Abstract
The subject of this paper is the explicit momentum construction of complete Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized by Hwang-Singer, we show that there are non-trivial complete conformally Kähler–Einstein metrics on certain Hermitian holomorphic vector bundles and their subbundles over complete Kähler–Einstein manifolds. In special cases, we give the explicit expressions of these metrics. These examples show that there are a compact Kähler manifold M and its subvariety N whose codimension is greater than 1 such that there is a complete conformally Kähler–Einstein metric on M–N.
Similar content being viewed by others
References
Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: The geometry of weakly self-dual Kähler surfaces. Compositio Math. 135, 279–322 (2003)
Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: Hamiltonian 2-forms in Kähler geometry, I general theory. J. Differ. Geom. 73, 359–412 (2006)
Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: Ambitoric geometry I, Einstein metrics and extremal ambikähler structures. J. Reine Angew. Math. 721, 109–147 (2016)
Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: Ambitoric geometry II, extremal toric surfaces and Einstein 4-orbifolds. Ann. Sci. Éc. Norm. Supér. 48(4), 1075–1112 (2015)
Apostolov, V., Gauduchon, P.: Selfdual Einstein hermitian four-manifolds. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 1(1), 203–243 (2002)
Apostolov, V., Maschler, G.: conformally Kähler, Einstein-Maxwell geometry. J. Eur. Math. Soc. 21, 1319–1360 (2019)
Apostolov, V., Maschler, G., Tønnesen-Friedman, C.W.: Weighted extremal Kähler metrics and the Einstein-Maxwell geometry of projective bundles, arXiv:1808.02813
Bérard-Bergery, L.: Sur de nouvelles variétés riemanniennes d’Einstein, (French) [Some new Einstein Riemannian manifolds] Institut Élie Cartan, 6, 1-60, Inst. Élie Cartan, 6, Univ. Nancy, Nancy (1982)
Besse, A.L.: Einstein Manifolds, Ergebnisse (3) 10. Springer, Berlin (1987)
Bryant, R.: Bochner-Kähler metrics. J. Amer. Math. Soc. 14(3), 623–715 (2001)
Chen, X.X., LeBrun, C., Weber, B.: On conformally Kähler, Einstein manifolds. J. Amer. Math. Soc. 21, 1137–1168 (2008)
Derdziński, A.: Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compositio Math. 49, 405–433 (1983)
Derdziński, A.: Hermitian Einstein metrics, Global Riemannian Geometry, Durham, 1983 In: Willmore, T.J., Hitchin, N. Ellis H. (eds.), pp. 105-114. Chichester (1984)
Derdziński, A., Maschler, G.: Local classification of conformally-Einstein Kähler metrics in Higher demensions. Proc. Lond. Math. Soc. 87(3), 779–819 (2003)
Derdziński, A., Maschler, G.: Special Kähler-Ricci potentials on compact Kähler manifolds. J. Reine Angew. Math. 593, 73–116 (2006)
Derdziński, A., Maschler, G.: A moduli curve for compact conformally-Einstein Kähler manifolds. Compositio Math. 141, 1029–1080 (2005)
Dixon, K.: Regular ambitoric \(4\)-manifolds: from Riemannian Kerr to a complete classification. Commun. Anal. Geom. 29(3), 629–679 (2021)
Feng, Z.M.: The first two coefficients of the Bergman function expansions for Cartan-Hartogs domains. Int. J. Math. 29, 1850043 (2018)
Feng, Z.M.: Rotationally symmetric conformal Kähler, Einstein-Maxwell metrics. N. Y. J. Math. 26, 334–361 (2020)
Fu, J., Yau, S.T., Zhou, W.: On complete constant scalar curvature Kähler metrics with Poincaré-Mok-Yau asymptotic property. Commun. Anal. Geom. 24(3), 521–557 (2016)
Futaki, A., Ono, H.: Volume minimization and conformally Kähler, Einstein-Maxwell geometry. J. Math. Soc. Jpn. 70(4), 1493–1521 (2018)
Futaki, A., Ono, H.: conformally Einstein-Maxwell Kähler metrics and structure of the automorphism group. Mathematische Zeitschrift 292, 571–589 (2019)
Gao, P., Yau, S.T., Zhou, W.: Nonexistence for complete Kähler-Einstein metrics on some noncompact manifolds. Math. Ann. 369, 1271–1282 (2017)
Gordon, W.B.: An analytical criterion for the completeness of Riemannian manifolds. Proc. Am. Math. Soc. 37(1), 221–225 (1973)
Gordon, W.B.: Corrections to an analytical criterion for the completeness of riemannian manifolds. Proc. Am. Math. Soc. 45(1), 130–131 (1974)
Hwang, A., Singer, M.: A momentum construction for circle-invariant Kähler metrics. Trans. Am. Math. Soc. 354, 2285–2325 (2002)
Koca, C., Tønnesen-Friedman, C.W.: Strongly Hermitian Einstein-Maxwell solutions on ruled surfaces. Ann. Glob. Ann. Geom. 50, 29–46 (2016)
Lahdili, A.: Automorphisms and deformations of conformally Kähler, Einstein-Maxwell metrics. J. Geom. Anal. 29, 542–568 (2019)
Lahdili, A.: conformally Kähler, Einstein-Maxwell metrics and boundedness of the modified Mabuchi functional, Int. Math. Res. Notices (IMRN), rny239 (2018)
Lahdili, A.: Kähler metrics with constant weighted scalar curvature and weighted K-stability. Proc. Lond. Math. Soc. 119(4), 1065–1114 (2019)
LeBrun, C.R.: The Einstein-Maxwell equations, Kähler metrics, and Hermitian geometry. J. Geom. Phys. 91, 163–171 (2015)
LeBrun, C.R.: The Einstein-Maxwell equations and conformally Kähler geometry. Commun. Math. Phys. 344, 621–653 (2016)
Page, D.: A compact rotating gravitational instanton. Phys. Lett. B 79, 235–238 (1978)
Wang, A., Yin, W.P., Zhang, L.Y., Roos, G.: The Kähler-Einstein metric for some Hartogs domains over bounded symmetric domains. Sci. China Ser. A Math. 49(9), 1175–1210 (2006)
Acknowledgements
The author would like to thank the referee for many helpful suggestions and editor’s comments. The author was supported in part by the National Natural Science Foundation of China (No.12071354) and the Scientific Research Fund of Leshan Normal University (No.DGZZ202024).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Feng, Z. Globally conformally Kähler Einstein metrics on certain holomorphic bundles. Annali di Matematica 202, 1087–1129 (2023). https://doi.org/10.1007/s10231-022-01272-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-022-01272-0