Extremal Kähler Metrics II

Associated with a Hamiltonian holomorphic vector field on a compact Kähler manifold, a nice functional on a space of Kähler metrics will be constructed as an integration of the bilinear pairing in [FM] contracted with the Hamiltonian holomorphic vector field. As applications, we have functionals whose critical points are extremal Kähler metrics or “Kähler-Einstein metrics” in the sense of [M4], respectively. Finally, the same method as used by [G1] allows us to obtain, from the convexity of , the uniqueness of “Kähler-Einstein metrics” on nonsingular toric Fano varieties possibly with nonvanishing Futaki character.

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Ambitoric geometry I: Einstein metrics and extremal ambikähler structures

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2014-0060 ◽

2016 ◽

Vol 2016 (721) ◽

Cited By ~ 5

Author(s):

Vestislav Apostolov ◽

David M. J. Calderbank ◽

Paul Gauduchon

Keyword(s):

Maxwell Equations ◽

Einstein Metrics ◽

Natural Extension ◽

Torus Action ◽

Local Geometry ◽

Kähler Metrics ◽

Kahler Metrics ◽

Bach Tensor ◽

Extremal Kähler Metrics

AbstractWe present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kähler metrics which are toric with respect to a common 2-torus action. In the generic case, these “ambitoric” structures have an intriguing local geometry depending on a quadratic polynomialWe use this description to classify 4-dimensional Einstein metrics which are hermitian with respect to both orientations, as well as a class of solutions to the Einstein–Maxwell equations including riemannian analogues of the Plebański–Demiański metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors.These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kähler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kähler metrics. For both the Einstein–Maxwell and the extremal ambitoric structures,

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