Abstract
Various estimates of the lower bound of the holomorphic invariant α(M), defined in [T], are given here by using branched coverings, potential estimates and Lelong numbers of positive,d-closed (1, 1) currents of certain type, etc. These estimates are then applied to produce Kähler-Einstein metrics on complex surfaces withC 1>0, in particular, we prove that there are Kähler-Einstein structures withC 1>0 on any manifold of differential type\(CP^2 \# \overline {nCP^2 } (3 \leqq n \leqq 8)\).
Similar content being viewed by others
References
Burns, D.: Stability of vector bundles and some curvature equations (preprint)
Demailly, J.P.: Formules de Jensen en plusieurs variables et applications arithmétiques. Bull. Soc. Math. France110, 75–102 (1982)
Griffith, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1987
Gilberg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin, Heidelberg, New York: Springer 1977
Hartshorne, R.: Algebraic geometry. Graduate texts in mathematics, Vol. 52. Gehring, F.W. (ed.). Berlin, Heidelberg, New York: Springer 1977
Lelong, P.: Fonctions plurisousharmoniques et formes différentielles positives. New York: Gordon and Breach et Paris: Dunod 1969
L'Vovskii, S.M.: Boundedness of the degree of Fano threefolds. Math. USSR Izvestija, Vol. 19, 3 (1982)
Matsusaka, T.: Polarized varieties with a given Hilbert polynomial. Am. J. Math.94, 1027–1077 (1972)
Schoen, R.: Conformation deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom.20, 479–496 (1984)
Siu, Y.-T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math.27, 53–156 (1974)
Siu, Y.-T.: The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group (preprint)
Skoda, H.: Sous-ensembles analytiques d'ordre fini on infini dansC n. Bull. Soc. Math. France100, 353–408 (1972)
Skoda, H.: Prolongement des courants, positifs, fermés de masse finie. Invent. Math.66, 361–376 (1982)
Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math.113, 1–24 (1981)
Tian, G.: On Kähler-Einstein metrics on certain Kähler-manifolds withC 1>0. Invent. Math. (to appear)
Trudinger, N.S.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Sc. Norm. Super. Pisa (3)22, 265–274 (1986)
Author information
Authors and Affiliations
Additional information
Communicated by E. Lieb
Dedicated to Walter Thirring on his 60th birthday
Research supported in part by Alfred P. Sloan Fellowship for doctoral dissertation
Research supported in part by NSF grant # DMS 84-08447 and ONR contract # N-00014-85-K-0367
Rights and permissions
About this article
Cite this article
Tian, G., Yau, ST. Kähler-Einstein metrics on complex surfaces withC 1>0. Commun.Math. Phys. 112, 175–203 (1987). https://doi.org/10.1007/BF01217685
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01217685