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Degenerate J-flow on compact Kähler manifolds

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Abstract

In this note, we study a degenerate twisted J-flow on compact Kähler manifolds. We show that it exists for all time, it is unique and converges to a weak solution of a degenerate twisted J-equation. In particular, this confirms an expectation formulated by Song–Weinkove for the J-flow. As a consequence, we establish the properness of the Mabuchi K-energy twisted by a certain semi-positive closed (1,1)-form for Kähler classes in a certain subcone.

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References

  1. Chen, G.: The J-equation and the supercritical deformed Hermitian–Yang–Mills equation. Invent. math. 225, 529–602 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, X.X.: On the lower bound of the Mabuchi K-energy and its application. Int. Math. Res. Not. 12, 607–623 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Collins, T., Székelyhidi, G.: Convergence of the J-flow on toric manifolds. J. Diff. Geom. 107(1), 47–81 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Dervan, R.: Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds. Ann. Fac. Sci. Toulouse Sér. 6(25), 919–934 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Donaldson, S.K.: Moment maps and diffeomorphisms. Asian J. Math. 3(1), 1–15 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fang, H., Lai, M., Song, J., Weinkove, B.: The J-flow on Kähler surfaces: a boundary case. Anal. PDE 7(1), 215–226 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fine, J.: Constant scalar curvature Kähler metrics on fibred complex surfaces. J. Differ. Geom. 68(3), 397–432 (2004)

    MATH  Google Scholar 

  10. Fine, J.: Fibrations with constant scalar curvature Kähler metrics and the CM-line bundle. Math. Res. Lett. 14(2), 239–247 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, reprint of the 1998 ed., (2001)

  12. Guan, B.: Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds. Duke Math. J. 163(8), 1491–1524 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guedj, V., Zeriahi, A.: Degenerate complex Monge-Ampère equations, EMS Tracts in Math. (2017), 496p

  14. Jian, W., Shi, Y., Song, J.: A remark on constant scalar curvature Kähler metrics on minimal models. Proc. Am. Math. Soc. 147, 3507–3513 (2019)

    Article  MATH  Google Scholar 

  15. Lejmi, M., Székelyhidi, G.: The J-flow and stability. Adv. Math. 274, 404–431 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, H.-Z., Shi, Y.L., Yao, Y.: A criterion for properness of the K-energy in a general Kähler class. Math. Ann. 361(1–2), 135–156 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mabuchi, T.: Some symplectic geometry on compact Kähler manifolds I. Osaka J. Math. 24, 227–252 (1987)

    MathSciNet  MATH  Google Scholar 

  18. Phong, D.H., Sturm, J.: The Dirichlet problem for degenerate complex Monge-Ampère equations. Comm. Anal. Geom. 18(1), 145–170 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Phong, D.H., Tô, T.D.: Fully non-linear parabolic equations on compact Hermitian manifolds. Ann. Sci. Éc. Norm. Supér. 54(3), 793–832 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sjöström, Z.: Dyrefelt, Optimal lower bounds for Donaldson’s J-functional. Adv. Math. 374, 107271 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sjöström, Z.: Dyrefelt,Existence of cscK metrics on smooth minimal models, to appear in Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

  22. Song, J.: Nakai-Moishezon criterions for complex Hessian equations,arXiv:2012.07956

  23. Song, J., Weinkove, B.: The degenerate J-flow and the Mabuchi energy on minimal surfaces of general type. Univ. Iagel. Acta Math. 2, 89–106 (2013)

    MathSciNet  MATH  Google Scholar 

  24. Song, J., Weinkove, B.: On the convergence and singularities of the J-flow with applications to the Mabuchi energy. Comm. Pure Appl. Math. 61, 210–229 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Székelyhidi, G.: Fully non-linear elliptic equations on compact Hermitian manifolds. J. Differ. Geom. 109(2), 337–378 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Tian, G.: Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130(1), 1–37 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tian, G.: Canonical metrics in Kähler geometry, Lectures in Mathematics ETH Zurich, Birkhauser Verlag, Basel (2000)

  28. Tso, K.: On an Aleksandrov-Bakel’Man type maximum principle for second-order parabolic equations. Comm. Partial Differ. Equ. 10(5), 543–553 (1985)

    Article  MATH  Google Scholar 

  29. Tsuji, H.: Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type. Math. Ann. 281(1), 123–133 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  30. Weinkove, B.: Convergence of the J-flow on Kähler surfaces. Comm. Anal. Geom. 12(4), 949–965 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Weinkove, B.: On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy. J. Diff. Geom. 73(2), 351–358 (2006)

    MathSciNet  MATH  Google Scholar 

  32. Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Comm. Pure Appl. Math. 31, 339–411 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zheng, K.: I-properness of Mabuchi’s K-energy. Calc. Var. Partial Differ. Equ. 54(3), 2807–2830 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zheng, K.: Existence of constant scalar curvature Kähler cone metrics, properness and geodesic stability, arXiv:1803.09506

Download references

Acknowledgements

The author is grateful to Vincent Guedj for support, suggestions and encouragement. We also would like to thank Duong Hong Phong and Zakarias Sjöström Dyrefelt for very useful discussions. The author is partially supported by ANR-21-CE40-0011-01 (research project MARGE) and ANR-11-LABX-0040 (research project HERMETIC). The author would like to thank the referee for useful comments and suggestions.

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  1. Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, 4 place Jussieu, 75005, Paris, France

    Tat Dat Tô

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Tô, T.D. Degenerate J-flow on compact Kähler manifolds. Math. Z. 303, 97 (2023). https://doi.org/10.1007/s00209-023-03247-0

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