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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.

DOI: https://doi.org/10.4213/rm9235

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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

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    This publication is cited in the following articles:
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    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
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    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
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    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
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    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
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    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
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    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
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    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
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