Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



The $\alpha$-invariant on certain surfaces with symmetry groups

Author: Jian Song
Translated by:
Journal: Trans. Amer. Math. Soc. 357 (2005), 45-57
MSC (2000): Primary 53-XX
Published electronically: January 29, 2004
MathSciNet review: 2098086
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The global holomorphic $\alpha$-invariant introduced by Tian is closely related to the existence of Kähler-Einstein metrics. We apply the result of Tian, Yau and Zelditch on polarized Kähler metrics to approximate plurisubharmonic functions and compute the $\alpha$-invariant on $CP^2\char93 n\overline{CP^2}$ for $n=1,2,3$.

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

  • 1. Ben Abdesselem, A., Équations de Monge-Ampère d'origine géométrique sur certaines variétés algébriques, J.Funct. Anal. 149 (1997), 102-134. MR 98i:32020
  • 2. Ding, W and Tian, G., The generalized Moser-Trudinger Inequality, Proceedings of Nankai International Conference on Nonlinear Analysis, 1993.
  • 3. Lu, Z., On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math. 122 (2000), no. 2, 235-273. MR 2002d:32034
  • 4. Phong, D. H. and Sturm, J., Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions, Ann. of Math. (2) 152 (2000), no. 1, 277-329. MR 2002f:11180
  • 5. Siu, Y.T., The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group, Ann. of Math. (2) 127 (1988), no. 3, 585-627. MR 89e:58032
  • 6. Tian, G. and Yau, S.T., Kähler-Einstein metrics on complex surfaces with $C\sb 1>0$, Comm. Math. Phys. 112 (1987), no.1, 175-203. MR 88k:32070
  • 7. Tian, G., On Kähler-Einstein metrics on certain Kähler manifolds with $C_ {1}(M)>0$, Invent. Math. 89 (1987), no. 2, 225-246. MR 88e:53069
  • 8. Tian, G., On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geometry 32 (1990), 99-130. MR 91j:32031
  • 9. Tian, G., On Calabi's conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101-172. MR 91d:32042
  • 10. Tian, G., Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), no. 1, 1-37. MR 99e:53065
  • 11. Tian, G. and Zhu, X.H., A nonlinear inequality of Moser-Trudinger type, Calc. Var. Partial Differential Equations 10 (2000), no. 4, 349-354. MR 2001f:32044
  • 12. Yau, S.T., On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Comm. Pure Appl. Math. 31 (1978), 339-411. MR 81d:53045
  • 13. Zelditch, S., Szegö Kernels and a Theorem of Tian, IMRN1998, No. 6, 317-331. MR 99g:32055

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53-XX

Retrieve articles in all journals with MSC (2000): 53-XX

Additional Information

Jian Song
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027

Received by editor(s): May 1, 2003
Published electronically: January 29, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society