Skip to main content

Introduction

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

Abstract

This book is the first comprehensive reference on the Kähler–Ricci flow. It provides an introduction to fully non-linear parabolic equations, to the Kähler–Ricci flow in general and to Perelman’s estimates in the Fano case, and also presents the connections with the Minimal Model program.

Keywords

  • Einstein Metrics
  • Ricci Flow
  • Fano Manifold
  • Holomorphic Vector Field
  • Minimal Model Program

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   69.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   89.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    We follow the convention to specify a Kähler metric g on a complex manifold by the associated closed (1,1)-form ω.

  2. 2.

    The equivalence between K-polystability and the existence of a Kähler–Einstein metric has recently been announced by Chen–Donaldson–Sun and Tian, independently.

References

  1. R. Berman, S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, Kähler-Einstein metrics and the Kähler–Ricci flow on log Fano varieties (2011). Preprint [arXiv.1111.7158v2]

    Google Scholar 

  2. A. Besse, in Einstein Manifolds. Erg. der Math. 3. Folge, vol. 10 (Springer, Berlin, 1987)

    Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. J.P. Demailly, Complex analytic and differential geometry (2009), OpenContent book available at http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

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

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. W. Hodge, The Theory and Applications of Harmonic Integrals (Cambridge University Press, Cambridge, 1941)

    Google Scholar 

  10. E. Kähler, Über eine bemerkenswerte Hermitesche metrik. Abh. Math. Semin. Univ. Hambg. 9, 173–186 (1933)

    CrossRef  Google Scholar 

  11. S. Lefschetz, L’Analysis Situs et la Géométrie Algébrique (Gauthier-Villars, Paris, 1924)

    MATH  Google Scholar 

  12. J. Song, G. Tian, The Kähler–Ricci flow through singularities (2009). Preprint [arXiv:0909.4898]

    Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. J. Song, B. Weinkove, Contracting exceptional divisors by the Kähler–Ricci flow (2010). Preprint (arXiv:1003.0718 [math.DG])

    Google Scholar 

  15. G. Székelyhidi, V. Tosatti, Regularity of weak solutions of a complex Monge-Ampère equation. Anal. PDE 4, 369–378 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. A. Weil, Introduction à l’étude des variétés kählériennes (Hermann, Paris, 1957)

    Google Scholar 

  18. 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)

    CrossRef  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sébastien Boucksom .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Boucksom, S., Eyssidieux, P., Guedj, V. (2013). Introduction. 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_1

Download citation