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On the Asymptotic Behavior of Complete Kähler Metrics of Positive Ricci Curvature
Published online by Cambridge University Press: 20 November 2018
Abstract
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Let 
$\left( X,\,g \right)$ be a complete noncompact Kähler manifold, of dimension 
$n\,\ge \,2$, 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 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 
${{H}^{i}}(\bar{X},\,\Omega \frac{1}{X}(\log \,D))$.
MSC classification
- Type
- Research Article
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- Copyright
- Copyright © Canadian Mathematical Society 2010
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