A Remark on ’some Numerical Results in Complex Differential Geometry’

In order to find explicit numerical approximation of Kähler-Einstein metric of projective manifolds, Donaldson introduced in [3] various operators with good properties to approximate classical operators. See the discussions in Section 4.2 of [3] for more details related to our discussion. In this note we verify certain statement of Donaldson about the operator QK in Section 4.2 by using the full asymptotic expansion of Bergman kernel derived in [2, Theorem 4.18] and [4, §3.4]. Such statement is needed for the convergence of the approximation procedure. Let (X,ω, J) be a compact Kähler manifold of dimC X = n, and let (L, h ) be a holomorphic Hermitian line bundle on X. Let ∇L be the holomorphic Hermitian connection on (L, h) with curvature R. We assume that √ −1 2π R = ω. (1)