Holomorphic vector fields and perturbed extremal Kahler metrics精编.pdfVIP

HOLOMORPHIC VECTOR FIELDS AND PERTURBED ¨ 7 EXTREMAL KAHLER METRICS 0 0 2 AKITO FUTAKI b e Abstract. We prove a theorem which asserts that the Lie algebra of all holo- F morphic vector fields on a compact K¨ahler manifold with a perturbed extremal 4 metric has the structure similar to the case of an unperturbed extremal K¨ahler 2 metric proved by Calabi. ] G D 1. Introduction . h Let M be a compact symplectic manifold with symplectic form ω . On the space t a J of all ω-compatible complex structures J there is a natural symplectic form with m respect to which the scalar curvature S(J) of the K¨ahler manifold (M, ω, J) becomes [ a moment map for the action of the group of all Hamiltonian diffeomorphisms of (M, ω) acting on J (c.f. [3], [4]). This means that the problem of finding extremal 1 v K¨ahler metrics can be set in the framework of stability in the sense of geometric 1 invariant theory. It was shown in [7] that, perturbing the symplectic form on J and 2 the scalar curvature incorporating with the higher Chern classes and with a small 7 real parameter t, the perturbed scalar curvature S(J, t) becomes a moment map 2 with respect to the perturbed symplectic form on J . Note that the unperturbed 0 7 scalar curvature is