Toric Kähler metrics: cohomogeneity one examples of constant scalar curvature in action-angle coordinates (Q983098)

As the author states in the introduction these notes present Calabi's general four parameter family of \(U(n)\)-invariant extremal Kähler metrics (constructed using complex coordinates [\textit{E. Calabi}, Semin. differential geometry, Ann. Math. Stud. 102, 259--290 (1982; Zbl 0487.53057)]) in local symplectic action-angle coordinates [see \textit{M. Abreu}, Int. J. Math. 9, No.~6, 641--651 (1998; Zbl 0932.53043)]. By doing he is able to show that the family contains a wide variety of interesting cohomogeneity one Kähler metrics as special cases. Among them constructions in [\textit{E. Calabi}, Ann. Sci. Éc. Norm. Supér. (4) 12, 269--294 (1978; Zbl 0431.53056)], [\textit{C. LeBrun}, Commun. Math. Phys. 118, No.~4, 591--596 (1988; Zbl 0659.53050)], [\textit{H. Pedersen} and \textit{Y. S. Poon}, Commun. Math. Phys. 136, No.~2, 309--326 (1991; Zbl 0792.53065)] and [\textit{S. R. Simanca}, ``Kähler metrics of constant scalar curvature on bundles over \({\mathbb C}{\mathbb P}^{n-1}\)'', Math. Ann. 291, No.~2, 239--246 (1991; Zbl 0725.53066)]. As is appropriate for notes from a conference mini course this one contains a brief introduction to symplectic geometry, toric symplectic manifolds and toric Kähler metrics, with a good number of examples.