Note on Poincar\'e type K\"ahler metrics and Futaki characters

Abstract

A Poincar\'e type K\"ahler metric on the complement X\D of a simple normal crossing divisor D, in a compact K\"ahler manifold X, is a K\"ahler 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 Poincar\'e type K\"ahler class, to the classical Futaki character for the relative smooth class. As an application we express a numerical obstruction to the existence of extremal Poincar\'e type K\"ahler metrics, in terms of mean scalar curvatures and Futaki characters.