Extremal Kähler metrics and complex deformation theory

Abstract

Let (M, J, g) be a compact Kahler manifold of constant scalar curvature. Then the Kahler class [ω] has an open neighborhood inH 1,1 (M, ℝ) consisting of classes which are represented by Kahler forms of extremal Kahler metrics; a class in this neighborhood is represented by the Kahler 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 Kahler metrics of constant scalar curvature. We then apply these results to prove new existence theorems for extremal Kahler metrics on certain compact complex surfaces.