Toric Kähler metrics: cohomogeneity one examples of constant scalar curvature in action-angle coordinates. (English) Zbl 1210.53068

Abreu, Miguel

J. Geom. Symmetry Phys. 17, 1-33 (2010).

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 [E. Calabi, Semin. differential geometry, Ann. Math. Stud. 102, 259–290 (1982; Zbl 0487.53057)]) in local symplectic action-angle coordinates [see 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 [E. Calabi, Ann. Sci. Éc. Norm. Supér. (4) 12, 269–294 (1978; Zbl 0431.53056)], [C. LeBrun, Commun. Math. Phys. 118, No. 4, 591–596 (1988; Zbl 0659.53050)], [H. Pedersen and Y. S. Poon, Commun. Math. Phys. 136, No. 2, 309–326 (1991; Zbl 0792.53065)] and [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.

Reviewer: James G. Timourian (Edmonton)

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
53C80	Applications of global differential geometry to the sciences
53C99	Global differential geometry
53D05	Symplectic manifolds (general theory)
14M25	Toric varieties, Newton polyhedra, Okounkov bodies
32Q25	Calabi-Yau theory (complex-analytic aspects)

Keywords:

Kähler metrics; extremal metric; Kähler-Einstein metrics; symplectic geometry; toric manifolds

Citations:

Zbl 0487.53057; Zbl 0932.53043; Zbl 0431.53056; Zbl 0659.53050; Zbl 0792.53065; Zbl 0725.53066