Geometric and Functional Analysis

, Volume 27, Issue 1, pp 78–129 | Cite as

Kähler–Einstein metrics on group compactifications

  • Thibaut Delcroix


We obtain a necessary and sufficient condition of existence of a Kähler–Einstein metric on a G × G-equivariant Fano compactification of a complex connected reductive group G in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the continuity method and its translation into a real Monge–Ampère equation, using the invariance under the action of a maximal compact subgroup K × K.


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  1. AB04.
    Alexeev, V., Brion, M.: Stable reductive varieties. II. Projective case. Adv. Math. 184(2), 380–408 (2004)MathSciNetzbMATHGoogle Scholar
  2. Abr98.
    Abreu, M.: Kähler geometry of toric varieties and extremal metrics. Int. J. Math. 9(6), 641–651 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. AK05.
    Alexeev, V., Katzarkov, L.: On \(K\)-stability of reductive varieties. Geom. Funct. Anal. 15(2), 297–310 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. AL92.
    H. Azad and J.-J. Loeb. Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces. Indag. Math. (N.S.), 3(4):365–375, 1992Google Scholar
  5. Aub76.
    T. Aubin. Équations du type Monge-Ampère sur les variétés kähleriennes compactes. C. R. Acad. Sci. Paris Sér. A-B, 283(3):Aiii, A119–A121, 1976Google Scholar
  6. Bie04.
    Bielawski, R.: Prescribing Ricci curvature on complexified symmetric spaces. Math. Res. Lett. 11(4), 435–441 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. BK05.
    Brion, M., Kumar, S.: Frobenius splitting methods in geometry and representation theory. Progress in Mathematics, vol. 231. Birkhäuser Boston Inc, Boston, MA (2005)zbMATHGoogle Scholar
  8. Bla56.
    Blanchard, A.: Sur les variétés analytiques complexes. Ann. Sci. Ecole Norm. Sup. 3(73), 157–202 (1956)zbMATHGoogle Scholar
  9. Bor91.
    A. Borel. Linear algebraic groups, volume 126 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991Google Scholar
  10. Bri89.
    Brion, M.: Groupe de Picard et nombres caractéristiques des variétés sphériques. Duke Math. J. 58(2), 397–424 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Bri07.
    Brion, M.: The total coordinate ring of a wonderful variety. J. Algebra 313(1), 61–99 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. CDS15a.
    Chen, X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Amer. Math. Soc. 28(1), 183–197 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. CDS15b.
    Chen, X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than \(2\pi \). J. Amer. Math. Soc. 28(1), 199–234 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. CDS15c.
    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. J. Amer. Math. Soc. 28(1), 235–278 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. CSW.
    X. Chen, S. Sun, B. Wang. Kähler-Ricci flow, Kähler-Einstein metric and K-stability. arXiv:1508.04397
  16. DCP83.
    C. De Concini and C. Procesi. Complete symmetric varieties. In Invariant theory (Montecatini, 1982), volume 996 of Lecture Notes in Math., pages 1–44. Springer, Berlin, 1983Google Scholar
  17. Dela.
    T. Delcroix. K-stability of Fano spherical varieties. arXiv:1608.01852
  18. Delb.
    T. Delcroix. Log canonical thresholds on group compactifications. arXiv:1510.05079v1
  19. Del15.
    T. Delcroix. Métriques de Kähler-Einstein sur les compactifications de groupes. Theses, Université Grenoble Alpes, October 2015.Google Scholar
  20. Don08.
    S. K. Donaldson. Kähler geometry on toric manifolds, and some other manifolds with large symmetry. In Handbook of geometric analysis. No. 1, volume 7 of Adv. Lect. Math. (ALM), pages 29–75. Int. Press, Somerville, MA, 2008Google Scholar
  21. DR.
    T. Darvas and Y. Rubinstein. Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics. arXiv:1506.07129v2
  22. DS.
    V. Datar and G. Székelyhidi. Kähler-Einstein metrics along the smooth continuity method. arXiv:1506.07495
  23. Ful93.
    W. Fulton. Introduction to toric varieties, volume 131 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in GeometryGoogle Scholar
