Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Kähler-Einstein metrics for some quasi-smooth log del Pezzo surfaces


Author: Carolina Araujo
Journal: Trans. Amer. Math. Soc. 354 (2002), 4303-4312
MSC (2000): Primary 14Q10, 32Q20
Published electronically: July 2, 2002
MathSciNet review: 1926877
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Recently Johnson and Kollár determined the complete list of anticanonically embedded quasi-smooth log del Pezzo surfaces in weighted projective $3$-spaces. They also proved that many of those surfaces admit a Kähler-Einstein metric, and that some of them do not have tigers.

The aim of this paper is to settle the question of the existence of Kähler-Einstein metrics and tigers for those surfaces for which the question was left open. In order to do so, we will use techniques developed earlier by Nadel, Demailly and Kollár.


References [Enhancements On Off] (What's this?)

  • [BGN1] C. P. Boyer, K. Galicki and M. Nakamaye: Sasakian-Einstein Structures on $9\char93 (S^2\times S^3)$. Preprint DG/0102181 (2001).
  • [BGN2] C. P. Boyer, K. Galicki and M. Nakamaye: On the Geometry of Sasakian-Einstein 5-manifolds. Preprint DG/0012047 (2001).
  • [DK] Jean-Pierre Demailly and János Kollár, Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 4, 525–556 (English, with English and French summaries). MR 1852009, 10.1016/S0012-9593(01)01069-2
  • [Dol] Igor Dolgachev, Weighted projective varieties, Group actions and vector fields (Vancouver, B.C., 1981) Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 34–71. MR 704986, 10.1007/BFb0101508
  • [Fle] Alessio Corti and Miles Reid (eds.), Explicit birational geometry of 3-folds, London Mathematical Society Lecture Note Series, vol. 281, Cambridge University Press, Cambridge, 2000. MR 1798978
  • [JK] J. M. Johnson and J. Kollár, Kähler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-spaces, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 1, 69–79 (English, with English and French summaries). MR 1821068
  • [KMcK] Seán Keel and James McKernan, Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc. 140 (1999), no. 669, viii+153. MR 1610249, 10.1090/memo/0669
  • [Kol] János Kollár, Singularities of pairs, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221–287. MR 1492525
  • [KM] János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959
  • [Na] Alan Michael Nadel, Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Ann. of Math. (2) 132 (1990), no. 3, 549–596. MR 1078269, 10.2307/1971429

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14Q10, 32Q20

Retrieve articles in all journals with MSC (2000): 14Q10, 32Q20


Additional Information

Carolina Araujo
Affiliation: Mathematics Department, Princeton University, Princeton, New Jersey 08544
Email: caraujo@math.princeton.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03081-7
Received by editor(s): December 12, 2001
Published electronically: July 2, 2002
Additional Notes: Partial financial support was provided by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brazil)
Article copyright: © Copyright 2002 American Mathematical Society