Note on Poincaré type Kähler metrics and Futaki characters

2013/12/31 by Hugues Auvray, Auvray, Hugues
Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1401.0128

openalex publication_date 2013/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A Poincaré type Kähler metric on the complement X\D of a simple normal crossing divisor D, in a compact Kähler manifold X, is a Kähler metric on X\D with cusp singularity along D. We relate the Futaki character for holomorphic vector fields parallel to the divisor, defined for any fixed Poincaré type Kähler class, to the classical Futaki character for the relative smooth class. As an application we express a numerical obstruction to the existence of extremal Poincaré type Kähler metrics, in terms of mean scalar curvatures and Futaki characters.

Citations

• Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof
• Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities

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• The uniqueness of Poincaré type constant scalar curvature Kähler metric
• The space of Poincar 'e type K "ahler metrics on the complement of a\n divisor
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• Extremal almost-Kähler metrics