Mathematische Annalen

, Volume 372, Issue 3–4, pp 1239–1276 | Cite as

K-stability of smooth del Pezzo surfaces

  • Jihun Park
  • Joonyeong WonEmail author


In a new algebro-geometric way we completely determine whether smooth del Pezzo surfaces are K-(semi)stable or not.

In the present article, all varieties are defined over an algebraically closed field k of characteristic 0.


  1. 1.
    Blum, H., Jonsson, M.: Thresholds, valuations, and K-stability. arXiv:1706.04548
  2. 2.
    Cheltsov, I.: Log canonical thresholds of del Pezzo surfaces. Geom. Funct. Anal. 11, 1118–1144 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cheltsov, I., Park, J.: Global log-canonical thresholds and generalized Eckardt points. Sb. Math. 193(5–6), 779–789 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cheltsov, I., Park, J.: Sextic double solids. In: Bogomolov, F., Tschinkel, Yu. (eds.) Cohomological and geometric approaches to rationality problems, Progr. Math., vol. 282, pp. 75–132. Birkhäuser, Boston (2010)Google Scholar
  5. 5.
    Cheltsov, I., Park, J., Won, J.: Log canonical thresholds of certain Fano hypersurfaces. Math. Z. 276(1–2), 51–79 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cheltsov, I., Shramov, K.: Log-canonical thresholds for nonsingular Fano threefolds. With an appendix by J.-P. Demailly. Russ. Math. Surv. 63(5), 859–958 (2008)CrossRefGoogle Scholar
  7. 7.
    Chen, X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds I: approximation of metrics with cone singularities. J. Am. Math. Soc. 28(1), 183–197 (2015)CrossRefGoogle Scholar
  8. 8.
    Chen, X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds, II: limits with cone angle less than \(2\pi \). J. Am. Math. Soc. 28(1), 199–234 (2015)CrossRefGoogle Scholar
  9. 9.
    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. Am. Math. Soc. 28(1), 235–278 (2015)CrossRefGoogle Scholar
  10. 10.
    Fujita, K.: On K-stability and the volume functions of \(\mathbb{Q}\)-Fano varieties. Proc. Lond. Math. Soc. 113(5), 541–582 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fujita, K.: K-stability of Fano manifolds with not small alpha invariants. To appear in J. Inst. Math. Jussieu. doi: 10.1017/S1474748017000111
  12. 12.
    Fujita, K.: On Berman–Gibbs stability and K-stability of \(\mathbb{Q}\)-Fano varieties. Compositio Math. 152, 288–298 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fujita, K., Odaka, Yu.: On the K-stability of Fano varieties and anticanonical divisors. To appear in Tohoku Math. J. arXiv:1602.01305
  14. 14.
    Hacking, P., Prokhorov, Yu.: Smoothable del Pezzo surfaces with quotient singularities. Compositio Math. 146(1), 169–192 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hwang, J.-M., Kim, H., Lee, Y., Park, J.: Slopes of smooth curves on Fano manifolds. Bull. Lond. Math. Soc. 43(5), 827–839 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kempf, G.: Instability in invariant theory. Ann. Math. (2) 108(2), 299–316 (1978)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kollár, J.: Singularities of pairs. In: Kollr, J., Lazarsfeld, R., Morrison, D. (eds.) Algebraic geometry (Santa Cruz, 1995) Part 1, Proc. Sympos. Pure Math., vol. 62, pp. 221–287. Amer. Math. Soc. (1997)Google Scholar
  18. 18.
    Manetti, M.: Normal degenerations of the complex projective plane. J. Reine Angew. Math. 419, 89–118 (1991)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Manetti, M.: Normal projective surfaces with \(\rho =1\), \(P_{-1}\ge 5\). Rend. Sem. Mat. Univ. Padova 89, 195–205 (1993)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Odaka, Yu., Sano, Yu.: Alpha invariant and K-stability of \(\mathbb{Q}\)-Fano varieties. Adv. Math. 229(5), 2818–2834 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Odaka, Yu., Spotti, C., Sun, S.: Compact moduli spaces of del Pezzo surfaces and Kähler–Einstein metrics. J. Differ. Geom. 102(1), 127–172 (2016)CrossRefGoogle Scholar
  22. 22.
    Panov, D., Ross, J.: Slope stability and exceptional divisors of high genus. Math. Ann. 343(1), 79–101 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Prokhorov, Yu.: A note on degenerations of del Pezzo surfaces. Ann. Inst. Fourier (Grenoble) 65(1), 369–388 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ross, J., Thomas, R.: A study of the Hilbert–Mumford criterion for the stability of projective varieties. J. Algebraic Geom. 16(2), 201–255 (2007)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tian, G.: On Kähler–Einstein metrics on certain Kähler manifolds with \(c_1(M)>0\). Invent. Math. 89, 225–246 (1987)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Tian, G.: K-stability and Kähler–Einstein metrics. Commun. Pure Appl. Math. 68 (7), 1085–1156 (2015). [Corrigendum: K-Stability and Kähler–Einstein Metrics. 68 (11), 2082–2083 (2015)]Google Scholar

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© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Center for Geometry and PhysicsInstitute for Basic Science (IBS)PohangKorea
  2. 2.Department of MathematicsPOSTECHPohangKorea

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