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The Kähler–Ricci flow through singularities

Abstract

We prove the existence and uniqueness of the weak Kähler–Ricci flow on projective varieties with log terminal singularities. We also show that the weak Kähler–Ricci flow can be uniquely continued through divisorial contractions and flips if they exist. Finally we propose an analytic version of the minimal model program with Ricci flow.

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References

  1. 1.

    Aubin, T.: Equations du type Monge–Ampère sur les variétés Kähleriennes compacts. Bull. Sci. Math. 102, 119–121 (1976)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Birkar, C., Cascini, P., Hacon, C., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Blocki, Z., Kolodziej, S.: On regularization of plurisubharmonic functions on manifolds. Proc. Am. Math. Soc. 135(7), 2089–2093 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

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

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Chen, X.X., Ding, W.: Ricci flow on surfaces with degenerate initial metrics. J. Partial Differ. Equ. 20, 193–202 (2007)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Chen, X.X., Tian, G.: Geometry of Kähler metrics and foliations by holomorphic discs. Publ. Math. Inst. Hautes Études Sci. 107, 1–107 (2008)

    Article  MATH  Google Scholar 

  7. 7.

    Chen, X.X., Tian, G., Zhang, Z.: On the weak Kähler–Ricci flow. Trans. Am. Math. Soc. 363(6), 2849–2863 (2011)

    Article  MATH  Google Scholar 

  8. 8.

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

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Debarre, O.: Higher-dimensional algebraic geometry, xiv+233 pp. Universitext. Springer, New York (2001)

  10. 10.

    Demailly, J.-P., Pali, N.: Degenerate complex Monge–Ampère equations over compact Kähler manifolds. Int. J. Math. 21(3), 357–405 (2010)

    Article  MATH  Google Scholar 

  11. 11.

    Dinew, S., Zhang, Z.: Stability of bounded solutions for degenerate complex Monge–Ampère equations. Adv. Math. 225(1), 367–388 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62(2), 289–349 (2002)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler–Einstein metrics. J. Am. Math. Soc. 22(3), 607–639 (2009)

    Article  MATH  Google Scholar 

  14. 14.

    Eyssidieux, P., Guedj, V., Zeriahi, A.: A priori \(L^{\infty }\)-estimates for degenerate complex Monge–Ampère equations. Int. Math. Res. Not. IMRN Art. ID rnn 070, 8 pp (2008)

  15. 15.

    Eyssidieux, P., Guedj, V., Zeriahi, A.: Viscosity solutions to degenerate complex Monge–Ampère equations. Commun. Pure Appl. Math. 64(8), 1059–1094 (2011)

    Article  MATH  Google Scholar 

  16. 16.

    Fornaess, J.E., Narasimhan, R.: The Levi problem on complex spaces with singularities. Math. Ann. 248(1), 47–72 (1980)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Hacon, C., McKernen, J.: Existence of minimal models for varieties of log general type II. J. Am. Math. Soc. 23(2), 469–490 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Kawamata, Y., Matsuda, K., K. Matsuki, K.: Introduction to the minimal model problem. In: Oda,T. (ed.) Algebraic Geometry (Sendai, 1985), vol. 10, pp. 283–360. Adv. Stud. Pure Math., Amsterdam (1987)

  20. 20.

    Kolodziej, S.: The complex Monge–Ampère equation. Acta Math. 180(1), 69–117 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Kolodziej, S.: The Monge–Ampr̀e equation on compact Kähler manifolds. Indiana Univ. Math. J. 52(3), 667–686 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Li, P., Yau, S.T.: Estimates of eigenvalues of a compact Riemannian manifold. In: Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), pp. 205–239, Proc. Sympos. Pure Math., XXXVI. Amer. Math. Soc., Providence (1980)

  23. 23.

    Lazarafeld, J.: Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. In: A Series of Modern Surveys in Mathematics, vol. 48, xviii+387 pp. Springer, Berlin (2004)

  24. 24.

    Phong, D.H., Sturm, J.: On the Kähler–Ricci flow on complex surfaces. Pure Appl. Math. Q. 1(2), 405–413 (2005). (part 1)

    Article  MATH  Google Scholar 

  25. 25.

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

    MATH  Google Scholar 

  26. 26.

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

    Article  MATH  Google Scholar 

  27. 27.

    Phong, D.H., Song, J., Sturm, J., Weinkove, B.: The Kähler–Ricci flow and the \({{\bar{\partial }}}\) operator on vector fields. J. Differ. Geom. 81(3), 631–647 (2009)

    MATH  Google Scholar 

  28. 28.

    Phong, D.H., Song, J., Sturm, J., Weinkove, B.: The modified Kähler–Ricci flow and solitons. Comment. Math. Helv. 86(1), 91–112 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

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

    Article  MATH  Google Scholar 

  30. 30.

