General information
Latest issue
Impact factor
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Uspekhi Mat. Nauk:

Personal entry:
Save password
Forgotten password?

Uspekhi Mat. Nauk, 2008, Volume 63, Issue 5(383), Pages 73–180 (Mi umn9235)  

This article is cited in 49 scientific papers (total in 49 papers)

Log canonical thresholds of smooth Fano threefolds

I. A. Cheltsov, K. A. Shramov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The complex singularity exponent is a local invariant of a holomorphic function determined by the integrability of fractional powers of the function. The log canonical thresholds of effective $\mathbb{Q}$-divisors on normal algebraic varieties are algebraic counterparts of complex singularity exponents. For a Fano variety, these invariants have global analogues. In the former case, it is the so-called $\alpha$-invariant of Tian; in the latter case, it is the global log canonical threshold of the Fano variety, which is the infimum of log canonical thresholds of all effective $\mathbb{Q}$-divisors numerically equivalent to the anticanonical divisor. An appendix to this paper contains a proof that the global log canonical threshold of a smooth Fano variety coincides with its $\alpha$-invariant of Tian. The purpose of the paper is to compute the global log canonical thresholds of smooth Fano threefolds (altogether, there are 105 deformation families of such threefolds). The global log canonical thresholds are computed for every smooth threefold in 64 deformation families, and the global log canonical thresholds are computed for a general threefold in 20 deformation families. Some bounds for the global log canonical thresholds are computed for 14 deformation families. Appendix A is due to J.-P. Demailly.


Full text: PDF file (1406 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2008, 63:5, 859–958

Bibliographic databases:

Document Type: Article
UDC: 512.76
MSC: Primary 14J45; Secondary 14J17, 32Q20
Received: 26.07.2008

Citation: I. A. Cheltsov, K. A. Shramov, “Log canonical thresholds of smooth Fano threefolds”, Uspekhi Mat. Nauk, 63:5(383) (2008), 73–180; Russian Math. Surveys, 63:5 (2008), 859–958

Citation in format AMSBIB
\by I.~A.~Cheltsov, K.~A.~Shramov
\paper Log canonical thresholds of smooth Fano threefolds
\jour Uspekhi Mat. Nauk
\yr 2008
\vol 63
\issue 5(383)
\pages 73--180
\jour Russian Math. Surveys
\yr 2008
\vol 63
\issue 5
\pages 859--958

