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Kähler–Einstein metrics along the smooth continuity method

Geometric and Functional Analysis volume 26, pages 975–1010 (2016)Cite this article

Abstract

We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kähler–Einstein metric. This is a strengthening of the solution of the Yau–Tian–Donaldson conjecture for Fano manifolds by Chen–Donaldson–Sun (Int Math Res Not (8):2119–2125, 2014), and can be used to obtain new examples of Kähler–Einstein manifolds. We also give analogous results for twisted Kähler–Einstein metrics and Kahler–Ricci solitons.

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References

  1. Alexeev V.: Complete moduli in the presence of semiabelian group action. Annals of Mathematics (2) 155(3), 611–708 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  2. Alexeev V., Brion M.: Stable reductive varieties, I. Affine varieties. Inventiones Mathematicae 157(2), 227–274 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  3. Alexeev V., Brion M.: Stable reductive varieties, II. Projective case. Advances in Mathematics 184(2), 380–408 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  4. Alexeev V., Katzarkov L.: On K-stability of reductive varieties. Geometric and Functional Analysis 15(2), 297–310 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  5. M. Anderson. Convergence and rigidity of manifolds under Ricci curvature bounds. Inventiones Mathematicae, 97 (1990), 429–445.

    MathSciNet  MATH  Article  Google Scholar 

  6. Aubin T.: Réduction de cas positif de l’équation de Monge-Ampère sur les variétés kählériennes compactes à la démonstration d’une inégalité. Journal of functional Analysis 57(2), 143–153 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  7. S. Bando and T. Mabuchi. Uniqueness of Einstein Kähler metrics modulo connected group actions. In: Algebraic geometry, Sendai. Advanced Studies in Pure Mathematics, Vol. 10 (1985), pp. 11–40.

  8. Bedford E., Taylor B.A.: The Dirichlet problem for a complex Monge–Ampère equation. Inventiones Mathematicae 37(1), 1–44 (1976)

    MathSciNet  MATH  Article  Google Scholar 

  9. R.J. Berman, S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi. Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties. arXiv:1111.7158.

  10. R.J. Berman and D.W. Nystrom. Complex optimal transport and the pluripotential theory of Kähler–Ricci solitons. arXiv:1401.8264.

  11. Berndtsson B.: A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry. Inventiones Mathematicae 200(1), 149–200 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  12. Berndtsson B.: Curvature of vector bundles associated to holomorphic fibrations. Annals of Mathematics (2) 169(2), 531–560 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  13. Błocki Z., Kołodziej S.: On regularization of plurisubharmonic functions on manifolds. Proceedings of the American Mathematical Society 135(7), 2089–2093 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  14. Cheeger J., Colding T.: On the structure of spaces with Ricci curvature bounded below. I. Journal of Differential Geometry 46(3), 406–480 (1997)

    MathSciNet  MATH  Google Scholar 

  15. Cheeger J., Colding T., Tian G.: On the singularities of spaces with bounded Ricci curvature. Geometric and Functional Analysis 12(5), 873–914 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  16. J. Cheeger and A. Naber. Regularity of Einstein manifolds and the codimension 4 conjecture. Annals of Mathematics (2), (3)182 (2015), 1093–1165.

  17. X. Chen, S. Donaldson, and S. Sun. Kähler–Einstein metrics and stability. International Mathematics Research Notices, (8) (2014), 2119–2125.

    MathSciNet  MATH  Google Scholar 

  18. Chen X., Donaldson S., Sun S.: Kähler–Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. Journal of the American Mathematical Society 28(1), 183–197 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  19. Chen X., Donaldson S., Sun S.: Kähler–Einstein metrics on Fano manifolds. II: Limits with cone angle less than \({2\pi}\) . Journal of the American Mathematical Society 28(1), 199–234 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  20. Chen X., Donaldson S., Sun S.: Kähler–Einstein metrics on Fano manifolds. III: Limits as cone angle approaches \({2\pi}\) and completion of the main proof. Journal of the American Mathematical Society 28(1), 235–278 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  21. Colding T.H.: Ricci curvature and volume convergence. Annals of Mathematics (2) 145(3), 477–501 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  22. J.-P. Demailly. Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines. Mémoires de la Société Mathématique France (N. S.), 19 (1985), 1–125.

  23. Demailly J.-P.: Regularization of closed positive currents and intersection theory. The Journal of Algebraic Geometry 1(3), 361–409 (1992)

    MathSciNet  MATH  Google Scholar 

  24. R. Dervan. Uniform stability of twisted constant scalar curvature Kähler metrics. arXiv:1412.0648.

  25. S.K. Donaldson. Stability, birational transformations and the Kähler–Einstein problem. Surveys in differential geometry. Vol. XVII, 203–228, Surv. Differ. Geom., 17, Int. Press, Boston, MA, 2012.

  26. S.K. Donaldson. Scalar curvature and stability of toric varieties. Journal of Differential Geometry, 62 (2002), 289–349.

