Skip to main content
SpringerLink
Log in

Existence of Einstein Metrics on Fano Manifolds

  • Conference paper
  • First Online:
Metric and Differential Geometry

Part of the book series: Progress in Mathematics ((PM,volume 297))

Abstract

It is a long-standing problem to establish the existence of Kähler- Einstein metrics on Fano manifolds since Yau’s solution for the Calabi conjecture in late 70s. It is also one of driving forces in today’s study in Kähler geometry. In this paper, we discuss a program I started more than twenty years ago on this famous problem. It includes some of my results and speculations on the existence of Kähler-Einstein metrics on Fano manifolds, such as, holomorphic invariants, the K-stability, the compactness theorem for Kähler- Einstein manifolds, the partial CO-estimates and their variations. I will also discuss some related problems as well as some recent advances.

Mathematics Subject Classification (2000). 14Jxx, 58Jxx

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [An90] Anderson,M.: Ricci curvature bounds and Einstein metrics on compact manifolds. J. Amer. Math. Soc., 3 (1990), 355–374.

    Article  MathSciNet  Google Scholar 

  2. [ArPa05] Arezzo, C. and Pacard, F.: Blowing up and desingularizing constant scalar curvature Kähler manifolds. Acta Math. 196 (2006), 179–228.

    Article  MathSciNet  MATH  Google Scholar 

  3. [ALV09] Arezzo, C., Lanave, G. and Vedova, A.: Singularity, test configurations and constant scalar curvature Kähler metrics. Preprint, 2009.

    Google Scholar 

  4. [Au76] Aubin, T.: Equations du type de Monge-Amp`ere sur les variétés Kähleriennes compactes. C. R. Acad. Sci. Paris, 283 (1976), 119–121.

    MathSciNet  MATH  Google Scholar 

  5. [Au83] Aubin, T.: Réduction du cas positif de l’équation de Monge-Amp`ere sur les variétés Kähleriennes compactes `a la demonstration d’une inégalité. J. Funct. Anal., 57 (1984), 143–153.

    Article  MathSciNet  MATH  Google Scholar 

  6. [BM86] Bando, S. and Mabuchi, T.: Uniqueness of Einstein Kähler metrics modulo connected group actions. Algebraic Geometry, Adv. Studies in Pure Math., 10 (1987).

    Google Scholar 

  7. [Cal82] Calabi, E.: Extremal Kähler metrics. Seminar on Diff. Geom., Ann. of math. Stud., 102, Princeton Univ. Press, 1982.

    Google Scholar 

  8. [Cao86] Cao, H.D.: Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds. Inv. Math., 81 (1985), 359–372.

    Article  MATH  Google Scholar 

  9. [CCT95] Cheeger, J., Colding, T. and Tian, G.: Constraints on singularities under Ricci curvature bounds. C. R. Acad. Sci. Paris Sér. I, Math. 324 (1997), 645–649.

    Google Scholar 

  10. [Ch07] Cheltsov, I.: Log canonical thresholds of Del Pezzo surfaces, Geometric Analysis and Functional Analysis, 11 (2008), 1118–1144.

    Article  MathSciNet  Google Scholar 

  11. [Ch08] Cheltsov, I. and Shramov, C.: Log canonical thresholds of smooth Fano threefolds. With an appendix by Jean-Pierre Demailly. Preprint, math.DG/0806.2107.

    Google Scholar 

  12. [ChTi04] Chen, X.X. and Tian, G.: Geometry of Kähler metrics and foliations by holomorphic discs. Publ. Math. Inst. Hautes ´ Etudes Sci. 107 (2008), 1–107.

    Google Scholar 

  13. [ChWa09] Chen, X.X. and Wang, B.: Kähler Ricci flow on Fano manifolds (I). Preprint, math.DG/0909.2391.

    Google Scholar 

  14. [DeKo01] Demailly, P. and Kollar, J.: Semicontinuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Ann. Ec. Norm. Sup 34 (2001), 525–556.

    MathSciNet  MATH  Google Scholar 

  15. [DT91] Ding,W. and Tian, G.: The generalized Moser-Trudinger inequality, in: Nonlinear Analysis and Microlocal Analysis: Proceedings of the International Conference at Nankai Institute of Mathematics (K.C. Chang et al., eds.), World Scientific, 1992, 57–70.

    Google Scholar 

  16. [DT92] Ding, W. and Tian, G.: Kähler-Einstein metrics and the generalized Futaki invariants. Invent. Math., 110 (1992), 315–335.

    Article  MathSciNet  MATH  Google Scholar 

  17. [Do00] Donaldson, S.: Scalar curvature and projective embeddings. I. J. Diff. Geom. 59 (2001), 479–522.

    MathSciNet  MATH  Google Scholar 

  18. [Do02] Donaldson, S.: Scalar curvature and stability of toric varieties. J. Diff. Geom., 62 (2002), 289–349.

    MathSciNet  MATH  Google Scholar 

  19. [Do05] Donaldson, S.: Lower bounds on the Calabi functional, J. Diff. Geom., 70 (2005), 453–472.

    MathSciNet  MATH  Google Scholar 

  20. [Fut83] Futaki, A.: An obstruction to the existence of Einstein-Kähler metrics. Inv. Math., 73 (1983), 437–443.

    Article  MathSciNet  MATH  Google Scholar 

  21. [Fut90] Futaki, A.: Kähler-Einstein Metrics and Integral Invariants. Lecture Notes in Mathematics, 1314, Springer-Verlag.

    Google Scholar 

  22. [Ma86] Mabuchi, T.: Some symplectic geometry on compact Kähler manifolds. Osaka J. Math., bf 24 (1987), 227–252.

