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Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons

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Abstract

Mabuchi solitons generalize Kähler–Einstein metrics on Fano manifolds, which constitute a Yau–Tian–Donaldson type correspondence with relative Ding stability. Comparing with Kähler–Ricci solitons, there is a distinct necessary condition for the existence. We show this condition can be implied by the uniformly relative Ding stability. For this, we study the inner product of \(\mathbb {C}^{*}\)-actions on equivariant test-configurations and obtain an integration formula over the total space. To analyze the uniform stability, by adapting Okounkov body construction to the setting of torus action, we give a convex-geometry description for the reduced non-Archimedean J-functionals.

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References

  1. Atiyah, M.: Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 14(1), 1–15 (1982)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Berman, R.: K-polystability of Q-Fano varieties admitting Kähler–Einstein metrics. Invent. Math. 203(3), 973–1025 (2016)

    Article  MathSciNet  Google Scholar 

  4. Berman, R., Boucksom, S., Jonsson, M.: A variational approach to the Yau–Tian–Donaldson conjecture. To appear in J. Amer. Math. Soc. arXiv:1509.04561v3

  5. Berman, R., Witt Nyström, D.: Complex optimal transport and the pluripotential theory of Kähler–Ricci solitons. arXiv: 1401.8264

  6. Boucksom, S., Chen, H.: Okounkov bodies of filtered linear series. Compos. Math. 147(4), 1205–1229 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Boucksom, S., Hisamoto, T., Jonsson, M.: Uniform K-stability and asymptotics of energy functionals in Kähler geometry. J. Eur. Math. Soc. 21(9), 2905–2944 (2019)

    Article  MathSciNet  Google Scholar 

  9. Chen, X., Cheng, J.: On the constant scalar curvature Kähler metrics, general automorphism group. arXiv:1801.05907

  10. Chu, J., Tosatti, V., Weinkove, B.: C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays. Commun. Partial Differ. Equ. 43(2), 292–312 (2018)

    Article  Google Scholar 

  11. Collins, T., Hisamoto, T., Takahashi, R.: The inverse Monge–Ampère flow and applications to Kähler–Einstein metrics. To appear in J. Differ. Geom. arXiv:1712.01685v2

  12. Dervan, R.: Relative K-stability for Kähler manifolds. Math. Ann. 372(3–4), 859–889 (2018)

    Article  MathSciNet  Google Scholar 

  13. Donaldson, S.: Lower bounds on the Calabi functional. J. Differ. Geom. 70(3), 453–472 (2005)

    Article  MathSciNet  Google Scholar 

  14. Donaldson, S.: The Ding functional, Berndtsson Convexity and Moment Maps. Geometry, Analysis and Probability (In Honor of Jean-Michel Bismut). Birkhäuser, Cham (2017)

    Google Scholar 

  15. Futaki, A., Mabuchi, T.: Bilinear forms and extremal Kähler vector fields associated with Kähler classes. Math. Ann. 301(1), 199–210 (1995)

    Article  MathSciNet  Google Scholar 

  16. Han, J., Li, C.: On the Yau-Tian-Donaldson conjecture for generalized Kähler–Ricci soliton equations. arXiv:2006.00903v3

  17. Harder, G.: Lectures on Algebraic Geometry I. Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces. Springer, Wiesbaden (2011)

    MATH  Google Scholar 

  18. Hartshorne, R.: Algebraic Geometry. Springer, New York (1977)

    Book  Google Scholar 

  19. He, W.: On Calabi’s extremal metric and properness. Trans. Am. Math. Soc. 372(8), 5595–5619 (2019)

    Article  MathSciNet  Google Scholar 

  20. Hisamoto, T.: On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold. J. Reine Angew. Math. 713, 129–148 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Hisamoto, T.: Orthogonal projection of a test configuration to vector fields. arXiv: 1610.07158v3

