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Invariant scalar-flat Kähler metrics on line bundles over generalized flag varieties

Part of: Global differential geometry

Published online by Cambridge University Press:  13 May 2024

Qi Yao
Qi Yao*
Affiliation:
Mathematics Department, Stony Brook University, 100 Nicolls Road, Stony Brook, NY 11794, United States
*
e-mail: qi.yao@stonybrook.edu
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Abstract

Let G be a simply connected semisimple compact Lie group, let X be a simply connected compact Kähler manifold homogeneous under G, and let L be a negative holomorphic line bundle over X. We prove that all G-invariant Kähler metrics on the total space of L arise from the Calabi ansatz. Using this, we show that there exists a unique G-invariant scalar-flat Kähler metric in each G-invariant Kähler class of L. The G-invariant scalar-flat Kähler metrics are automatically asymptotically conical.

Keywords

Canonical metricsCalabi ansatzinvariant scalar-flat Kähler metricsasymptotically conical Kähler metrics

MSC classification

Primary: 53C30: Homogeneous manifolds 53C55: Hermitian and Kählerian manifolds
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 26
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work is completed while the author is partly supported by graduate assistant fellowship in Graduate Center, CUNY, and also funded by the DFG under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure, and by the CRC 1442, Geometry: Deformations and Rigidity, of the DFG.

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