    Google Scholar 

  23. [MaMu93] Mabuchi, T. and Mukai, S.: Stability and Einstein-Kähler metric of a quartic del Pezzo surface. Einstein metrics and Yang-Mills connections (Sanda, 1990), 133–160, Lecture Notes in Pure and Appl. Math., 145, Dekker, New York, 1993.

    Google Scholar 

  24. [Na88] Nakajima, H.: Hausdorff convergence of Einstein 4-manifolds. J. Fac. Sci. Univ. 35 (1988), 411–424.

    MathSciNet  MATH  Google Scholar 

  25. [Paul08] Paul, S.: Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics. Preprint math.DG/ 0811.2548.

    Google Scholar 

  26. [PT04] Paul, S. and Tian, G.: CM Stability and the Generalized Futaki Invariant I. math.DG/ 0605.278.

    Google Scholar 

  27. [Pe03] Perelman, G.: Private communication.

    Google Scholar 

  28. [PhSt00] Phong, D. and Sturm, J.: Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions. Annals of Mathematics, 152 (2000), 277–329.

    Article  MathSciNet  MATH  Google Scholar 

  29. [PSSW06] Phong, D.; Song, J.; Sturm, J. and Weinkove, B.: The Moser-Trudinger inequality on Kähler-Einstein manifolds. Amer. J. Math. 130 (2008), 1067–1085.

    Google Scholar 

  30. [Ru07] Rubinstein, Y.: On energy functionals, Kähler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood, J. Funct. Anal. 255 (2008), 2641–2660.

    Article  MathSciNet  MATH  Google Scholar 

  31. [Ru08] Rubinstein, Y.: On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow. Trans. Amer. Math. Soc. 361 (2009), 5839–5850.

    Article  MathSciNet  MATH  Google Scholar 

  32. [SeTi07] Sesum, N. and Tian, G.: Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman). J. Inst. Math. Jussieu, 7 (2008), 575–587.

    Article  MathSciNet  MATH  Google Scholar 

  33. [Sho92] Shokurov, V.: Three-dimensional log perestroikas. Izvestiya: Mathematics, 56 (1992), 105-203.

    Google Scholar 

  34. [Shi09] Shi, Yalong: On the a-Invariants of Cubic Surfaces with Eckardt Points. Preprint, arXiv:0902.3203.

    Google Scholar 

  35. [Sto07] Stoppa, J.: K-stability of constant scalar curvature Kähler manifolds. Adv. Math. 221 (2009), 1397–1408.

    Article  MathSciNet  MATH  Google Scholar 

  36. [Ti87] Tian, G.: On Kähler-Einstein metrics on certain Kähler Manifolds with Ci(M) > 0. Invent. Math., 89 (1987), 225–246.

    Article  MathSciNet  MATH  Google Scholar 

  37. [Ti88] Tian, G.: Harvard PhD Thesis, May, 1988.

    Google Scholar 

  38. [Ti89] Tian, G.: On Calabi’s conjecture for complex surfaces with positive first Chern class. Inv. Math. 101, (1990), 101–172.

    Article  MATH  Google Scholar 

  39. [Ti90] Tian, G.: Kähler-Einstein on algebraic manifolds. Proc. of ICM, Kyoto, 1990.

    Google Scholar 

  40. [Ti97] Tian, G.: Kähler-Einstein metrics with positive scalar curvature. Invent. Math., 130 (1997), 1–39.

    Article  MathSciNet  MATH  Google Scholar 

  41. [Ti98] Tian, G.: Canonical Metrics on Kähler Manifolds. Lectures in Mathematics ETH ZÜrich, Birkhäuser Verlag, 2000.

    Google Scholar 

  42. [TV05] Tian, G. and Viaclovsky, J.: Moduli spaces of critical Riemannian metrics in dimension four. Adv. Math. 196 (2005), 346–372.

    Article  MathSciNet  MATH  Google Scholar 

  43. [TY87] Tian, G. and Yau, S.T.: Kähler-Einstein metrics on complex surfaces with Ci(M) positive. Comm. Math. Phys., 112 (1987).

    Google Scholar 

  44. [TZ97] Tian, G. and Zhu, X.H.: A nonlinear inequality of Moser-Trudinger type. Calc. Var. Partial Differential Equations, 10 (2000), 349–354.

    Article  MathSciNet  MATH  Google Scholar 

  45. [TZ00] Tian, G. and Zhu, X.H.: Uniqueness of Kähler-Ricci solitons. Acta Math. 184 (2000), no. 2, 271–305.

    Article  MathSciNet  MATH  Google Scholar 

  46. [TZ07] Tian, G. and Zhu, X.H.: Convergence of Kähler-Ricci flow. J. Amer. Math. Soc., 20 (2007), 675–699.

    Article  MathSciNet  MATH  Google Scholar 

  47. [Ya76] Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Amp`ere equation, I. Comm. Pure Appl. Math., 31 (1978), 339–441.

    Article  MATH  Google Scholar 

  48. [Ye07] Ye, R.G.: The logarithmic Sobolev inequality along the Ricci flow. Preprint, math.DG/0707.2424.

    Google Scholar 

  49. [QZ07] Zhang, Qi: A uniform Sobolev inequality under Ricci flow. Preprint Math. DG/0706.1594.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Beijing University and Princeton University, Beijing, China

    Gang Tian

Authors
  1. Gang Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of California, South Hall 6511, Santa Barbara, 93106, New Jersey, USA

    Xianzhe Dai

  2. , Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, 08854, New Jersey, USA

    Xiaochun Rong

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Basel

About this paper

Cite this paper

Tian, G. (2012). Existence of Einstein Metrics on Fano Manifolds. In: Dai, X., Rong, X. (eds) Metric and Differential Geometry. Progress in Mathematics, vol 297. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0257-4_5

Download citation

Publish with us

Policies and ethics

Search

Navigation