The J-equation and the supercritical deformed Hermitian–Yang–Mills equation


In this paper, we prove that for any Kähler metrics \(\omega _0\) and \(\chi \) on M, there exists a Kähler metric \(\omega _\varphi =\omega _0+\sqrt{-1}\partial {\bar{\partial }}\varphi >0\) satisfying the J-equation \({\mathrm {tr}}_{\omega _\varphi }\chi =c\) if and only if \((M,[\omega _0],[\chi ])\) is uniformly J-stable. As a corollary, we find a sufficient condition for the existence of constant scalar curvature Kähler metrics with \(c_1<0\). Using the same method, we also prove a similar result for the supercritical deformed Hermitian–Yang–Mills equation.

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  1. 1.

    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. 173(3), 547–601 (2008)

  2. 2.

    Aubin, T.: Réduction du cas positif de l’équation de Monge-Ampère sur les variétés kählériennes compactes à la démonstration d’une inégalité. J. Funct. Anal. 57(2), 143–153 (1984)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Błocki, Z., Kołodziej, S.: On regularization of plurisubharmonic functions on manifolds. Proc. Am. Math. Soc. 135(7), 2089–2093 (2007)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge–Ampère equation. Invent. Math. 37(1), 1–44 (1976)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chen, X., Cheng, J.: On the constant scalar curvature Kähler metrics, apriori estimates, arXiv e-prints (2017). arXiv:1712.06697

  6. 6.

    Chen, X., Cheng, J.: On the constant scalar curvature Kähler metrics, existence results. arXiv e-prints (2018). arXiv:1801.00656

  7. 7.

    Chen, X., Cheng, J.: On the constant scalar curvature Kähler metrics, general automorphism group. arXiv e-prints (2018), arXiv:1801.05907

  8. 8.

    Chen, X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Am. Math. Soc. 28(1), 183–197 (2015)

    Article  Google Scholar 

  9. 9.

    Chen, X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than \(2\pi \). J. Am. Math. Soc. 28(1), 199–234 (2015)

    Article  Google Scholar 

  10. 10.

    Chen, X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches \(2\pi \) and completion of the main proof. J. Am. Math. Soc. 28(1), 235–278 (2015)

    Article  Google Scholar 

  11. 11.

    Chen, X.: On the lower bound of the Mabuchi energy and its application. Int. Math. Res. Notices 12, 607–623 (2000)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Collins, T.C., Jacob, A., Yau, S.-T.: \((1,1)\) forms with specified Lagrangian phase: a priori estimates and algebraic obstructions. Camb. J. Math. 8(2), 407–452 (2020)

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

    Collins, T.C., Yau, S.-T.: Moment maps, nonlinear PDE, and stability in mirror symmetry. arXiv e-prints (2018). arXiv:1811.04824

  15. 15.

    Darvas, T.: The Mabuchi geometry of finite energy classes. Adv. Math. 285, 182–219 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Demailly, J.P.: Complex analytic and differential geometry. Université de Grenoble I, (2012)

  17. 17.

    Donaldson, S.K.: Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. (3) 50(1), 1–26 (1985)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Donaldson, S.K.: Moment maps and diffeomorphisms, vol. 3, 1999, Sir Michael Atiyah: a great mathematician of the twentieth century, pp. 1–15

  19. 19.

    Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62(2), 289–349 (2002)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Demailly, J.-P., Paun, M.: Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. Math. (2) 159(3), 1247–1274 (2004)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Datar, V.V., Pingali, V.P.: A numerical criterion for generalised Monge–Ampere equations on projective manifolds. arXiv e-prints (2020). arXiv:2006.01530

  22. 22.

    Dervan, R., Ross, J.: K-stability for Kähler manifolds. Math. Res. Lett. 24(3), 689–739 (2017)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Evans, L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35(3), 333–363 (1982)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Evans, L.C.: Classical solutions of the Hamilton–Jacobi–Bellman equation for uniformly elliptic operators. Trans. Am. Math. Soc. 275(1), 245–255 (1983)

    MathSciNet  Article  Google Scholar 

  25. 25.

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

  26. 26.

