Extremal Kähler Poincaré Type Metrics on Toric Varieties

Abstract

We develop a general theory for the existence of extremal Kahler metrics of Poincare type in the sense of Auvray (J Reine Angew Math 722:1–64, 2017), defined on the complement of a torus invariant divisor of a smooth compact toric variety. In the case when the divisor is smooth, we obtain a list of necessary conditions which must be satisfied for such a metric to exist. Using the explicit methods of Apostolov et al. (Ann Sci Ecole Norm Supp (4) 48:1075–1112, 2015; J Reine Angew Math 721:109–147, 2016, https://doi.org/10.1515/crelle-2014-0060 ) together with the computational approach of Sektnan (N Y J Math 24:317–354, 2018), we show that on a Hirzebruch complex surface the necessary conditions are also sufficient. In particular, on such a complex surface the complement of the infinity section admits an extremal Kahler metric of Poincare type, whereas the complement of a fibre fixed by the torus action admits a complete ambitoric extremal Kahler metric which is not of Poincare type.