Advertisement

Communications in Mathematical Physics

, Volume 331, Issue 3, pp 927–973 | Cite as

Conical Kähler–Einstein Metrics Revisited

  • Chi LiEmail author
  • Song Sun
Article

Abstract

In this paper we introduce the “interpolation–degeneration” strategy to study Kähler–Einstein metrics on a smooth Fano manifold with cone singularities along a smooth divisor that is proportional to the anti-canonical divisor. By “interpolation” we show the angles in (0, 2π] that admit a conical Kähler–Einstein metric form a connected interval, and by “degeneration” we determine the boundary of the interval in some important cases. As a first application, we show that there exists a Kähler–Einstein metric on \({\mathbb{P}^2}\) with cone singularity along a smooth conic (degree 2) curve if and only if the angle is in (π/2, 2π]. When the angle is 2π/3 this proves the existence of a Sasaki–Einstein metric on the link of a three dimensional A 2 singularity, and thus answers a question posed by Gauntlett–Martelli–Sparks–Yau. As a second application we prove a version of Donaldson’s conjecture about conical Kähler–Einstein metrics in the toric case using Song–Wang’s recent existence result of toric invariant conical Kähler–Einstein metrics.

Keywords

Cone Angle Exceptional Divisor Einstein Metrics Bergman Kernel Reeb Vector 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arezzo C., Ghigi A., Pirola G.P.: Symmetries, quotients and Kähler–Einstein metrics. J. Reine Angew. Math. 591, 177–200 (2006)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Aubin T.: Réduction du cas positif de l’équation de Monge–Ampère sur les variétés Kählériennes compactes à la démonstration d’une inégalité. J. Func. Anal. 57, 143–153 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Aubin T.: Equations du type Monge–Ampère sur les variétés compactes. C. R. Acad. Sci. Paris 283, 119–121 (1976)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bando, S., Mabuchi, T.: Uniqueness of Einstein Kähler metrics modulo connected group actions, Algebraic geometry, Sendai 1985. Adv. Stud. Pure Math. 10, 11–40 (1987)Google Scholar
  5. 5.
    Bedford E., Taylor B.A.: The Dirichlet Problem for a complex Monge–Ampère equation. Invent. Math. 37, 1–44 (1976)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Berman, R.J: A thermodynamic formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics. Adv. Math. 248, 1254–1297 (2013)Google Scholar
  7. 7.
    Berman, R.J.: K-polystability of Q-Fano varieties admitting Kähler–Einstein metrics. arXiv:1205.6214
  8. 8.
    Berman, R.J., Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A.: Kähler–Ricci flow and Ricci iteration on log-Fano varieties. arXiv:1111.7158
  9. 9.
    Berndtsson B.: Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains. Ann. Inst. Fourier Grenoble 56(6), 1633–1662 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Berndtsson B.: Curvature of vector bundles associated to holomorphic fibrations. Ann. Math. 169, 531–560 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Berndtsson, B.: A Brunn–Minkowski type inequality for Fano manifolds and the Bando–Mabuchi uniqueness theorem. arXiv:1103.0923
  12. 12.
    Berndtsson B., Paun M.: Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J. 145(2), 341–378 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Błocki, Z.: On geodesics in the space of Kähler metric, to appear in the Proceedings of the “Conference in Geometry” dedicated to Shing-Tung Yau (Warsaw, April 2009)Google Scholar
  14. 14.
    Błocki Z., Kołodziej S.: Regularization of plurisubharmonic functions on manifolds. Proc. Am. Math. Soc. 135(7), 2089–2093 (2007)zbMATHCrossRefGoogle Scholar
  15. 15.
    Caffarelli L., Kohn J.J., Nirenberg L., Spruck J.: The Dirichlet problem for non-linear second order elliptic equations II: Complex Monge–Ampère, and uniformly elliptic equations. Commun. Pure Appl. Math. 38, 209–252 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Cheltsov, I.A., Rubinstein, Y.A.: Asymptotically log Fano varieties. arXiv:1308.2503
  17. 17.
    Chen X.: Space of Kähler metric. J. Differ. Geom. 56, 189–234 (2000)zbMATHGoogle Scholar
  18. 18.
    Chen, X.: Space of Kähler metrics (IV)—On the lower bound of the K-energy. arXiv:0809.4081
