Note on Poincaré Type Kähler Metrics and Futaki Characters

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 onX\D with cusp singularity alongD. 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. Introduction A basic fact in Kähler geometry is the independence of the de Rham class of the Ricci form from the background metric on a compact Kähler manifold: it is always −2πc1(K), with c1(K) the first Chern class of the canonical line bundle. This topological invariance constitutes the first obstacle for a compact Kähler manifold to admit a Kähler-Einstein metric: the Chern class in question must then have a sign, which, if definite, forces Kähler-Einstein metrics to lie in a consequently fixed Kähler class. When c1(K) > 0, a (unique) Kähler-Einstein metric was obtained by Aubin and Yau, and Bochner’s technique then rules out the existence of non-trivial holomorphic vector fields . Conversely, in the opposite case c1(K) < 0, the so-called “Fano case”, non-trivial holomorphic vector fields may exist, and the existence of a Kähler-Einstein metric, which does not always hold, is noticeably more involved. More precisely, in this case – and, respectively, on any compact Kähler manifold – non-trivial holomorphic vector fields bring a constraint to the existence of a Kähler-Einstein metric – respectively, of a constant scalar curvature metric Kähler metric in a fixed Kähler class. If indeed such a canonical metric exists, a numerical