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


Published:20091204
Printed: Feb 2010
Boudjemâa Anchouche,
Qaboos University, Oman
Abstract
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 quasiprojective
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 FubiniStudy
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 $H^{i}( \overline{X},
\Omega_{\overline{X}}^{1}( \log D) )$.