Hostname: page-component-5c6d5d7d68-wbk2r Total loading time: 0 Render date: 2024-08-15T15:19:39.890Z Has data issue: false hasContentIssue false
  • English
  • Français

Einstein metrics, complex surfaces, and symplectic 4-manifolds

Published online by Cambridge University Press:  01 July 2009

CLAUDE LeBRUN
CLAUDE LeBRUN*
Affiliation:
Department of Mathematics, SUNY, Stony Brook, NY 11794-3651, U.S.A. e-mail: claude@math.sunysb.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Which smooth compact 4-manifolds admit an Einstein metric with non-negative Einstein constant? A complete answer is provided in the special case of 4-manifolds that also happen to admit either a complex structure or a symplectic structure.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 1 , July 2009 , pp. 1 - 8
Copyright
Copyright © Cambridge Philosophical Society 2009

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

REFERENCES

[1]Atiyah, M. F., Hitchin, N. J. and Singer, I. M.Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London Ser. A, 362 (1978), pp. 425–461.Google Scholar
[2]Aubin, T.Equations du type Monge-Ampère sur les variétés kählériennes compactes, C. R. Acad. Sci. Paris, 283A (1976), pp. 119–121.Google Scholar
[3]Barth, W., Peters, C. and Van de Ven, A. Compact complex surfaces. Ergeb. Math. Grenzgeb. vol. 4 (3) [Results in Mathematics and Related Areas (3)]. (Springer-Verlag, 1984).Google Scholar
[4]Besse, A. L.Einstein manifolds. Ergeb. Math. Grenzgeb. vol. 10 (3) [Results in Mathematics and Related Areas (3)]. (Springer-Verlag, 1987).CrossRefGoogle Scholar
[5]Bieberbach, L.Über die Bewegungsgruppen der Euklidischen Räume II. Math. Ann. 72 (1912), pp. 400–412.CrossRefGoogle Scholar
[6]Bourguignon, J.-P.Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d'Einstein. Invent. Math. 63 (1981), pp. 263–286.CrossRefGoogle Scholar
[7]Buchdahl, N.Compact Kähler surfaces with trivial canonical bundle. Ann. Global Anal. Geom. 23 (2003), pp. 189–204.CrossRefGoogle Scholar
[8]Cheeger, J. and Gromoll, D.The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differential Geom. 6 (1971), pp. 119–128.CrossRefGoogle Scholar
[9]Chen, X. X., LeBrun, C. and Weber, B.On conformally Kähler, Einstein manifolds. J. Amer. Math. Soc. 21 (2008), pp. 1137–1168.CrossRefGoogle Scholar
[10]Demazure, M. Surfaces de del Pezzo, II, III, IV, V. in Séminaire sur les Singularités des Surfaces. Lecture Notes in Mathematics. vol. 777. (Springer, 1980), pp. 21–69.CrossRefGoogle Scholar
[11]Derdziński, A.Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compositio Math. 49 (1983), pp. 405–433.Google Scholar
[12]Friedman, R. and Morgan, J.Algebraic surfaces and Seiberg-Witten invariants. J. Alg. Geom. 6 (1997), pp. 445–479.Google Scholar
[13]Griffiths, P. and Harris, J.Principles of Algebraic Geometry (Wiley-Interscience, 1978).Google Scholar
[14]Hitchin, N. J.On compact four-dimensional Einstein manifolds. J. Differential Geom. 9 (1974), pp. 435–442.CrossRefGoogle Scholar
[15]Kodaira, K.On the structure of compact complex analytic surfaces. I. Amer. J. Math. 86 (1964), pp. 751–798.CrossRefGoogle Scholar
[16]LeBrun, C.On the topology of self-dual 4-manifolds. Proc. Amer. Math. Soc. 98 (1986), pp. 637–640.CrossRefGoogle Scholar
[17]LeBrun, C.Einstein metrics and Mostow rigidity. Math. Res. Lett. 2 (1995), pp. 1–8.CrossRefGoogle Scholar
[18]LeBrun, C.On the scalar curvature of complex surfaces. Geom. Funct. Anal. 5 (1995), pp. 619–628.CrossRefGoogle Scholar
[19]LeBrun, C.Four-manifolds without Einstein metrics. Math. Res. Lett. 3 (1996), pp. 133–147.CrossRefGoogle Scholar
[20]LeBrun, C.Ricci curvature, minimal volumes and Seiberg-Witten theory. Invent Math. 145 (2001), pp. 279–316.CrossRefGoogle Scholar
[21]Liu, A.-K.Some new applications of general wall crossing formula, Gompf's conjecture and its applications. Math. Res. Lett. 3 (1996), pp. 569–585.CrossRefGoogle Scholar
[22]Manin, Y. I.Cubic Forms: Algebra, Geometry, Arithmetic (North-Holland Publishing Co., 1974). Translated from the Russian by Hazewinkel, M.Google Scholar
[23]Matsushima, Y.Sur la structure du groupe d'homéomorphismes d'une certaine varieté Kählérienne. Nagoya Math. J. 11 (1957), pp. 145–150.CrossRefGoogle Scholar
[24]Ohta, H. and Ono, K.Notes on symplectic 4-manifolds with b +2 = 1. II. Internat. J. Math. 7 (1996), pp. 755–770.CrossRefGoogle Scholar
[25]Page, D.A compact rotating gravitational instanton. Phys. Lett. 79B (1979), pp. 235–238.Google Scholar
[26]Siu, Y.-T.The existence of Kähler-Einstein metrics on manifolds with positive anti-canonical line bundle and suitable finite symmetry group. Ann. Math. 127 (1988), pp. 585–627.CrossRefGoogle Scholar
[27]Taubes, C. H.The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett. 1 (1994), pp. 809–822.CrossRefGoogle Scholar
[28]Thorpe, J. A.Some remarks on the Gauss-Bonnet formula. J. Math. Mech. 18 (1969), pp. 779–786.Google Scholar
[29]Thurston, W. P. Three-dimensional geometry and topology. Vol. 1. Princeton Mathematical Series. vol. 35. (Princeton University Press, 1997). Edited by Levy, SilvioGoogle Scholar
[30]Tian, G.On Calabi's conjecture for complex surfaces with positive first Chern class. Inv. Math. 101 (1990), pp. 101–172.CrossRefGoogle Scholar
[31]Tian, G. and Yau, S.-T.Kähler-Einstein metrics on complex surfaces with c1 > 0. Comm. Math. Phys. 112 (1987), pp. 175–203.CrossRefGoogle Scholar
[32]Yau, S.-T.Calabi's conjecture and some new results in algebraic geometry. Proc. Nat. Acad. USA. 74 (1977), pp. 1789–1799.CrossRefGoogle ScholarPubMed
[33]Yau, S.-T.On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. Comm. Pure Appl. Math. 31 (1978), pp. 339–411.CrossRefGoogle Scholar