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

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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 -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.

**[BGN1]**C. P. Boyer, K. Galicki and M. Nakamaye: Sasakian-Einstein Structures on . 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

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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