Kähler Metrics of Constant Scalar Curvature on Hirzebruch Surfaces

It is shown that a Hirzebruch surface admits a Kähler metric (possibly indefinite) of constant scalar curvature if and only if its degree equals zero. There have been many extensive studies for positive-definite Kähler metrics of constant scalar curvature, especially, Kähler Einstein metrics and scalar-flat Kähler metrics, on existence, uniqueness, obstructions, and relationships with other notions, for example, certain stabilities of polarized Kähler manifolds (e.g., Donaldson [7] and references therein). An indefinite counterpart of the notion of Kähler metrics of constant scalar curvature is defined in a natural way similar to that in the positive-definite case, and includes the notions of indefinite Kähler-Einstein metrics and scalarflat indefinite Kähler metrics. The existence problem for such metrics has its own task in the geometry of pseudo-Riemannian manifolds. The lowest real dimension of an indefinite Kähler manifold (M, g) must be four and the signature of the metric g must be (2, 2). Matsushita [20] studied the existence problem for metrics of signature (2, 2), from a point of view of that for fields of two-planes. Indefinite Kähler metrics of signature (2, 2) also appear in mathematical physics. Motivated by the work of Ooguri-Vafa [23] on string theory, Petean [24] studied the existence problem for indefinite Kähler Einstein metrics on compact complex surfaces and obtained a complete classification of compact complex surfaces that admit Ricci-flat indefinite Kähler metrics. On the other hand, concerning the existence of scalar-flat indefinite Kähler metrics on compact complex surfaces (or equivalently, self-dual Kähler metrics of neutral signature), several results have been known. For example, many scalar-flat (nonRicci-flat) indefinite Kähler metrics have been constructed explicitly on the product S × S of two two-dimensional spheres, by an indefinite analogue of LeBrun’s ansatz (Tod [26], Kamada [14]). By construction, each of these metrics has an obvious (Hamiltonian) S-symmetry. Conversely, it has been shown in [14] that a compact scalar-flat indefinite Kähler surface (more generally, a ∗2000 Mathematics Subject Classification: 14J26, 53C55, 53C50. †