Extremal K\"ahler Poincar\'e type metrics on toric varieties

We develop a general theory for the existence of extremal K"ahler metrics of Poincar'e 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 K"ahler metric of Poincar'e type whereas the complement of a fibre admits a complete ambitoric extremal K"ahler metric which is not of Poincar'e type.