Geometric & Functional Analysis GAFA

, Volume 4, Issue 3, pp 298–336

Extremal Kähler metrics and complex deformation theory

  • C. LeBrun
  • S. R. Simanca

DOI: 10.1007/BF01896244

Cite this article as:
LeBrun, C. & Simanca, S.R. Geometric and Functional Analysis (1994) 4: 298. doi:10.1007/BF01896244


Let (M, J, g) be a compact Kähler manifold of constant scalar curvature. Then the Kähler class [ω] has an open neighborhood inH1,1 (M, ℝ) consisting of classes which are represented by Kähler forms of extremal Kähler metrics; a class in this neighborhood is represented by the Kähler form of a metric of constant scalar curvature iff the Futaki invariant of the class vanishes. If, moreover, the derivative of the Futaki invariant at [ω] is “nondegenerate,” every small deformation of the complex manifold (M, J) also carries Kähler metrics of constant scalar curvature. We then apply these results to prove new existence theorems for extremal Kähler metrics on certain compact complex surfaces.

Copyright information

© Birkhäuser Verlag 1994

Authors and Affiliations

  • C. LeBrun
    • 1
  • S. R. Simanca
    • 1
  1. 1.Institute of Mathematical SciencesState University of New York at Stony BrookStony BrookUSA

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