Regularizing Properties of the Kähler–Ricci Flow

  • Sébastien BoucksomEmail author
  • Vincent Guedj
Part of the Lecture Notes in Mathematics book series (LNM, volume 2086)


These notes present a general existence result for degenerate parabolic complex Monge–Ampère equations with continuous initial data, slightly generalizing the work of Song and Tian on this topic. This result is applied to construct a Kähler–Ricci flow on varieties with log terminal singularities, in connection with the Minimal Model Program. The same circle of ideas is also used to prove a regularity result for elliptic complex Monge–Ampère equations, following Székelyhidi–Tosatti.


Line Bundle Local Potential Ricci Flow Smooth Path Quotient Singularity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [Aub78]
    T. Aubin, Equation de type Monge-Ampère sur les variétés kählériennes compactes. Bull. Sci. Math. 102, 63–95 (1978)MathSciNetzbMATHGoogle Scholar
  2. [BM87]
    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. [BBGZ13]
    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. [Blo03]
    Z. Błocki, Uniqueness and stability for the complex Monge-Ampère equation on compact Kähler manifolds. Indiana Univ. Math. J. 52(6), 1697–1701 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Bło09]
    Z. Błocki, A gradient estimate in the Calabi-Yau theorem. Math. Ann. 344, 317–327 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [Bou04]
    S. Boucksom, Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. Ecole Norm. Sup. (4) 37(1), 45–76 (2004)Google Scholar
  7. [BEGZ10]
    S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, Monge-Ampère equations in big cohomology classes. Acta Math. 205, 199–262 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [Cao85]
    H.D. Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds. Invent. Math. 81(2), 359–372 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Dem85]
    J.P. Demailly, Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines. Mém. Soc. Math. Fr. 19, 124 p. (1985)Google Scholar
  10. [Dem92]
    J.P. Demailly, Regularization of closed positive currents and intersection theory. J. Algebr. Geom. 1(3), 361–409 (1992)MathSciNetzbMATHGoogle Scholar
  11. [Dem09]
    J.P. Demailly, Complex analytic and differential geometry (2009), OpenContent book available at
  12. [DemPaun04]
    J.-P. Demailly, M. Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. Math. (2) 159(3), 1247–1274 (2004)Google Scholar
  13. [DZ10]
    S. Dinew, Z. Zhang, On stability and continuity of bounded solutions of degenerate complex Monge-Ampère equations over compact Kähler manifolds. Adv. Math. 225(1), 367–388 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [EGZ09]
    P. Eyssidieux, V. Guedj, A. Zeriahi, Singular Kähler-Einstein metrics. J. Am. Math. Soc. 22, 607–639 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [EGZ11]
    P. Eyssidieux, V. Guedj, A. Zeriahi, Viscosity solutions to degenerate complex Monge-Ampère equations. Comm. Pure Appl. Math. 64, 1059–1094 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [FS90]
    A. Fujiki, G. Schumacher, The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics. Publ. Res. Inst. Math. Sci. 26(1), 101–183 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [Gill11]
    M. Gill, Convergence of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds. Comm. Anal. Geom. 19(2), 277–303 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [GR56]
    H. Grauert, R. Remmert, Plurisubharmonische Funktionen in komplexen Räumen. Math. Z. 65, 175–194 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [GZ12]
    V. Guedj, A. Zeriahi, Stability of solutions to complex Monge-Ampère equations in big cohomology classes. Math. Res. Lett. 19(5), 1025–1042 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [KolMori98]
    J. Kollár, S. Mori, in Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998)Google Scholar
  21. [Kol98]
    S. Kołodziej, The complex Monge-Ampère equation. Acta Math. 180(1), 69–117 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [Kol03]
    S. Kołodziej, The Monge-Ampère equation on compact Kähler manifolds. Indiana Univ. Math. J. 52(3), 667–686 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Lieb96]
    G.M. Lieberman, Second Order Parabolic Differential Equations (World Scientific, River Edge, 1996)CrossRefzbMATHGoogle Scholar
  24. [Siu87]
    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
  25. [ST09]
    J. Song, G. Tian, The Kähler–Ricci flow through singularities (2009). Preprint [arXiv:0909.4898]Google Scholar
  26. [SzTo11]
    G. Székelyhidi, V. Tosatti, Regularity of weak solutions of a complex Monge-Ampère equation. Anal. PDE 4, 369–378 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [Tzha06]
    G. Tian, Z. Zhang, On the Kähler–Ricci flow on projective manifolds of general type. Chin. Ann. Math. Ser. B 27(2), 179–192 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Tsu88]
    H. Tsuji, Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type. Math. Ann. 281(1), 123–133 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  29. [Yau78]
    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)CrossRefzbMATHGoogle Scholar
  30. [Zer01]
    A. Zeriahi, Volume and capacity of sublevel sets of a Lelong class of psh functions. Indiana Univ. Math. J. 50(1), 671–703 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Institut de Mathématiques de JussieuCNRS-Université Pierre et Marie CurieParisFrance
  2. 2.Institut de Mathématiques de Toulouse and Institut Universitaire de FranceUniversité Paul SabatierToulouse Cedex 9France

Personalised recommendations