Invariant scalar-flat Kähler metrics on \(\mathcal {O} (- \ell )\) (Q2000021)

The author studies several extremal Kähler metrics, in a unified framework based on the momentum profile by \textit{A. D. Hwang} and \textit{M. A. Singer} [Trans. Am. Math. Soc. 354, No. 6, 2285--2325 (2002; Zbl 0987.53032)]. In the first part, the author focuses on $\mathrm{U}(m)$-invariant scalar-flat Kähler metrics on $\mathcal{O}(-\ell)$ over $\mathbb{P}^{m-1}$, which include the Burns-Simanca metrics, see [\textit{D. Burns}, Twistors and harmonic maps. AMS conference talk (unpublished) (1986); \textit{S. R. Simanca}, Math. Ann. 291, No. 2, 239--246 (1992; Zbl 0725.53066)], and \textit{C. LeBrun}'s metrics, see [Commun. Math. Phys. 118, No. 4, 591--596 (1988; Zbl 0659.53050)]. Then, he also considers the extremal Kähler metrics on ruled manifolds over $\mathbb P^{m-1}$ by \textit{E. Calabi} [Ann. Math. Stud. 102, 259--290 (1982; Zbl 0487.53057)].