Skip to main content
SpringerLink
Log in

Uniqueness of tangent cone of Kähler-Einstein metrics on singular varieties with crepant singularities

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

Let (X, L) be a polarized Calabi-Yau variety (or canonical polarized variety) with crepant singularities. Suppose \(\omega _{KE}\in c_1(L)\) (or \(\omega _{KE}\in c_1(K_X))\) is the unique Ricci flat current (or Käher-Einstein current with negative scalar curvature) with local bounded potential constructed in (Eyssidieux in J Am Math Soc 22: 607-639, 2009), we show that the local tangent at any point \(p\in X\) of metric \(\omega _{KE}\) is unique.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, M.: T convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math. 102(2), 429–445 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  2. Cheeger, J., Colding, T.H.: On the structure of space with Ricci curvature bounded below I. J. Differential. Geom. 46, 406–480 (1997)

    Article  MathSciNet  Google Scholar 

  3. Cheeger, J., Colding, T.H.: On the structure of space with Ricci curvature bounded below II. J. Differ. Geom. 52, 13–35 (1999)

    MathSciNet  Google Scholar 

  4. Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. III. J. Differ. Geom. 54(1), 37–74 (2000)

    MathSciNet  Google Scholar 

  5. Cheeger, J., Naber, A.: Regularity of Einstein manifolds and the codimension 4 conjecture. Ann. Math. 182(3), 1093–1165 (2015)

    Article  MathSciNet  Google Scholar 

  6. Cheeger, J., Tian, G.: On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay. Invent. Math. 118(3), 493–571 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  7. Cheeger, J., Colding, T.H., Tian, G.: On the singularities of spaces with bounded Ricci curvature. Geom. Funct. Anal. 12, 873–914 (2002)

    Article  MathSciNet  Google Scholar 

  8. Cheeger, J., Jiang, W.S., Naber, A.: Rectifiability of singular sets of noncollapsed limit spaces with Ricci curvature bounded below. Ann. Math. 193(2), 407–538 (2021)

    Article  MathSciNet  Google Scholar 

  9. Chen, X.X., Donaldson, S.K., 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)

    Article  Google Scholar 

  10. Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28(3), 333–354 (1975)

    Article  MathSciNet  Google Scholar 

  11. Colding, T.H.: Ricci curvature and volume convergence. Ann. Math. 145(3), 477–501 (1997)

    Article  MathSciNet  Google Scholar 

  12. Colding, T.H., Minicozzi, W.P., II.: On uniqueness of tangent cones for Einstein manifolds. Invent. Math. 196(3), 515–588 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  13. Collins, T.C, Sebastien Picard, S., Yau, S-T.: Stability of the tangent bundle through conifold transitions, preprint arXiv:2102.11170

  14. Demailly, J.P.: Analytic methods in algebraic geometry, https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/analmeth.pdf. Accessed 14 Dec 2009

  15. Demailly, J.P.: Complex analytic and differential geometry. online book available at http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf. Accessed 21 June 2012

  16. Donaldson, S.K., Sun, S.: Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, preprint arXiv:1206.2609

  17. Donaldson, S.K., Sun, S.: Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, II. J. Differ. Geom. 107(2), 327–371 (2016)

    Google Scholar 

  18. Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler-Einstein metrics. J. Am. Math. Soc. 22, 607–639 (2009)

    Article  Google Scholar 

  19. Gauntlett, J., Martelli, D., Sparks, J., Yau, S.-T.: Obstructions to the existence of Sasaki-Einstein metrics. Comm. Math. Phys. 273(3), 803–827 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  20. Jiang, W.S., Naber, A.: L2 curvature bounds on manifolds with bounded Ricci curvature. Ann. Math. 193(1), 107–222 (2021)

    Article  MathSciNet  Google Scholar 

  21. Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem in Algebraic geometry, Sendai, 1985, 283–360, Adv. Stud. Pure Math., 10, NorthHolland, Amsterdam, (1987)

  22. Li, C., Xu, C.Y.: Stability of valuations: higher rational rank. Peking Math. J. 1(1), 1–79 (2018)

    Article  MathSciNet  Google Scholar 

  23. Li, C., Wang, X.W., Xu, C.Y.: Algebraicity of the metric tangent cones and equivariant K-stability. J. Am. Math. Soc. 34(4), 1175–1214 (2021)

