Hirzebruch Surfaces and Weighted Projective Planes

Abstract

For any positive integer, we show that the standard self-dual orbifold Kähler structure of the weighted projective surface ℙ_1,1,k can be realized as a limit of the Hirzebruch surface F _k , 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 ℙ_1,1,k and F _k .

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 F _k is the blow-up of the weighted projective plane $\mathbb{P}_k^2$ 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 $\mathbb{P}_k^2$, equipped with its standard self-dual orbifold Kähler metric — cf. Section 1 — can be viewed as a limit of the Hirzebruch surface F _k , 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 F _k and $\mathbb{P}_k^2$.

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 surfaces (Section 2), and of their toric structures (Section 3). The limiting process itself is firstly described in the toric setting in Section 3, then, in a more precise formulation — cf. Theorem 2 — in Section 4.