### Abstract

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_{c}X = 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)).

Original language | English |
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Pages (from-to) | 3-18 |

Number of pages | 16 |

Journal | Canadian Journal of Mathematics |

Volume | 62 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2010 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**On the asymptotic behavior of complete kähler metrics of positive Ricci curvature.** / Anchouche, Boudjemâa.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On the asymptotic behavior of complete kähler metrics of positive Ricci curvature

AU - Anchouche, Boudjemâa

PY - 2010/2

Y1 - 2010/2

N2 - 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 Kxq 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 dimcX = 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)).

AB - 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 Kxq 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 dimcX = 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)).

UR - http://www.scopus.com/inward/record.url?scp=76749135828&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=76749135828&partnerID=8YFLogxK

U2 - 10.4153/CJM-2010-001-0

DO - 10.4153/CJM-2010-001-0

M3 - Article

VL - 62

SP - 3

EP - 18

JO - Canadian Journal of Mathematics

JF - Canadian Journal of Mathematics

SN - 0008-414X

IS - 1

ER -