Abstract
Let (M, J, g) be a compact Kähler manifold of constant scalar curvature. Then the Kähler class [ω] has an open neighborhood inH 1,1 (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.
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The first author is supported in part by NSF grant DMS 92-04093.
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LeBrun, C., Simanca, S.R. Extremal Kähler metrics and complex deformation theory. Geometric and Functional Analysis 4, 298–336 (1994). https://doi.org/10.1007/BF01896244
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DOI: https://doi.org/10.1007/BF01896244