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Cheltsov I., “Log canonical thresholds of del Pezzo surfaces”, Geom. Funct. Anal., 18:4 (2008), 1118–1144  crossref  mathscinet  zmath  isi  scopus
    2. I. A. Cheltsov, K. A. Shramov, “Extremal Metrics on del Pezzo Threefolds”, Proc. Steklov Inst. Math., 264 (2009), 30–44  mathnet  crossref  mathscinet  isi  elib  elib
    3. Cheltsov I., “On singular cubic surfaces”, Asian J. Math., 13:2 (2009), 191–214  crossref  mathscinet  zmath  isi  scopus
    4. A. V. Pukhlikov, “Existence of the Kähler–Einstein Metric on Certain Fano Complete Intersections”, Math. Notes, 88:4 (2010), 552–558  mathnet  crossref  crossref  mathscinet  isi
    5. Cheltsov I., Park Jihun, Shramov C., “Exceptional del Pezzo hypersurfaces”, J. Geom. Anal., 20:4 (2010), 787–816  crossref  mathscinet  zmath  isi  elib  scopus
    6. Shi Yalong, “On the alpha-invariants of cubic surfaces with Eckardt points”, Adv. Math., 225:3 (2010), 1285–1307  crossref  mathscinet  zmath  isi  elib  scopus
    7. Park Jihun, Won Joonyeong, “Log-canonical thresholds on del Pezzo surfaces of degrees $\ge 2$”, Nagoya Math. J., 200 (2010), 1–26  crossref  mathscinet  zmath  isi  elib  scopus
    8. Park Jihun, Won Joonyeong, “Log canonical thresholds on Gorenstein canonical del Pezzo surfaces”, Proc. Edin. Math. Soc. (2), 54:1 (2011), 187–219  crossref  mathscinet  zmath  isi  scopus
    9. Nill B., Paffenholz A., “Examples of Kähler–Einstein toric Fano manifolds associated to non-symmetric reflexive polytopes”, Beitr. Algebra Geom., 52:2 (2011), 297–304  crossref  mathscinet  zmath  elib  scopus
    10. Cheltsov I., Shramov C., “On exceptional quotient singularities”, Geom. Topol., 15:4 (2011), 1843–1882  crossref  mathscinet  zmath  isi  elib  scopus
    11. Cheltsov I., Shramov C., “Six-dimensional exceptional quotient singularities”, Math. Res. Lett., 18:6 (2011), 1121–1139  crossref  mathscinet  zmath  isi  elib  scopus
    12. Sakovics D., “Weakly-exceptional quotient singularities”, Cent. Eur. J. Math., 10:3 (2012), 885–902  crossref  mathscinet  zmath  isi  elib  scopus
    13. Odaka Yu., Sano Yu., “Alpha invariant and K-stability of Q-Fano varieties”, Adv. Math., 229:5 (2012), 2818–2834  crossref  mathscinet  zmath  isi  elib  scopus
    14. Tosatti V., “Kähler-Einstein metrics on Fano surfaces”, Expo. Math., 30:1 (2012), 11–31  crossref  mathscinet  zmath  isi  scopus
    15. Cheltsov I., Shramov C., “Three embeddings of the Klein simple group into the Cremona group of rank three”, Transform. Groups, 17:2 (2012), 303–350  crossref  mathscinet  zmath  isi  elib  scopus
    16. Mustata M., “Impanga Lecture Notes on Log Canonical Thresholds”, Contribution to Algebraic Geometry: Impanga Lecture Note, EMS Ser. Congr. Rep., ed. Pragacz P., Eur. Math. Soc., 2012, 407–442  mathscinet  zmath  isi
    17. Sano Yu., “Multiplier Ideal Sheaves and the Kahler-Ricci Flow on Toric Fano Manifolds with Large Symmetry”, Commun. Anal. Geom., 20:2 (2012), 341–368  crossref  mathscinet  zmath  isi  elib  scopus
    18. Cheltsov I., Wilson A., “Del Pezzo surfaces with many symmetries”, J. Geom. Anal., 23:3 (2013), 1257–1289  crossref  mathscinet  isi  elib  scopus
    19. Odaka Y., Okada T., “Birational superrigidity and slope stability of Fano manifolds”, Math. Z., 275:3-4 (2013), 1109–1119  crossref  mathscinet  zmath  isi  scopus
    20. Süß H., “Kähler–Einstein metrics on symmetric Fano $T$-varieties”, Adv. Math., 246 (2013), 100–113  crossref  mathscinet  zmath  isi  elib  scopus
    21. Cheltsov I., Shramov C., “Del Pezzo Zoo”, Exp. Math., 22:3 (2013), 313–326  crossref  mathscinet  zmath  isi  scopus
    22. I. A. Cheltsov, K. A. Shramov, “Sporadic simple groups and quotient singularities”, Izv. Math., 77:4 (2013), 846–854  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    23. Cheltsov I., Kosta D., “Computing $\alpha$-invariants of singular del Pezzo surfaces”, J. Geom. Anal., 24:2 (2014), 798–842  crossref  zmath  isi  scopus
    24. Cheltsov I. Shramov C., “Five Embeddings of One Simple Group”, Trans. Am. Math. Soc., 366:3 (2014), 1289–1331  crossref  mathscinet  zmath  isi  scopus
    25. Haozhao Li, Yalong Shi, Yi Yao, “A criterion for the properness of the $K$-energy in a general Kähler class”, Math. Ann, 2014  crossref  mathscinet  isi  scopus
