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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: [email protected]
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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

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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