    Harvey, R., Lawson, H.B., Jr.: Calibrated geometries. Acta Math. 148, 47–157 (1982)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Harvey, F.R, Jr. Lawson, H.B: Pseudoconvexity for the special Lagrangian potential equation. Calc. Var. Partial Differ. Equ. 60(1), Paper No. 6, 37 (2021)

  28. 28.

    Jacob, A., Yau, S.-T.: A special Lagrangian type equation for holomorphic line bundles. Math. Ann. 369(1–2), 869–898 (2017)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Krylov, N.V.: Boundedly nonhomogeneous nonlinear elliptic and parabolic equations in the plane. Uspehi Mat. Nauk 24(4)(148), 201–202 (1969)

    MathSciNet  Google Scholar 

  30. 30.

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

    MathSciNet  Article  Google Scholar 

  31. 31.

    Leung, N.C., Yau, S.-T., Zaslow, E.: From special Lagrangian to Hermitian–Yang–Mills via Fourier-Mukai transform. Adv. Theor. Math. Phys. 4(6), 1319–1341 (2000)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Mariño, M., Minasian, R., Moore, G., Strominger, A.: Nonlinear instantons from supersymmetric \(p\)-branes. J. High Energy Phys., no. 1, Paper 5, 32 (2000)

  33. 33.

    Pingali, V.P.: A note on the deformed Hermitian Yang–Mills PDE. Complex Var. Elliptic Equ. 64(3), 503–518 (2019)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Pingali, V.P.: The deformed Hermitian Yang-Mills equation on three-folds. arXiv e-prints (2019). arXiv:1910.01870

  35. 35.

    Ross, J., Richard, T.: An obstruction to the existence of constant scalar curvature Kähler metrics. J. Differ. Geom. 72(3), 429–466 (2006)

    Article  Google Scholar 

  36. 36.

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

    MathSciNet  Article  Google Scholar 

  37. 37.

    Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Song, J.: Nakai–Moishezon criterions for complex Hessian equations. arXiv e-prints (2020). arXiv:2012.07956

  39. 39.

    Spruck, J.: Geometric aspects of the theory of fully nonlinear elliptic equations, Global theory of minimal surfaces. Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, pp. 283–309 (2005)

  40. 40.

    Song, J., Weinkove, B.: On the convergence and singularities of the \(J\)-flow with applications to the Mabuchi energy. Commun. Pure Appl. Math. 61(2), 210–229 (2008)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is \(T\)-duality. Nuclear Phys. B 479(1–2), 243–259 (1996)

    MathSciNet  Article  Google Scholar 

  42. 42.

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

    MathSciNet  Article  Google Scholar 

  43. 43.

    Takahashi, R.: Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow. Int. J. Math. 31(14), 2050116,26 (2020)

  44. 44.

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

    MathSciNet  Article  Google Scholar 

  45. 45.

    Neil, S.: Fully nonlinear, uniformly elliptic equations under natural structure conditions. Trans. Am. Math. Soc. 278(2), 751–769 (1983)

    MathSciNet  Article  Google Scholar 

  46. 46.

    Uhlenbeck, K., Yau, S.-T.: On the existence of Hermitian-Yang-Mills connections in stable vector bundles, vol. 39, 1986. Frontiers of the Mathematical Sciences (New York, 1985), pp. S257–S293 (1985)

  47. 47.

    Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I. Commun. Pure Appl. Math. 31(3), 339–411 (1978)

    Article  Google Scholar 

  48. 48.

    Yau, S.-T.: Open problems in geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, : Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc. Providence 1993, 1–28 (1990)

  49. 49.

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

    MathSciNet  Article  Google Scholar 

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Correspondence to Gao Chen.

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The author wishes to thank Xiuxiong Chen for suggesting this problem and providing valuable comments. The author is also grateful to Jingrui Cheng for pointing out a gap in the first version of this paper; to Helmut Hofer for a discussion about symplectic geometry; to anonymous referees for useful comments that made this article more readable; and to Simone Calamai, Jiyuan Han, Long Li, Yaxiong Liu, Vamsi Pingali, and Ryosuke Takahashi for minor suggestions. Sections 1–4 were based upon work supported by the National Science Foundation under Grant No. 1638352 and by a fund from the S. S. Chern Foundation for Mathematics Research when the author was a member of the Institute for Advanced Study. Section 5 was supported by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation when the author was an assistant professor of the University of Wisconsin-Madison.

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

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