Indefinite Kähler metrics of constant scalar curvature on Hirzebruch surfaces

Abstract

It is known that a Hirzebruch surface admits a positive-definite Kähler metric of constant scalar curvature if and only if its degree is zero, that is, it is biholomorphic to the product of two complex projective lines. This result was shown usually by using the Matsushima–Lichnerowicz obstruction for the existence of such metrics; however, an indefinite analogue of this obstruction is still unknown. In this article, we introduce the Bando–Calabi–Futaki obstruction for Kähler classes of arbitrary signature on any compact toric manifold. As an application, we prove, from a unified aspect, that the result stated above holds in both positive-definite and indefinite cases.