  24. GH15.
    Gagliardi, G., Hofscheier, J.: Gorenstein spherical Fano varieties. Geom. Dedicata 178, 111–133 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. GR13.
    Gandini, J., Ruzzi, A.: Normality and smoothness of simple linear group compactifications. Math. Z. 275(1–2), 307–329 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Hel78.
    Sigurdur Helgason. Differential geometry, Lie groups, and symmetric spaces, volume 80 of Pure and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978Google Scholar
  27. Hoc65.
    Hochschild, G.: The structure of Lie groups. Holden-Day Inc, San Francisco-London-Amsterdam (1965)zbMATHGoogle Scholar
  28. Joh48.
    John, F.: Extremum problems with inequalities as subsidiary conditions. In Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, pages 187–204. Interscience Publishers Inc., New York, N. Y. (1948)Google Scholar
  29. Kaz87.
    Kazarnovskiĭ, B.Y.: Newton polyhedra and Bezout's formula for matrix functions of finite-dimensional representations. Funktsional. Anal. i Prilozhen. 21(4), 73–74 (1987)MathSciNetGoogle Scholar
  30. Kna02.
    A. W. Knapp. Lie groups beyond an introduction, volume 140 of Progress in Mathematics. Birkhäuser Boston Inc, Boston, MA, second edition, 2002Google Scholar
  31. Li11.
    Li, C.: Greatest lower bounds on Ricci curvature for toric Fano manifolds. Adv. Math. 226(6), 4921–4932 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Mab87.
    Mabuchi, T.: Einstein-K ähler forms, Futaki invariants and convex geometry on toric Fano varieties. Osaka J. Math. 24(4), 705–737 (1987)MathSciNetzbMATHGoogle Scholar
  33. Mat57.
    Matsushima, Y.: Sur la structure du groupe d'homéomorphismes analytiques d'une certaine variété kählérienne. Nagoya Math. J. 11, 145–150 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Oda88.
    T. Oda. Convex bodies and algebraic geometry, volume 15 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties, Translated from the JapaneseGoogle Scholar
  35. Per14.
    Perrin, N.: On the geometry of spherical varieties. Transform. Groups 19(1), 171–223 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. Pez09.
    Pezzini, G.: Automorphisms of wonderful varieties. Transform Groups 14(3), 677–694 (2009)MathSciNetCrossRefGoogle Scholar
  37. PS10.
    Podestà, F., Spiro, A.: Kähler-Ricci solitons on homogeneous toric bundles. J. Reine Angew. Math. 642, 109–127 (2010)MathSciNetzbMATHGoogle Scholar
  38. Ruz12.
    Ruzzi, A.: Fano symmetric varieties with low rank. Publ. Res. Inst. Math. Sci. 48(2), 235–278 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  39. Siu88.
    Y. T. Siu. The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group. Ann. of Math. (2), 127(3):585–627, 1988Google Scholar
  40. Szé11.
    Székelyhidi, G.: Greatest lower bounds on the Ricci curvature of Fano manifolds. Compos. Math. 147(1), 319–331 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  41. Tia87.
    Tian, G.: On Kähler-Einstein metrics on certain Kähler manifolds with \(C_1(M) > 0\). Invent. Math. 89(2), 225–246 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  42. Tia15.
    Tian, G.: K-stability and Kähler-Einstein metrics. Comm. Pure Appl. Math. 68(7), 1085–1156 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  43. Tim11.
    D. A. Timashev. Homogeneous spaces and equivariant embeddings, volume 138 of Encyclopaedia of Mathematical Sciences. Springer, Heidelberg, 2011. Invariant Theory and Algebraic Transformation Groups, 8Google Scholar
  44. WZ04.
    Wang, X.-J., Zhu, X.: Kähler-Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188(1), 87–103 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  45. Yau78.
    Yau, S.T.: 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
  46. ZZ14.
    Zhang, X., Zhang, X.: Generalized Kähler-Einstein metrics and energy functionals. Canad. J. Math. 66(6), 1413–1435 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Département de mathématiques et applicationsÉcole normale supérieureParis Cedex 05France

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