  22. Hisamoto, T.: Stability and coercivity for toric polarizations. arXiv: 1610.07998v3

  23. Hisamoto, T.: Mabuchi’s soliton metric and relative D-stability. arXiv: 1905.05948v2

  24. Huybrechts, D.: Complex Geometry: An Introduction. Springer, Berlin (2005)

    MATH  Google Scholar 

  25. Lazarsfeld, R.: Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series. Springer, Berlin (2004)

    MATH  Google Scholar 

  26. Lazarsfeld, R., Mustaţă, M.: Convex bodies associated to linear series. Ann. Sci. Éc. Norm. Supér. 42(5), 783–835 (2009)

    Article  MathSciNet  Google Scholar 

  27. Li, C.: G-uniform stability and Kähler–Einstein metrics on Fano varieties. To appear in Invent. Math. https://doi.org/10.1007/s00222-021-01075-9 (2021)

  28. Li, C.: Geodesic rays and stability in the cscK problem. arXiv: 2001.01366

  29. Li, Y., Zhou, B.: Mabuchi metrics and properness of the modified Ding functional. Pac. J. Math. 302(2), 659–692 (2019)

    Article  MathSciNet  Google Scholar 

  30. Mabuchi, T.: Kähler–Einstein metrics for manifolds with nonvanishing Futaki character. Tohoku Math. J. 53(2), 171–182 (2001)

    Article  MathSciNet  Google Scholar 

  31. Mabuchi, T.: A theorem of Calabi–Matsushima’s type. Osaka J. Math. 39, 49–57 (2002)

    MathSciNet  MATH  Google Scholar 

  32. Mabuchi, T.: Multiplier Hermitian structures on Kähler manifolds. Nagoya Math. J. 170, 73–115 (2003)

    Article  MathSciNet  Google Scholar 

  33. Meinrenken, E.: On Riemann–Roch Formulas for Multiplicities. J. Am. Math. Soc. 9(2), 373–389 (1996)

    Article  MathSciNet  Google Scholar 

  34. Nitta,Y., Saito S., Yotsutani, N.: Relative Ding stability of toric Fano manifolds in low dimensions. arXiv: 1712.01131v3

  35. Okounkov, A.: Brunn–Minkowski inequality for multiplicities. Invent. Math. 125(3), 405–411 (1996)

    Article  MathSciNet  Google Scholar 

  36. Phong, D., Sturm, J.: Test configurations for K-stability and geodesic rays. J. Symplectic Geom. 5(2), 221–247 (2007)

    Article  MathSciNet  Google Scholar 

  37. Ross, J., Thomas, R.: A study of the Hilbert–Mumford criterion for the stability of projective varieties. J. Algebraic Geom. 16(2), 201–255 (2007)

    Article  MathSciNet  Google Scholar 

  38. Sjöström Dyrefelt, Z.: K-semistability of cscK manifolds with transcendental cohomology class. J. Geom. Anal. 28(4), 2927–2960 (2018)

    Article  MathSciNet  Google Scholar 

  39. Székelyhidi, G.: Extremal metrics and K-stability. Bull. Lond. Math. Soc. 39(1), 76–84 (2007)

    Article  MathSciNet  Google Scholar 

  40. Wang, X., Zhu, X.: Kähler–Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188(1), 87–103 (2004)

    Article  MathSciNet  Google Scholar 

  41. Witt Nyström, D.: Test configurations and Okounkov bodies. Compos. Math. 148(6), 1736–1756 (2012)

    Article  MathSciNet  Google Scholar 

  42. Yao, Y.: Mabuchi solitons and relative Ding stability of toric Fano varieties. To appear in Int. Math. Res. Not. IMRN. https://doi.org/10.1093/imrn/rnab226 (2021)

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Acknowledgements

The author would like to thank Feng Wang, Jia-xiang Wang, and Naoto Yotsutani for helpful discussions, and specially thank Mingchen Xia for valuable revision suggestions. The author is supported by Grants: National Natural Science Foundation of China (NSFC) (No. 751203123) and Fundamental Research Funds for the Central Universities (No. 531118010149).

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  1. School of Mathematics, Hunan University, Changsha, 410082, China

    Yi Yao

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Yao, Y. Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons. J Geom Anal 32, 105 (2022). https://doi.org/10.1007/s12220-021-00858-z

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