    Siu, Y.-T.: Techniques for the Analytic Proof of the finite Generation of the Canonical Ring. In: Jerison, D., et al. (eds.) Current Developments in Mathematics, 2007, 177–219. Int. Press, Somerville (2009)

    Google Scholar 

  31. 31.

    Song, J.: Finite time extinction of the Kähler–Ricci flow. Math. Res. Lett. 21(6), 1435–1449 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Song, J., Tian, G.: The Kähler–Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170(3), 609–653 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Song, J., Tian, G.: Canonical measures and Kähler–Ricci flow. J. Am. Math. Soc. 25(2), 303–353 (2012)

    Article  MATH  Google Scholar 

  34. 34.

    Song, J., Weinkove, B.: The Kähler–Ricci flow on Hirzebruch surfaces. J. Reine Angew. Math. 659, 141–168 (2011)

    MATH  Google Scholar 

  35. 35.

    Song, J., Weinkove, B.: Lecture notes on the Kähler-Ricci flow. In: Boucksom, S. et al. (eds.) An Introduction to the Kähler-Ricci Flow, Lecture Notes in Math., vol. 2086, pp. 89–188. Springer, Cham (2013)

  36. 36.

    Song, J., Yuan, Y.: Convergence of the Kähler-Ricci flow on singular Calabi-Yau varietie. In: Ji, L. et al. (eds.) Advances in Geometric Analysis, Adv. Lect. Math. (ALM), vol. 21, pp. 119–137. Int. Press, Somerville (2012)

  37. 37.

    Szekelyhidi, G.: The Kähler–Ricci flow and K-stability. Am. J. Math. 132(4), 1077–1090 (2010)

    Article  MATH  Google Scholar 

  38. 38.

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

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Tian, G.: On a set of polarized Kähler metrics on algebraic manifolds. J. Differ. Geom. 32, 99–130 (1990)

    MATH  Google Scholar 

  40. 40.

    Tian, G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math. 130(1), 1–37 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Tian, G.: New progress and problems on Kähler–Ricci flow. Différ. Phys. Math. Math. Soc. II. Astérisque 322, 71–92 (2008)

    MATH  Google Scholar 

  42. 42.

    Tian, G., Zhang, Z.: On the Kähler–Ricci flow on projective manifolds of general type. Chin. Ann. Math. Ser. B 27(2), 179–192 (2006)

    Article  MATH  Google Scholar 

  43. 43.

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

    Article  MATH  Google Scholar 

  44. 44.

    Tosatti, V.: Kähler–Ricci flow on stable Fano manifolds. J. Reine Angew. Math. 640, 67–84 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  45. 45.

    Tsuji, H.: Existence and degeneration of Kähler–Einstein metrics on minimal algebraic varieties of general type. Math. Ann. 281, 123–133 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  46. 46.

    Ueno, K.: Classification theory of algebraic varieties and compact complex spaces, notes written in collaboration with P. Cherenack. In: Lecture Notes in Mathematics, vol. 439, xix+278 pp. Springer, Berlin (1975)

  47. 47.

    Yau, S.-T.: A general Schwarz lemma for Kähler manifolds. Am. J. Math. 100(1), 197–203 (1978)

    Article  MATH  Google Scholar 

  48. 48.

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

    Article  MATH  Google Scholar 

  49. 49.

    Yau, S.-T.: Open problems in geometry. Proc. Symp. Pure Math. 54, 1–28 (1993). (problem 65)

    MathSciNet  Article  MATH  Google Scholar 

  50. 50.

    Zhang, Z.: On degenerate Monge–Ampère equations over closed Kähler manifolds. Int. Math. Res. Not. Art. ID 63640, 18 pp (2006)

  51. 51.

    Zhang, Z.: Scalar curvature bound for Kähler–Ricci flows over minimal manifolds of general type. Int. Math. Res. Not. (2009). doi:10.1093/imrn/rnp073

  52. 52.

    Zhang, Z.: Scalar curvature behavior for finite time singularity of Kähler–Ricci flow. Michigan Math. J. 59(2), 419–433 (2010)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgments

The authors would particularly like to thank Chenyang Xu for many inspiring discussions and for bringing MMP with scaling to the authors’ attention. They thank Valentino Tosatti for a number of helpful suggestions on a previous draft of the paper. They The first named author is grateful to D. H. Phong for his advice, encouragement and support. He also wants to thank Yuan Yuan for some helpful discussions and comments. Finally, both authors would like to thank the referee for the careful review and many valuable comments.

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Correspondence to Jian Song.

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Research supported in part by National Science Foundation Grants DMS-0847524 and DMS-0804095. Jian Song is also supported in part by a Sloan Foundation Fellowship.

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Song, J., Tian, G. The Kähler–Ricci flow through singularities. Invent. math. 207, 519–595 (2017). https://doi.org/10.1007/s00222-016-0674-4

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