    MathSciNet  MATH  Google Scholar 

  27. Donaldson S., Sun S.: Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry. Acta Mathematica 213(1), 63–106 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  28. Ding W.Y., Tian G.: Kähler–Einstein metrics and the generalized Futaki invariant. Inventiones Mathematicae 110(2), 315–335 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  29. Harvey R., Polking J.: Extending analytic objects. Communications on Pure and Applied Mathematics, 28(6), 701–727 (1975)

    MathSciNet  MATH  Article  Google Scholar 

  30. W. He. F-functional and geodesic stability. arXiv:1208.1020

  31. N. Ilten and H. Süss. K-stability for varieties with torus action of complexity one. arXiv:1507.04442

  32. W. Jian, F. Wang, and X. Zhu. Bergman kernels for a sequence of almost Kähler–Ricci solitons. arXiv:1401.6542

  33. M. Lejmi and G. Székelyhidi. The J-flow and stability. Advances in Mathematics, 274 (2015), 404–431.

  34. Li C.: Greatest lower bounds on Ricci curvature for toric Fano manifolds. Advances in Mathematics 226(6), 4921–4932 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  35. D. Luna. Slices étales. In: Sur les groupes algébriques. Bull. Soc. Math. France, Paris, Mémoire, Vol. 33. Soc. Math. France, Paris (1973), pp. 81–105.

  36. Y. Matsushima. Sur la structure du groupe d'homéomorphismes analytiques d'une certaine variété kählérienne. Nagoya Mathematical Journal, 11 (1957), 145–150

    MathSciNet  MATH  Article  Google Scholar 

  37. D.H. Phong, J. Song, and J. Sturm. Degenerations of Kähler–Ricci solitons on Fano manifolds. Universitatis Iagellonicae Acta Mathematica, 52 (2015), 29–43.

  38. F. Podestà and A. Spiro. Kähler–Ricci solitons on homogeneous toric bundles. Journal Fur Die Reine Und Angewandte Mathematik, 642 (2010), 109–127.

    MATH  Google Scholar 

  39. Shiffman B., Zelditch S.: Distribution of zeros of random and quantum chaotic sections of positive line bundles. Communications in Mathematical Physics 200(3), 661–683 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  40. J. Song and X. Wang. The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality. Geometry Topology, (1)20 (2016), 49–102.

  41. H. Süss. Kähler–Einstein metrics on symmetric Fano t-varieties. Advances in Mathematics, 246 (2013), 100–113.

    MathSciNet  MATH  Article  Google Scholar 

  42. H. Süss. Fano threefolds with 2-torus action: a picture book. Documenta Mathematica, 19 (2014), 905–940.

    MathSciNet  MATH  Google Scholar 

  43. G. Székelyhidi. Greatest lower bounds on the Ricci curvature of Fano manifolds. Compositio Mathematica, 147 (2011), 319–331.

    MathSciNet  MATH  Article  Google Scholar 

  44. G. Székelyhidi. A remark on conical Kähler–Einstein metrics. Mathematical Research Letters, (3) 20 (2013), 581–590.

  45. G. Székelyhidi. The partial \(C^0\)-estimate along the continuity method. Journal of the American Mathematical Society, (2) 29 (2016), 537–560.

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

    MathSciNet  MATH  Article  Google Scholar 

  47. G. Tian. Kähler–Einstein metrics with positive scalar curvature. Inventiones Mathematicae, 137 (1997), 1–37.

    MATH  Article  Google Scholar 

  48. G. Tian and Z. Zhang. Degeneration of Kähler–Ricci solitons. International Mathematics Research Notices, (5) (2012), 957–985.

    MATH  Google Scholar 

  49. Tian G., Zhu X.: Uniqueness of Kähler–Ricci solitons. Acta Mathematica 184(2), 271–305 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  50. F. Wang and X. Zhu. On the structure of spaces with Bakry–Émery Ricci curvature bounded below. arXiv:1304.4490

  51. X.-J. Wang and X. Zhu. Kähler–Ricci solitons on toric manifolds with positive first Chern class. Advances in Mathematics, 188 (2004), 87–103.

    MathSciNet  MATH  Article  Google Scholar 

  52. Wei G., Wylie W.: Comparison geometry for the Bakry–Emery Ricci tensor. Journal of Differential Geometry 83(2), 377–405 (2009)

    MathSciNet  MATH  Google Scholar 

  53. S.-T. Yau. On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation I. Communications on Pure and Applied Mathematics, 31 (1978), 339–411.

    MathSciNet  MATH  Article  Google Scholar 

  54. S.-T. Yau. Open problems in geometry. Proceedings of Symposia in Pure Mathematics, 54 (1993), 1–28.

    MathSciNet  MATH  Article  Google Scholar 

  55. L. Yi. A Bando–Mabuchi Uniqueness Theorem. arXiv:1301.2847.

  56. Z. Zhang. Degeneration of shrinking Ricci solitons. International Mathematics Research Notices, (21) (2010), 4137–4158.

    MathSciNet  MATH  Google Scholar 

  57. Zhu X.: Kähler–Ricci soliton typed equations on compact complex manifolds with C 1(M) > 0. The Journal of Geometric Analysis 10(4), 759–774 (2000)

    MathSciNet  MATH  Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Ved Datar

  2. Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN, 46556, USA

    Gábor Székelyhidi

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  1. Ved Datar
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  2. Gábor Székelyhidi
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Correspondence to Gábor Székelyhidi.

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Datar, V., Székelyhidi, G. Kähler–Einstein metrics along the smooth continuity method. Geom. Funct. Anal. 26, 975–1010 (2016). https://doi.org/10.1007/s00039-016-0377-4

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