Extremal Kähler Metrics of Toric Manifolds

An-Min Li , Li Sheng

Chinese Annals of Mathematics, Series B ›› 2023, Vol. 44 ›› Issue (6) : 827 -836.

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Chinese Annals of Mathematics, Series B ›› 2023, Vol. 44 ›› Issue (6) : 827 -836. DOI: 10.1007/s11401-023-0047-4
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Extremal Kähler Metrics of Toric Manifolds

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Abstract

This paper is a survey of some recent developments concerning extremal Kähler metrics on Toric Manifolds.

Keywords

Extremal Kähler metric / Toric manifolds / $\widehat K$-stability / Uniform stability

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An-Min Li, Li Sheng. Extremal Kähler Metrics of Toric Manifolds. Chinese Annals of Mathematics, Series B, 2023, 44(6): 827-836 DOI:10.1007/s11401-023-0047-4

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References

[1]

Abreu M. Kähler geometry of toric varieties and extremal metrics. Internat. J. Math., 1998, 9: 641-651

[2]

Apostolov V, Calderbank D M J, Gauduchon P, Tønnesen-Friedman C W. Hamiltonian 2-forms in Kähler geometry III, extremal metrics and stability. Invent. Math., 2008, 173(3): 547-601

[3]

Berman R, Darvas T, Lu C H. Regularity of weak minimizers of the K-energy and applications to properness and K-stability. Ann. Sci. Ec. Norm. Super., 2020, 53(4): 267-289

[4]

Boucksom S, Hisamoto T, Jonsson M. Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs. Ann. Inst. Fourier (Grenoble), 2017, 67(2): 743-841

[5]

Chen B, Han Q, Li A-M Prescribed scalar curvatures for homogeneous toric bundles. Differential Geometry and its Applications, 2019, 63: 186-211

[6]

Chen B, Li A-M, Sheng L. Uniform K-stability for extremal metrics on toric varieties. J. Diff. Equations, 2014, 257: 1487-1500

[7]

Chen B, Li A-M, Sheng L. Extremal metrics on toric surfaces. Adv. Math., 2018, 340: 363-405

[8]

Chen X, Cheng J. On the constant scalar curvature Kähler metrics (I)–A priori estimates. J. Amer. Math. Soc., 2021, 34(4): 909-936

[9]

Chen X, Cheng J. On the constant scalar curvature Kähler metrics (II)-Existence results. J. Amer. Math. Soc., 2021, 34(4): 937-1009

[10]

Darvas T, Lu C. Geodesic stability, the space of rays and uniform convexity in Mabuchi geometry. Geom. Topol., 2020, 24(4): 1907-1967

[11]

Delcroix, T. with an appendix by Yuji Odaka, Uniform K-stability of polarized spherical varieties, Epijournal Geom. Algebrique, 7, 2023, Art. 9, 27pp.

[12]

Donaldson S K. Scalar curvature and stability of toric varieties. J. Diff. Geom., 2002, 62: 289-349

[13]

Donaldson S K. Kähler geometry on toric manifolds, and some other manifolds with large symmetry. Handbook of Geometric Analysis, 2008, Boston: International Press 1

[14]

Donaldson S K. Constant scalar curvature metrics on toric surfaces. Geom. Funct. Anal., 2009, 19: 83-136

[15]

Donaldson S K. Stability of algebraic varieties and Kähler geometry. Algebraic Geometry: Salt Lake City 2015, 2018, Providence, RI: AMS

[16]

Donaldson S K. Extremal Kähler metrics and convex analysis. LMS Newsletter, 2022, 500: 32-36

[17]

Guillemin V. Kähler structures on toric varieties. J. Diff. Geom., 1994, 40: 285-309

[18]

Hisamoto, T., Stability and coercivity for toric polarizations, 2016, arXiv:1610.07998.

[19]

Li A-M, Lian Z, Sheng L. Extremal metrics on toric manifolds and homogeneous toric bundles. J. Reine. Angew. Math., 2023, 2023(798): 237-259

[20]

Li C. Geodesic rays and stability in the cscK problem. Ann. Sci. Éc. Norm. Supér., 2022, 55(6): 1529-1574

[21]

Nyström D W. Test configurations and Okounkov bodies. Compos. Math., 2012, 148(6): 1736-1756

[22]

Székelyhidi, G., Extremal metrics and K-stability, Ph. D. Thesis, 2006, arXiv: math/0611002.

[23]

Székelyhidi G. Filtrations and test-configurations. Math. Ann., 2015, 362: 451-484 with an appendix by Sebastien Boucksom

[24]

Tian G. Kähler-Einstein metrics with postive scalar curvature. Invent. Math., 1997, 130: 1-39

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