Abstract
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.
Similar content being viewed by others
References
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)
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)
Błocki, Z., Kołodziej, S.: On regularization of plurisubharmonic functions on manifolds. Proc. Am. Math. Soc. 135(7), 2089–2093 (2007)
Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge–Ampère equation. Invent. Math. 37(1), 1–44 (1976)
Chen, X., Cheng, J.: On the constant scalar curvature Kähler metrics, apriori estimates, arXiv e-prints (2017). arXiv:1712.06697
Chen, X., Cheng, J.: On the constant scalar curvature Kähler metrics, existence results. arXiv e-prints (2018). arXiv:1801.00656
Chen, X., Cheng, J.: On the constant scalar curvature Kähler metrics, general automorphism group. arXiv e-prints (2018), arXiv:1801.05907
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)
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)
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)
Chen, X.: On the lower bound of the Mabuchi energy and its application. Int. Math. Res. Notices 12, 607–623 (2000)
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)
Collins, T.C., Székelyhidi, G.: Convergence of the J-flow on toric manifolds. J. Differ. Geom. 107(1), 47–81 (2017)
Collins, T.C., Yau, S.-T.: Moment maps, nonlinear PDE, and stability in mirror symmetry. arXiv e-prints (2018). arXiv:1811.04824
Darvas, T.: The Mabuchi geometry of finite energy classes. Adv. Math. 285, 182–219 (2015)
Demailly, J.P.: Complex analytic and differential geometry. Université de Grenoble I, (2012)
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)
Donaldson, S.K.: Moment maps and diffeomorphisms, vol. 3, 1999, Sir Michael Atiyah: a great mathematician of the twentieth century, pp. 1–15
Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62(2), 289–349 (2002)
Demailly, J.-P., Paun, M.: Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. Math. (2) 159(3), 1247–1274 (2004)
Datar, V.V., Pingali, V.P.: A numerical criterion for generalised Monge–Ampere equations on projective manifolds. arXiv e-prints (2020). arXiv:2006.01530
Dervan, R., Ross, J.: K-stability for Kähler manifolds. Math. Res. Lett. 24(3), 689–739 (2017)
Evans, L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35(3), 333–363 (1982)
Evans, L.C.: Classical solutions of the Hamilton–Jacobi–Bellman equation for uniformly elliptic operators. Trans. Am. Math. Soc. 275(1), 245–255 (1983)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, Berlin, Reprint of the 1998 edition (2001)
Harvey, R., Lawson, H.B., Jr.: Calibrated geometries. Acta Math. 148, 47–157 (1982)
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)
Jacob, A., Yau, S.-T.: A special Lagrangian type equation for holomorphic line bundles. Math. Ann. 369(1–2), 869–898 (2017)
Krylov, N.V.: Boundedly nonhomogeneous nonlinear elliptic and parabolic equations in the plane. Uspehi Mat. Nauk 24(4)(148), 201–202 (1969)
Lejmi, M., Székelyhidi, G.: The J-flow and stability. Adv. Math. 274, 404–431 (2015)
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)
Mariño, M., Minasian, R., Moore, G., Strominger, A.: Nonlinear instantons from supersymmetric \(p\)-branes. J. High Energy Phys., no. 1, Paper 5, 32 (2000)
Pingali, V.P.: A note on the deformed Hermitian Yang–Mills PDE. Complex Var. Elliptic Equ. 64(3), 503–518 (2019)
Pingali, V.P.: The deformed Hermitian Yang-Mills equation on three-folds. arXiv e-prints (2019). arXiv:1910.01870
Ross, J., Richard, T.: An obstruction to the existence of constant scalar curvature Kähler metrics. J. Differ. Geom. 72(3), 429–466 (2006)
Dyrefelt, Z.S.: K-semistability of cscK manifolds with transcendental cohomology class. J. Geom. Anal. 28(4), 2927–2960 (2018)
Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)
Song, J.: Nakai–Moishezon criterions for complex Hessian equations. arXiv e-prints (2020). arXiv:2012.07956
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)
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)
Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is \(T\)-duality. Nuclear Phys. B 479(1–2), 243–259 (1996)
Székelyhidi, G.: Fully non-linear elliptic equations on compact Hermitian manifolds. J. Differ. Geom. 109(2), 337–378 (2018)
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)
Tian, G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math. 130(1), 1–37 (1997)
Neil, S.: Fully nonlinear, uniformly elliptic equations under natural structure conditions. Trans. Am. Math. Soc. 278(2), 751–769 (1983)
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)
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)
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)
Zheng, K.: \(I\)-properness of Mabuchi’s \(K\)-energy. Calc. Var. Partial Differ. Equ. 54(3), 2807–2830 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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.
Rights and permissions
About this article
Cite this article
Chen, G. The J-equation and the supercritical deformed Hermitian–Yang–Mills equation. Invent. math. 225, 529–602 (2021). https://doi.org/10.1007/s00222-021-01035-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-021-01035-3