On the Asymptotic Behavior of Complete Kähler Metrics of Positive Ricci Curvature

http://dx.doi.org/10.4153/CJM-2010-001-0
Canad. J. Math. 62(2010), 3-18

  Published:2009-12-04
 Printed: Feb 2010

Let $( X,g) $ be a complete noncompact Kähler manifold, of dimension $n\geq2,$ with positive Ricci curvature and of standard type (see the definition below). N. Mok proved that $X$ can be compactified, \emph{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 Kähler metric $g$ and of the Fubini--Study metric induced on $X$. In the case of $\dim_{\mathbb{C} }X=2,$ we establish a relation between the number of components of the divisor $D$ and the dimension of the groups $H^{i}( \overline{X}, \Omega_{\overline{X}}^{1}( \log D) )$.