  19. 19.
    Chen, X., Donaldson, S.K., Sun, S.: Kahler–Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof (to appear in J. Amer. Math. Soc.). arXiv:1302.0282
  20. 20.
    Chen, Q., Wang, W., Wu, Y., Xu, B.: Conformal metrics with constant curvature one and finite conical singularities on compact Riemann surfaces. arXiv:1302.6457
  21. 21.
    Conti, D.: Cohomogeneity one Einstein–Sasaki 5-manifolds. Commun. Math. Phys. 274(3), 751-774 (2007)Google Scholar
  22. 22.
    Demailly J.P.: Regularization of closed positive currents and intersection theory. J. Alg. Geom. 1(3), 361–409 (1992)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Ding W.Y.: Remarks on the existence problem of positive Kähler–Einstein metrics. Math. Ann. 282, 463–471 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Donaldson S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62, 289–349 (2002)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Donaldson, S.K.: Kähler metrics with cone singularities along a divisor. Essays in Mathematics and its applications, 49–79 (2012). arXiv:1102.1196
  26. 26.
    Donaldson, S.K., Sun, S.: Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry. arXiv:1206.2609
  27. 27.
    Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler–Einstein metrics. J. Ame. Math. Soc. 22(3), 607–639 (2009)Google Scholar
  28. 28.
    Futaki, A.: An obstruction to the existence of Einstein Kähler metrics. Invent. Math. 73(3), 437–443 (1983)Google Scholar
  29. 29.
    Gauntlett J., Martelli D., Sparks J., Yau S-T.: Obstructions to the existence of Sasaki–Einstein metrics. Commun. Math. Phys. 273(3), 803–827 (2007)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Ghigi A., János Kollár J.: Kähler–Einstein metrics on orbifolds and Einstein metrics on spheres. Comment. Math. Helv. 82(4), 877–902 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Griffiths B., Harris J.: Principles of algebraic geometry. Wiley, New York (1978)zbMATHGoogle Scholar
  32. 32.
    Guan B.: The Dirichlet problem for complex Monge–Ampère equations and regularity of the pluri-complex Green function. Commun. Anal. Geom. 6, 687–703 (1998)zbMATHGoogle Scholar
  33. 33.
    Hacking P., Prokhorov Y.: Smoothable del Pezzo surfaces with quotient singularities. Compos. Math. 146, 169–192 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Jeffres, T.D., Mazzeo, R., Rubinstein, Y.A.: Kähler–Einstein metrics with edge singularities, with an appendix by Li, C. and Rubinstein, Y.A. arXiv:1105.5216v3
  35. 35.
    Kołodziej S.: The complex Monge–Ampère equation. Acta Math. 180(1), 69–117 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Legendre, E.: Toric Kähler–Einstein metrics and convex compact polytopes. arXiv:1112.3239
  37. 37.
    Li, C.: Notes on Sean Paul’s paper. http://www.math.sunysb.edu/chili
  38. 38.
    Li C.: Greatest lower bounds on the Ricci curvature of toric Fano manifolds. Adv. Math. 226, 4921–2932 (2011)Google Scholar
  39. 39.
    Li, C.: On the limit behavior of metrics in continuity method to Kahler-Einstein problem in toric Fano case. Compos. Math. 148(06), 1985–2003. arXiv:1012.5229
  40. 40.
    Li, C.: Remarks on logarithmic K-stability. Commun. Contemp. Math. (to appear). arXiv:1104.0428
  41. 41.
    Li, C.: PhD thesis (2012). http://www.math.sunysb.edu/chili
  42. 42.
    Li, C.: Numerical solutions of Kähler–Einstein metrics on \({\mathbb{P}^2}\) with conical singularities along conic curve. J. Geom. Anal. (to appear). arXiv:1207.6592
  43. 43.
    Li, C.: Yau–Tian–Donaldson correspondence for K-semistable Fano manifolds. arXiv:1302.6681
  44. 44.
    Li, C., Xu, C.: Special test configurations and K-stability of Fano varieties. Ann. Math. 180, 197–232 (2014). arXiv:1111.5398
  45. 45.
    Luo F., Tian G.: Liouville equation and spherical convex polytopes. Proc. Am. Math. Soc. 116, 1119–1129 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Matsushima Y.: Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kähléreine. Nagoya Math. J. 11, 145–150 (1957)zbMATHMathSciNetGoogle Scholar
  47. 47.
