Extremal Kähler-Ricci solitons on Fano homogeneous toric bundles

Lian, Zhao

doi:10.1090/proc/16966

Extremal Kähler-Ricci solitons on Fano homogeneous toric bundles
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by Zhao Lian;

Proc. Amer. Math. Soc. 152 (2024), 4873-4880

DOI: https://doi.org/10.1090/proc/16966

Published electronically: September 24, 2024

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Abstract:

In this short note, we prove that an extremal Kähler-Ricci soliton on a Fano homogeneous toric bundle is Kähler-Einstein.

References

Andreas Arvanitoyeorgos, An introduction to Lie groups and the geometry of homogeneous spaces, Student Mathematical Library, vol. 22, American Mathematical Society, Providence, RI, 2003. Translated from the 1999 Greek original and revised by the author. MR 2011126, DOI 10.1090/stml/022
Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
Eugenio Calabi, Extremal Kähler metrics, Seminar on Differential Geometry, Ann. of Math. Stud., No. 102, Princeton Univ. Press, Princeton, NJ, 1982, pp. 259–290. MR 645743
Simone Calamai and David Petrecca, On Calabi extremal Kähler-Ricci solitons, Proc. Amer. Math. Soc. 144 (2016), no. 2, 813–821. MR 3430856, DOI 10.1090/proc12725
Simone Calamai and David Petrecca, Toric extremal Kähler-Ricci solitons are Kähler-Einstein, Complex Manifolds 4 (2017), no. 1, 179–182. MR 3739311, DOI 10.1515/coma-2017-0012
Bohui Chen, Qing Han, An-Min Li, Zhao Lian, and Li Sheng, Prescribed scalar curvatures for homogeneous toric bundles, Differential Geom. Appl. 63 (2019), 186–211. MR 3903688, DOI 10.1016/j.difgeo.2019.01.002
S. K. Donaldson, Interior estimates for solutions of Abreu’s equation, Collect. Math. 56 (2005), no. 2, 103–142. MR 2154300
Simon K. Donaldson, Kähler geometry on toric manifolds, and some other manifolds with large symmetry, Handbook of geometric analysis. No. 1, Adv. Lect. Math. (ALM), vol. 7, Int. Press, Somerville, MA, 2008, pp. 29–75. MR 2483362
Alexander Kirillov Jr., An introduction to Lie groups and Lie algebras, Cambridge Studies in Advanced Mathematics, vol. 113, Cambridge University Press, Cambridge, 2008. MR 2440737, DOI 10.1017/CBO9780511755156
An-Min Li, Zhao Lian, and Li Sheng, Extremal metrics on toric manifolds and homogeneous toric bundles, J. Reine Angew. Math. 798 (2023), 237–259. MR 4579705, DOI 10.1515/crelle-2023-0017
An-Min Li, Li Sheng, and Guosong Zhao, Differential inequalities on homogeneous toric bundles, J. Geom. 108 (2017), no. 2, 775–790. MR 3667253, DOI 10.1007/s00022-017-0372-4
Toshiki Mabuchi, Test configurations, stabilities and canonical Kähler metrics—complex geometry by the energy method, SpringerBriefs in Mathematics, Springer, Singapore, [2021] ©2021. MR 4248091, DOI 10.1007/978-981-16-0500-0
Yasufumi Nitta, On relations between Mabuchi’s generalized Kähler-Einstein metrics and various canonical Kähler metrics, Proc. Amer. Math. Soc. 151 (2023), no. 7, 3083–3087. MR 4579380, DOI 10.1090/proc/16385
T. Nyberg, Constant Scalar Curvature of Toric Fibrations, Ph.D. Thesis, Columbia University.
Fabio Podestà and Andrea Spiro, Kähler-Ricci solitons on homogeneous toric bundles, J. Reine Angew. Math. 642 (2010), 109–127. MR 2658183, DOI 10.1515/CRELLE.2010.038
A. Raza, Scalar Curvature and Multiplicity-Free Actions, Ph.D. Thesis, Imperial College London, 2005.