Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-06-12T09:37:10.540Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  13 May 2024

Qi Yao*
Mathematics Department, Stony Brook University, 100 Nicolls Road, Stony Brook, NY 11794, United States


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.

© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


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.


Abreu, M. and Sena-Dias, R., Scalar-flat Kähler metrics on non-compact symplectic toric 4-manifolds . Ann. Global Anal. Geom. 41(2012), no. 2, 209239.CrossRefGoogle Scholar
Adams, J. F., Lectures on Lie groups, University of Chicago Press, Chicago, IL, 1982.Google Scholar
Apostolov, V. and Cifarelli, C., Hamiltonian 2-forms and new explicit Calabi–Yau metrics and gradient steady Kähler–Ricci solitons on ${C}^n$ . Preprint, 2023. arXiv:2305.15626 Google Scholar
Auvray, H., The space of Poincaré type Kähler metrics on the complement of a divisor . J. Reine Angew. Math. 2017(2017), no. 722, 164.CrossRefGoogle Scholar
Berman, R. and Berndtsson, B., Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics . J. Amer. Math. Soc. 30(2017), no. 4, 11651196.CrossRefGoogle Scholar
Besse, A. L., Einstein manifolds, Springer, Berlin, 2007.Google Scholar
Borel, A. and Hirzebruch, F., Characteristic classes and homogeneous spaces, I . Amer. J. Math. 80(1958), no. 2, 458538.CrossRefGoogle Scholar
Borel, A. and Remmert, R., Über kompakte homogene Kählersche Mannigfaltigkeiten . Math. Ann. 145(1962), no. 5, 429439.CrossRefGoogle Scholar
Bott, R., Homogeneous vector bundles . Ann. Math. 66(1957), 203248.CrossRefGoogle Scholar
Calabi, E., Métriques kählériennes et fibrés holomorphes . Ann. Sci. Ec. Norm. Supér. (4) 12(1979), no. 2, 269294.CrossRefGoogle Scholar
Calderbank, D. M. J. and Singer, M. A., Einstein metrics and complex singularities . Invent. Math. 156(2004), no. 2, 405443.CrossRefGoogle Scholar
Chen, X., The space of Kähler metrics . J. Differential Geom. 56(2000), no. 2, 189234.CrossRefGoogle Scholar
Conlon, R. J. and Hein, H.-J., Asymptotically conical Calabi–Yau manifolds, I . Duke Math. J. 162(2013), no. 15, 28552902.CrossRefGoogle Scholar
Conlon, R. J. and Rochon, F., New examples of complete Calabi–Yau metrics on ${C}^n$ for $n\ge 3$ . Ann. Sci. Ec. Norm. Supér. (4) 54(2021), 259303.CrossRefGoogle Scholar
Dancer, A. and Wang, M. Y., Kähler–Einstein metrics of cohomogeneity one . Math. Ann. 312(1998), 503526.CrossRefGoogle Scholar
Donaldson, S., A generalised Joyce construction for a family of nonlinear partial differential equations . J. Gökova Geom. Topol. 3(2009), 18.Google Scholar
Eguchi, T. and Hanson, A. J., Self-dual solutions to Euclidean gravity . Ann. Physics 120(1979), no. 1, 82106.CrossRefGoogle Scholar
Han, J. and Viaclovsky, J. A., Deformation theory of scalar-flat kähler ale surfaces . Amer. J. Math. 141(2019), no. 6, 15471589.CrossRefGoogle Scholar
Honda, N., Deformation of LeBrun’s ALE metrics with negative mass . Comm. Math. Phys. 322(2013), no. 1, 127148.CrossRefGoogle Scholar
Honda, N., Scalar flat Kähler metrics on affine bundles over ${CP}^1$ . SIGMA. Symmetry,Integrability and Geometry: Methods and Applications 10(2014), 046.CrossRefGoogle Scholar
Humphreys, J. E., Linear algebraic groups. Vol. 21, Springer, New York, 2012.Google Scholar
Hwang, A. and Singer, M., A momentum construction for circle-invariant Kähler metrics . Trans. Amer. Math. Soc. 354(2002), no. 6, 22852325.CrossRefGoogle Scholar
Joyce, D. D., Explicit construction of self-dual $4$ -manifolds . Duke Math. J. 77(1995), no. 3, 519552.CrossRefGoogle Scholar
Knapp, A. W., Lie groups beyond an introduction. Vol. 140, Springer, Cambridge, MA, 2013.Google Scholar
LeBrun, C., Counter-examples to the generalized positive action conjecture . Comm. Math. Phys. 118(1988), no. 4, 591596.CrossRefGoogle Scholar
LeBrun, C., Explicit self-dual metrics on ${CP}^2$ # $\dots$ # ${CP}^2$ . J. Differential Geom. 34(1991), 223253.CrossRefGoogle Scholar
Li, L. and Zheng, K., Uniqueness of constant scalar curvature Kähler metrics with cone singularities. I: Reductivity . Math. Ann. 373(2019), 679718.CrossRefGoogle Scholar
Li, Y., A new complete Calabi–Yau metric on ${C}^3$ . Invent. Math. 217(2019), 134.CrossRefGoogle Scholar
Lock, M. T. and Viaclovsky, J. A., A smörgåsbord of scalar-flat Kähler ALE surfaces . J. Reine Angew. Math. 2019(2019), no. 746, 171208.CrossRefGoogle Scholar
Matsushima, Y., Sur les espaces homogènes Kählériens d’un groupe de Lie réductif . Nagoya Math. J. 11(1957), 5360.CrossRefGoogle Scholar
Popov, V. L., Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles . Math. Izv. 8(1974), no. 2, 301.CrossRefGoogle Scholar
Sena-Dias, R., Uniqueness among scalar-flat Kähler metrics on non-compact toric 4-manifolds . J. Lond. Math. Soc. 103(2021), no. 2, 372397.CrossRefGoogle Scholar
Serre, J.-P., Géométrie algébrique et géométrie analytique . Ann. Inst. Fourier 6(1956), 142.CrossRefGoogle Scholar
Stenzel, M. B., Ricci-flat metrics on the complexification of a compact rank one symmetric space . Manuscripta Math. 80(1993), no. 1, 151163.CrossRefGoogle Scholar
Székelyhidi, G., Degenerations of ${C}^n$ and Calabi-Yau metrics. Duke Mathematical Journal, Duke University Press, 168(2019), no. 14, 26512700.Google Scholar
Tian, G. and Yau, S.-T., Complete Kähler manifolds with zero Ricci curvature. I . J. Amer. Math. Soc. 3(1990), no. 3, 579609.Google Scholar
Tian, G. and Yau, S. T., Complete Kähler manifolds with zero Ricci curvature. II . Invent. Math. 106(1991), 2760.CrossRefGoogle Scholar
Wang, M., Einstein metrics from symmetry and bundle constructions . Surv. Differ. Geom. 6(1):287325, 2001.CrossRefGoogle Scholar
Wright, D., Compact anti-self-dual orbifolds with torus actions . Selecta Math. 17(2011), 223280.CrossRefGoogle Scholar
Yau, S.-T., On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampére equation, I . Comm. Pure Appl. Math. 31(1978), no. 3, 339411.CrossRefGoogle Scholar