Geodesic
On the Asymptotic Behavior of Complete Kähler Metrics of Positive Ricci Curvature
Canadian journal of mathematics, Tome 62 (2010) no. 1, pp. 3-18

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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 ${{H}^{i}}(\bar{X},\,\Omega \frac{1}{X}(\log \,D))$ . 
Anchouche, Boudjemâa. On the Asymptotic Behavior of Complete Kähler Metrics of Positive Ricci Curvature. Canadian journal of mathematics, Tome 62 (2010) no. 1, pp. 3-18. doi: 10.4153/CJM-2010-001-0
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