    26. Martinez-Garcia J., “Log Canonical Thresholds of Del Pezzo Surfaces in Characteristic P”, Manuscr. Math., 145:1-2 (2014), 89–110  crossref  mathscinet  zmath  isi  scopus
    27. Cheltsov I., Shramov C., “Weakly-Exceptional Singularities in Higher Dimensions”, J. Reine Angew. Math., 689 (2014), 201–241  crossref  mathscinet  zmath  isi  scopus
    28. I. A. Cheltsov, “Two local inequalities”, Izv. Math., 78:2 (2014), 375–426  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    29. In-Kyun Kim, Jihun Park, “Log Canonical Thresholds of Complete Intersection Log Del Pezzo Surfaces”, Proceedings of the Edinburgh Mathematical Society, 2015, 1  crossref  mathscinet  zmath  isi  scopus
    30. Dervan R., “Alpha Invariants and K-Stability For General Polarizations of Fano Varieties”, no. 16, 2015, 7162–7189  crossref  mathscinet  zmath  isi  scopus
    31. Delcroix T., “Alpha-Invariant of Toric Line Bundles”, 114, no. 1, 2015, 13–27  crossref  mathscinet  zmath  isi  scopus
    32. Cheltsov I.A., Rubinstein Ya.A., “Asymptotically log Fano varieties”, Adv. Math., 285 (2015), 1241–1300  crossref  mathscinet  zmath  isi  elib  scopus
    33. Song J., Wang X., “The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality”, Geom. Topol., 20:1 (2016), 49–102  crossref  mathscinet  zmath  isi  scopus
    34. Cheltsov I., Park J., Won J., “Affine cones over smooth cubic surfaces”, J. Eur. Math. Soc., 18:7 (2016), 1537–1564  crossref  mathscinet  zmath  isi  elib  scopus
    35. Spotti C., Sun S., Yao Ch., “Existence and deformations of Kähler–Einstein metrics on smoothable $\mathbb{Q}$ -Fano varieties”, Duke Math. J., 165:16 (2016), 3043–3083  crossref  mathscinet  zmath  isi  elib  scopus
    36. Tiep Ph.H., “the Alpha-Invariant and Thompson'S Conjecture”, Forum Math. Pi, 4 (2016), e5  crossref  mathscinet  zmath  isi
    37. Cheltsov I., Martinez-Garcia J., “Dynamic Alpha-invariants of Del Pezzo Surfaces”, Int. Math. Res. Notices, 2016, no. 10, 2994–3028  crossref  mathscinet  isi  elib  scopus
    38. Cheltsov I., Park J., “Birational Rigidity and Main Theorem”, Mem. Am. Math. Soc., 246:1167 (2017), 1+  mathscinet  isi
    39. Sławomir Dinew, Grzegorz Kapustka, Michał Kapustka, “Remarks on Mukai threefolds admitting $\mathbb C^*$ action”, Mosc. Math. J., 17:1 (2017), 15–33  mathnet  mathscinet
    40. Guedj V., Zeriahi A., “Degenerate Complex Monge-Ampere Equations”, Degenerate Complex Monge-Ampere Equations, Ems Tracts in Mathematics, 26, Eur. Math. Soc., 2017, 1–472  crossref  mathscinet  isi
    41. Smirnov E., “Singularities of Divisors on Flag Varieties Via Hwang'S Product Theorem”, Bull. Korean. Math. Soc., 54:5 (2017), 1773–1778  crossref  mathscinet  isi  scopus
    42. Jiang Ch., “K-Semistable Fano Manifolds With the Smallest Alpha Invariant”, Int. J. Math., 28:6 (2017), 1750044  crossref  mathscinet  zmath  isi  scopus
    43. Ahmadinezhad H., Cheltsov I., Schicho J., “On a Conjecture of Tian”, Math. Z., 288:1-2 (2018), 217–241  crossref  mathscinet  zmath  isi  scopus
    44. Paemurru E., “Del Pezzo Surfaces in Weighted Projective Spaces”, Proc. Edinb. Math. Soc., 61:2 (2018), 545–572  crossref  mathscinet  isi  scopus
    45. Keller J., Zheng K., “Construction of Constant Scalar Curvature Kahler Cone Metrics”, Proc. London Math. Soc., 117:3 (2018), 527–573  crossref  isi  scopus
    46. Pukhlikov A.V., “Canonical and Log Canonical Thresholds of Fano Complete Intersections”, Eur. J. Math., 4:1, 1, SI (2018), 381–398  crossref  mathscinet  zmath  isi  scopus
    47. Cheltsov I., Parka J., Shramov C., “Alpha-Invariants and Purely Log Terminal Blow-Ups”, Eur. J. Math., 4:3, 2, SI (2018), 845–858  crossref  mathscinet  isi  scopus
    48. Park J., Won J., “K-Stability of Smooth Del Pezzo Surfaces”, Math. Ann., 372:3-4 (2018), 1239–1276  crossref  mathscinet  zmath  isi  scopus
    49. Fujita K., Odaka Yu., “On the K-Stability of Fano Varieties and Anticanonical Divisors”, Tohoku Math. J., 70:4 (2018), 511–521  crossref  mathscinet  isi  scopus
  • Успехи математических наук Russian Mathematical Surveys
    Number of views:
    This page:608
    Full text:145
    First page:6

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019