    Mabuchi T.: K-energy maps integrating Futaki invariants. Tohoku Math. J. (2) 38(4), 575–593 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    McOwen R.C.: Point singularties and conformal metrics on Riemann surfaces. Proc. Am. Math. Soc. 103, 222–224 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Odaka, Y., Sun, S.: Testing log-K-stability by blowing up formalism. arXiv:1112.1353
  50. 50.
    Paul S.T.: Geometric analysis of Chow Mumford stability. Adv. Math. 182, 333–356 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Paul S.T.: Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics. Ann. Math. 175, 255–296 (2012)zbMATHCrossRefGoogle Scholar
  52. 52.
    Paul, S.T., Tian, G.: CM stability and the generalized Futaki invariant II. Astérisque No. 328, 339–354.Google Scholar
  53. 53.
    Phong, D.H., Sturm, J.: The Dirichlet problem for degenerate complex Monge–Ampere equations. Commun. Anal. Geom. 18(1), 147–170 (2010). arXiv:0904.1898
  54. 54.
    Phong, D.H., Song, J., Sturm, J., Weinkove, B.: The Moser–Trudinger inequality on Kähler–Einstein manifolds. Ame. J. Math. 130(4), 1067–1085 (2008)Google Scholar
  55. 55.
    Ross J., Thomas R.: A study of the Hilbert–Mumford criterion for the stability of projective varieties. J. Algebraic Geom. 16, 201–255 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    Ross J., Thomas R.: Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics. J. Differ. Geom. 88, 109–160 (2011)zbMATHMathSciNetGoogle Scholar
  57. 57.
    Shi, Y., Zhu, X.: Kähler–Ricci solitons on toric Fano orbifolds. Math. Z. 271(3–4), 1241–1251 (2012).Google Scholar
  58. 58.
    Song, J., Wang, X.: The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality. (in preprint)Google Scholar
  59. 59.
    Sun, S.: Note on K-stability of pairs. Math. Annalen 355(1), 259–272. arXiv:1108.4603
  60. 60.
    Sun, S., Wang, Y.: On the Kähler–Ricci flow near a Kähler–Einstein metric. J. für die reine und angewandte Mathematik (to appear). arxiv:1004.2018
  61. 61.
    Székelyhidi G.: Greatest lower bounds on the Ricci curvature of Fano manifolds. Compos. Math. 147, 319–331 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  62. 62.
    Székelyhidi, G., Tosatti, V.: Regularity of weak solutions of a complex Monge–Ampère equation. Anal. PDE 4(3), 369–378 (2011)Google Scholar
  63. 63.
    Tian G.: On Kähler–Einstein metrics on certain Kähler manifolds with c 1(M) > 0. Invent. Math. 89, 225–246 (1987)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    Tian G.: On stability of the tangent bundles of Fano varieties. Int. J. Math. 3(3), 401–413 (1992)zbMATHCrossRefGoogle Scholar
  65. 65.
    Tian, G.: Kähler–Einstein metrics on algebraic manifolds. In: Transcendental Methods in Algebraic Geometry (Cetraro 1994), Lecture Notes in Math. pp. 143–185 (1646)Google Scholar
  66. 66.
    Tian G.: Bott–Chern forms and geometric stability. Discret Contin. Dyn. Syst. 6(1), 211–220 (2000)zbMATHCrossRefGoogle Scholar
  67. 67.
    Tian G.: The K-energy on hypersurfaces and stability. Commun. Anal. Geom. 2(2), 239–265 (1994)zbMATHGoogle Scholar
  68. 68.
    Tian, G.: Kähler–Einstein metrics with positive scalar curvature. Invent. math. 137, 1–37 (1997)Google Scholar
  69. 69.
    Tian, G.: Canonical Metrics on Kähler Manifolds. In: Lectures in Mathematics ETH Zürich Birkhäuser Verlag, Basel (2000)Google Scholar
  70. 70.
    Tian G., Zhu X.: Convergence of Khler–Ricci flow. J. Am. Math. Soc. 20(3), 675–699 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  71. 71.
    Tian, G.: Existence of Einstein metrics on Fano manifolds. Prog. Math. 297, Part 1, 119–159 (2012)Google Scholar
  72. 72.
    Troyanov M.: Prescribing curvature on compact surfaces with conic singularities. Trans. Am. Math. Soc. 324, 793–821 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  73. 73.
    Wang X., Zhu X.: Kähler–Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188, 87–103 (2004)Google Scholar
  74. 74.
    Yau S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I. Commun. Pure Appl. Math. 31, 339–441 (1978)zbMATHCrossRefGoogle Scholar
  75. 75.
    Zhang S.: Heights and reductions of Semi-stable varieties. Compos. Math. 104, 77–105 (1996)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Mathematics DepartmentStony Brook UniversityStony BrookUSA
  3. 3.Department of MathematicsImperial CollegeLondonUK

Personalised recommendations