The space of Poincaré type Kähler metrics on the complement of a divisor

Abstract

Abstract Consider a divisor D with simple normal crossings in a compact Kähler manifold X. It has been known since the work by G. Tian and S. T. Yau that if K X ⁢ [ D ] $K_{X}[D]$ is ample there exists on X \ D $X\backslash D$ a unique Kähler–Einstein metric with cusp singularities along the divisor (implying completeness and finite volume). We show in this article that a Kähler metric in an arbitrary class with constant scalar curvature and singularities analogous to that constructed by Tian and Yau, is unique in this class when K X ⁢ [ D ] $K_{X}[D]$ is ample. This we do by generalizing Chen’s construction of approximate geodesics in the space of Kähler metrics and proving an approximate version of the Calabi–Yau theorem, both independently of the ampleness of K X ⁢ [ D ] $K_{X}[D]$ .