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

TL;DR

This paper develops a theory for extremal Poincaré type Kähler metrics on toric varieties, providing necessary conditions and explicit examples on Hirzebruch surfaces, including both Poincaré and non-Poincaré extremal metrics.

Contribution

It establishes necessary conditions for the existence of extremal Poincaré type Kähler metrics on toric varieties and confirms their sufficiency in specific cases like Hirzebruch surfaces.

Findings

Necessary conditions for existence are identified.

On Hirzebruch surfaces, conditions are also sufficient.

Examples of Poincaré and non-Poincaré extremal metrics are constructed.

Abstract

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.