2017/11/22 by Apostolov, Vestislav, Auvray, Hugues, Sektnan, Lars Martin · 1 citation
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1711.08424
We develop a general theory for the existence of extremal Kähler metrics of Poincaré type in the sense of Auvray, defined on the complement of a toric divisor of a polarized 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-Calderbank-Gauduchon together with the computational approach of Sektnan, 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 Kähler metric of Poincaré type whereas the complement of a fibre admits a complete ambitoric extremal Kähler metric which is not of Poincaré type.