Hirzebruch Surfaces and Weighted Projective Planes

For any positive integer, we show that the standard self-dual orbifold Kähler structure of the weighted projective surface ℙ1,1,k can be realized as a limit of the Hirzebruch surface Fk, equipped with a sequence of Calabi extremal Kähler metrics whose Kähler classes tend to the boundary of the Kähler cone, and that this collapsing process is compatible with the natural toric structures of ℙ1,1,k and Fk. In reference to [25], nontrivial (geometrically) ruled surfaces of genus zero are usually called Hirzebruch surfaces. The first Hirzebruch surface F1 is well-known to be the blow-up of the complex projective plane at one point; more generally, the k-th Hirzebruch surface Fk is the blow-up of the weighted projective plane\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{P}_k^2 $$\end{document} of weight k = (1,1,k) at its (unique) singular point, cf., e.g., [19]. The aim of this article is to show that, for any fixed positive integer k, the weighted projective plane \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{P}_k^2 $$\end{document}, equipped with its standard self-dual orbifold Kähler metric — cf. Section 1 — can be viewed as a limit of the Hirzebruch surface Fk, when the latter is equipped with a sequence of Calabi extremal Kähler metrics whose Kähler classes tend to the boundary of the Kähler cone. Moreover, we show that this limiting — or collapsing — process fits nicely with the natural toric structures of Fk and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{P}_k^2 $$\end{document}. Notice that our construction can be regarded as an illustration of the general weak compactness theorem recently established by X. Chen and B. Weber in [16], cf. also [15]. In order to make this paper reasonably self-contained, we included a somewhat detailed exposition of the Bochner-flat Kähler metrics of weighted projective spaces in general (Section 1), of Calabi extremal Kähler metrics on Hirzebruch surface