Existence of higher extremal Kähler metrics on a minimal ruled surface (Q6143514)

Let \(c_n(\omega)\) be the top Chern form of a Kähler metric \(\omega\) on a compact Kähler \(n\)-dimensional manifold \(M\) and let \(\lambda\) be a smooth real-valued function on \(M\) such that \[ c_n(\omega)=\lambda \omega^n . \] In [\textit{V. P. Pingali}, Trans. Am. Math. Soc. 370, No. 10, 6995--7010 (2018; Zbl 1396.53099)], the notions of higher constant scalar curvature Kähler metrics (for brevity hcscK) and higher extremal Kähler metrics are introduced: \(\omega\) is said to be hcscK if \(\lambda\) is constant, while \(\omega\) is said to be higher extremal if \((\bar\partial \lambda)^\#\) is a holomorphic vector field. Let \(L\) be a degree \(d\neq0\) holomorphic line bundle on a genus \(g\geq 2\) (compact) Riemann surface \(\Sigma\). Let \(\mathcal{O}\) be the trivial line bundle on \(\Sigma\). Minimal ruled (complex) surfaces defined as the vector bundle projectivization of \(L\oplus \mathcal{O}\) are called pseudo-Hirzebruch surfaces (see [\textit{C. W. Tønnesen-Friedman}, J. Reine Angew. Math. 502, 175--197 (1998; Zbl 0921.53033)]). In the present paper, by using the momentum profile method (see [\textit{A. D. Hwang} and \textit{M. A. Singer}, Trans. Am. Math. Soc. 354, No. 6, 2285--2325 (2002; Zbl 0987.53032)]), the author constructs a higher extremal non-hcscK metric in each Kähler class on a pseudo-Hirzebruch surface. Moreover, by means of some computations involving the top Bando-Futaki invariant, he proves that every higher extremal metrics in any Kähler class containing a hcscK metric, need also to be hcscK. Therefore, hcscK metrics cannot exist in any Kähler class on these surfaces. Both problems are firstly addressed in the particular case when \(g=2\) and \( d=-1\), then it is shown that the same analysis and all the arguments of that special case are also valid in the general setting. Finally, a comparison with the usual extremal setup is made.