, S.: The Kernel Function and Conformal Mapping. Americal Mathematical Society, Providence (1970)

    MATH  Google Scholar 

  3. Berndtsson, B.: Curvature of vector bundles associated to holomorphic fibrations, Ann. Math. 169, 531-560 (2009)

    Google Scholar 

  4. Berndtsson, B., Paun, M.: Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J. 145(2), 341–378 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caratheodory, C.: Uber die Abbildungen, die durch Systeme von analytischen Funktionen von mehreren Veranderlichen erzeugt werden. Math. Z. 34, 758–792 (1932)

    Article  MathSciNet  Google Scholar 

  6. Cheng, S.-Y., Yau, S.-T.: On the existence of a complete Kahler-Einstein metric on noncompact complex manifolds and the regularity of Fefferman’s equation. Com. Pure Appl. Math. 33, 507–544 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Demailly, J.P.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1(3), 361–409 (1992)

    MathSciNet  MATH  Google Scholar 

  8. Demailly, J.P., Peternell, T., Schneider, M.: Pseudo-effective line bundles on compact Kähler manifolds. Int. J. Math. 12, 689–742 (2001)

    Google Scholar 

  9. Hormander, L.: An Introduction to Complex Analysis in Severa! Variables. North-Holland Amsterdam (1973)

    Google Scholar 

  10. Kawamata, Y.: Kodaira dimension of Algebraic fiber spaces over curves. Invent. Math. 66, 57–71 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kobayashi, S.: Intrinsic distances, measures and geometric function theory. Bull. Am. Math. Soc. 82(3), 357–416 (1976)

    Article  MATH  Google Scholar 

  12. Lelong, P.: Fonctions Plurisousharmoniques et Formes Differentielles Positives, Gordon and Breach (1968)

    Google Scholar 

  13. Mok, N., Yau, S.-T.: Completeness of the Kähler-Einstein metric on bounded domains and characterization of domains of holomorphy by curvature conditions. In: The Mathematical Heritage of Henri Poincare. Proceedings of Symposia in Pure Mathematics, vol. 39, Part I, pp. 41-60 (1983)

    Google Scholar 

  14. Nadel, A.M.: Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature. Ann. Math. 132, 549–596 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ohsawa, T., Takegoshi, K.: \(L^{2}\)-extension of holomorphic functions. Math. Z. 195, 197–204 (1987)

    Google Scholar 

  16. Ohsawa, T.: On the extension of \(L^{2}\) holomorphic functions V, effects of generalization. Nagoya Math. J. 161, 1–21 (2001) (Erratum : Nagoya Math. J. 163(2001))

    Google Scholar 

  17. Siu, Y.-T.: Invariance of plurigenera. Invent. Math. 134, 661–673 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Siu, Y.-T.: Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. Collected papers Dedicated to Professor Hans Grauert (2002), pp. 223-277

    Google Scholar 

  19. Song, J., Tian, G.: Canonical Measures and Kähler-Ricci Flow, Math. arXiv:0802.2570 (2008)

  20. Song, J., Weinkove, B.: Construction of Kähler-Einstein metrics with negative scalar curvature. Math. Ann. 347, 59–79 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tsuji, H.: Analytic Zariski decomposition. Proc. Japan Acad. 61, 161–163 (1992)

    Article  MathSciNet  Google Scholar 

  22. Tsuji, H.: Existence and Applications of Analytic Zariski Decompositions. Trends in Math. Analysis and Geometry in Several Complex Variables, pp. 253-272 (1999)

    Google Scholar 

  23. Tsuji, H.: Dynamical construction of Kähler-Einstein metrics. Nagoya Math. J. 199, 107–122 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tsuji, H.: Canonical Volume Forms on Compact Kähler Manifolds. arXiv:0707.0111 (2007)

  25. Tsuji, H.: Canonical singular hermitian metrics on relative canonical bundles. Am. J. Math. 133(6), 1469–1501 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tsuji, H.: Canonical Measures and Dynamical Systems of Bergman Kernels, Math (2008). arXiv:0805.1829

  27. Tsuji, H.: Ricci Iterations and Canonical Kähler-Einstein Currents on LC Pairs, Math (2009). arXiv:0903.5445 (2009)

  28. Tsuji, H.: On the Extremal Measure on a Complex Manifold. to appear in Progress in Mathematics, Birkhäuser (2014)

    Google Scholar 

  29. Tsuji, H.: Stability of Pluricanonical Systems for Compact Kähler Manifolds (in preparation)

    Google Scholar 

  30. Yau, S.-T.: On the Ricci curvature of a compact Kahier manifold and the complex Monge-Ampere equations I. Commun. Pure Appi. Math. 31, 339-411 (1978)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sophia University, 7-1 Kioicho, Chiyoda-ku, 102–8554, Japan

    Hajime Tsuji

Authors
  1. Hajime Tsuji
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Correspondence to Hajime Tsuji .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Università di Roma Tor Vergata, Rome, Italy

    Filippo Bracci

  2. Dept. of Mathematics Education, Kyungnam University, Changwon, Korea (Republic of)

    Jisoo Byun

  3. Université Joseph Fourier Grenoble, Institut Fourier, Grenoble, France

    Hervé Gaussier

  4. Graduate School of Mathematical Sciences, The University of Tokyo, Meguro-ku, Tokyo, Japan

    Kengo Hirachi

  5. Department of Mathematics, POSTECH, Pohang, Korea (Republic of)

    Kang-Tae Kim

  6. Department of Mathematics, Bergische Universität Wuppertal, Wuppertal, Nordrhein-Westfalen, Germany

    Nikolay Shcherbina

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Tsuji, H. (2015). Some Dynamical Systems of Extremal Measures. In: Bracci, F., Byun, J., Gaussier, H., Hirachi, K., Kim, KT., Shcherbina, N. (eds) Complex Analysis and Geometry. Springer Proceedings in Mathematics & Statistics, vol 144. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55744-9_25

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