Hostname: page-component-5cf477f64f-xc2pj Total loading time: 0 Render date: 2025-04-02T18:14:21.115Z Has data issue: false hasContentIssue false
  • English
  • Français

Log Canonical Thresholds of Complete Intersection Log Del Pezzo Surfaces

Part of: Computational aspects in algebraic geometry Complex manifolds Special varieties Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  20 February 2015

In-Kyun Kim  and
Jihun Park
In-Kyun Kim
Affiliation:
Center for Geometry and Physics, Institute for Basic Science, 77 Cheongam-ro, Nam-gu, Pohang, Gyeongbuk 790-784, Republic of Korea Department of Mathematics, Pohang University of Science and Technology, 77 Cheongam-ro, Nam-gu, Pohang, Gyeongbuk 790-784, Republic of Korea, (wlog@postech.ac.kr; soulcraw@gmail.com)
Jihun Park
Affiliation:
Center for Geometry and Physics, Institute for Basic Science, 77 Cheongam-ro, Nam-gu, Pohang, Gyeongbuk 790-784, Republic of Korea Department of Mathematics, Pohang University of Science and Technology, 77 Cheongam-ro, Nam-gu, Pohang, Gyeongbuk 790-784, Republic of Korea, (wlog@postech.ac.kr; soulcraw@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

We compute the global log canonical thresholds of quasi-smooth well-formed complete intersection log del Pezzo surfaces of amplitude 1 in weighted projective spaces. As a corollary we show the existence of orbifold Kähler—Einstein metrics on many of them.

Keywords

global log canonical thresholdweighted complete intersectionα-invariantKähler—Einstein metricexceptional Fano varietyweakly exceptional Fano variety

MSC classification

Primary: 14M10: Complete intersections 14J45: Fano varieties 14J17: Singularities
Secondary: 14Q10: Surfaces, hypersurfaces 32Q20: Kähler-Einstein manifolds
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 2 , June 2015 , pp. 445 - 483
Copyright
Copyright © Edinburgh Mathematical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Araujo, C.Kähler-Einstein metrics for some quasi-smooth log del Pezzo surfaces, Trans. Am. Math. Soc. 354(11) (2002), 4303–4312.CrossRefGoogle Scholar
2.Boyer, C.Galicki, K. and Nakamaye, M.On the geometry of Sasakian-Einstein 5-manifolds, Math. Annalen 325(3) (2003), 485–524.CrossRefGoogle Scholar
3.Cheltsov, I.Log canonical thresholds of del Pezzo surfaces, Geom. Funct. Analysis 18(4) (2008), 1118–1144.CrossRefGoogle Scholar
4.Cheltsov, I. and Shramov, C.Log canonical thresholds of smooth Fano threefolds, with an appendix by Jean-Pierre Demailly, Russ. Math. Surv. 63(5) (2008), 859–958.CrossRefGoogle Scholar
5.Cheltsov, I. and Shramov, C.On exceptional quotient singularities, Geom. Topol. 15(4) (2011), 1843–1882.CrossRefGoogle Scholar
6.Cheltsov, I. and Shramov, C.Del Pezzo zoo, Exp. Math. 22(3) (2012), 313–326.CrossRefGoogle Scholar
7.Cheltsov, I.Park, J. and Shramov, C.Exceptional del Pezzo hypersurfaces, J. Geom. Analysis 20(4) (2010), 787–816.CrossRefGoogle Scholar
8.Chen, J.J.Chen, J.A. and Chen, M.On quasi-smooth weighted complete intersections, J. Alg. Geom. 20(2) (2011), 239–262.CrossRefGoogle Scholar
9.Demailly, J.-P. and Kollár, J.Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Annales Scient. Éc. Norm. Sup. 34(4) (2001), 525–556.CrossRefGoogle Scholar
10.Iano-Fletcher, A.R.Working with weighted complete intersections, in Explicit bira-tional geometry of 3-folds, London Mathematical Society Lecture Note Series, Volume 281, pp. 101–173 (Cambridge University Press, 2000).CrossRefGoogle Scholar
11.Johnson, J. and Kollár, J.Kähler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-spaces, Annales Inst. Fourier 51(1) (2001), 69–79.CrossRefGoogle Scholar
12.Keel, S. and McKernan, J.Rational curves on quasi-projective surfaces, Memoirs of the American Mathematical Society, Volume 669 (American Mathematical Society, Providence, RI, 1999).CrossRefGoogle Scholar
13.Kim, I.Log canonical thresholds of complete intersection log del Pezzo surfaces, PhD Thesis, Pohang University of Science and Technology, 2014 (available at http://postech.dcollection.net/jsp/common/DcLoOrgPer.jsp?sItemId=000001673786).Google Scholar
14.Kollár, J.Singularities of pairs, Proc. Symp. Pure Math. 62 (1997), 221–287.CrossRefGoogle Scholar
15.Kudryavtsev, S.On purely log terminal blow-ups, Mat. Zametki 69(6) (2001), 892–898.Google Scholar
16.Mori, S. and Kollár, J.Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, Volume 134 (Cambridge University Press, 1998).Google Scholar
17.Mustaţă, M.Jet schemes of locally complete intersection canonical singularities, Invent. Math. 145(3) (2001), 397–424.CrossRefGoogle Scholar
18.Prokhorov, Yu.Lectures on complements on log surfaces, Mathematical Society of Japan Memoirs, Volume 10 (Mathematical Society of Japan, Tokyo, 2001).CrossRefGoogle Scholar
19.Tian, G.On Kähler-Einstein metrics on certain Kähler manifolds with c1(m) > 0, Invent. Math. 89(2) (1987), 225–246.CrossRefGoogle Scholar
20.Tlan, G. and Yau, S.-T.Kähler-Einstein metrics on complex surfaces with c1 > 0, Commun. Math. Phys. 112(1) (1987), 175–203.Google Scholar
21.Watanabe, K.Rational singularities with k*-action, in Commutative algebra: Proc. Conf. Trento, Italy, 1981, Lecture Notes in Pure and Applied Mathematics, Volume 84, pp. 339–351 (Marcel Dekker, New York, 1983).Google Scholar