    Article  MathSciNet  Google Scholar 

  24. Liu, G., Szekelyhidi, G.: Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below II. Comm. Pure Appl. Math. 74(5), 909–931 (2021)

    Article  MathSciNet  Google Scholar 

  25. Liu, G., Szekelyhidi, G.: Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below I. Geom. Funct. Anal. 32, 236–279 (2022)

    Article  MathSciNet  Google Scholar 

  26. Martelli, D., Sparks, J., Yau, S.-T.: Sasaki-Einstein manifolds and volume minimisation. Commun. Math. Phys. 280, 611–673 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  27. Perelman, G.: A complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and nonunique asymptotic cone, In: Comparison Geometry, Berkeley, CA, 1993-94. Math. Sci. Res. Inst. Publ., vol. 30, pp. 165-166. Cambridge University Press, Cambridge (1997)

  28. Remmert, R.: Local theory of complex spaces. Several complex variables. VII. Sheaf-theoretical methods in complex analysis. Encyclopaedia of mathematical sciences. Springer-Verlag (1994)

    Google Scholar 

  29. Rong, X., Zhang, Y.: Continuity of extremal transitions and flops for Calabi-Yau manifolds. J. Differ. Geom. 82(2), 233–269 (2011)

    MathSciNet  Google Scholar 

  30. Ruan, W., Zhang, Y.: Convergence of Calabi-Yau manifolds. Adv. Math. 228(3), 1543–1589 (2011)

    Article  MathSciNet  Google Scholar 

  31. Simon, L.: Isolated singularities of extrema of geometric variational problems, Harmonic mappings and minimal immersions (Montecatini, 1984), 206–277, Lecture Notes in Math., 1161, Springer, Berlin, (1985)

  32. Song, J.: Riemannian geometry of Kähler-Einstein currents, preprint, arxiv:1404.0445

  33. Song, J.: On a conjecture of Candelas and de la Ossa. Comm. Math. Phys. 334(2), 697–717 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  34. Song, J., Tian, G.: Canonical measures and Kähler-Ricci flow. J. Am. Math. Soc. 25, 303–353 (2012)

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  36. Tian, G., Wang, B.: On the structure of almost Einstein manifolds. J. Am. Math. Soc. 28(4), 1169–1209 (2015)

    Article  MathSciNet  Google Scholar 

  37. Tosatti, V.: Limits of Calabi-Yau metrics when the Kähler class degenerates. J. Eur. Math. Soc. 11, 744–776 (2009)

    Article  Google Scholar 

  38. Tosatti, V.: Adiabatic limits of Ricci-flat Kähler metrics. J. Differ. Geom. 84(2), 427–453 (2010)

    Article  Google Scholar 

  39. Van Coevering, C.: Examples of asymptotically conical Ricci-flat Kähler manifolds. Math. Z. 267(1–2), 465–496 (2011)

    Article  MathSciNet  Google Scholar 

  40. Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and complex Monge-Ampère equation I. Comm. Pure Appl. Math. 31, 339–411 (1978)

    Article  MathSciNet  Google Scholar 

  41. Zhang, Y.: Convergence of Kähler manifolds and calibrated fibrations, Ph.D. thesis, Nankai Institute of Mathematics, (2006)

  42. Zhang, Z.: On degenerate Monge-Ampère equations over closed Kähler manifolds, Int. Math. Res. Not. Art.ID 63640, 18pp (2006)

Download references

Acknowledgements

We would like to thank Professor Jian Song for suggesting this problem and many useful discussions. We thank Gabor Szekelyhidi for many useful communications and comments. At last, we also thank Professor Chi Li, Bin Guo, Song Sun for answering questions.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92617, USA

    Xin Fu

Authors
  1. Xin Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xin Fu.

Ethics declarations

Conflict of interest

I acknowledge that there is no conflict of interest and there is no associated data in my work.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fu, X. Uniqueness of tangent cone of Kähler-Einstein metrics on singular varieties with crepant singularities. Math. Ann. 388, 3229–3258 (2024). https://doi.org/10.1007/s00208-023-02602-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-023-02602-0

Search

Navigation