On the asymptotic behavior of complete kähler metrics of positive Ricci curvature

Let (X, g) be a complete noncompact Kähler manifold, of dimension n ≥ 2, with positive Ricci curvature and of standard type (see the definition below). N. Mok proved that X can be compactified, i.e., X is biholomorphic to a quasi-projective variety. The aim of this paper is to prove that the L ² holomorphic sections of the line bundle K_x^q and the volume form of the metric g have no essential singularities near the divisor at infinity. As a consequence we obtain a comparison between the volume forms of the Kahler metric g and of the Fubini-Study metric induced on X. In the case of dim_cX = 2, we establish a relation between the number of components of the divisor D and the dimension of the groups H ⁱ(X̄,Ω1/X(log D)).