Skip to main content
Log in

A numerical criterion for generalised Monge-Ampère equations on projective manifolds

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We prove that generalised Monge-Ampère equations (a family of equations which includes the inverse Hessian equations like the J-equation, as well as the Monge-Ampère equation) on projective manifolds have smooth solutions if certain intersection numbers are positive. As corollaries of our work, we improve a result of Chen (albeit in the projective case) on the existence of solutions to the J-equation, and prove a conjecture of Székelyhidi in the projective case on the solvability of certain inverse Hessian equations. The key new ingredient in improving Chen’s result is a degenerate concentration of mass result. We also prove an equivariant version of our results, albeit under the assumption of uniform positivity. In particular, we can recover existing results on manifolds with large symmetry such as projective toric manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Notes

  1. Indeed, if \(dim(Z)=1\), then for large N, \(Y \cap Z\) is a non-empty collection of points and is hence a divisor in Z. If \(dim(Z)\ge 2\), then since \(dim(Y)+dim(Z)>dim(M)\) the analytic set \(Y\cap Z\) is a union of connected subvarieties by the Fulton-Hansen connectedness theorem. Hence, if we ensure that there exists a Y that intersects every component of Z in at least one smooth point transversally, we are done by Bertini’s theorem.

References

  • E. Bierstone and P.D. Milman. “Resolution of singularities.” Several complex variables (Berkeley, CA, 1995-1996) 37 (1999) : 43-78.

  • Z. Blocki and S. Kolodziej. “On regularization of plurisubharmonic functions on manifolds.” Proc. Amer. Math. Soc. 135 (2007): 2089-2093.

  • G. Chen. The J-equation and the supercritical deformed Hermitian-Yang-Mills equation. Invent. math. (2021). https://doi.org/10.1007/s00222-021-01035-3.

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

  • J. Chu, M.-C Lee, and R. Takahashi. “A Nakai-Moishezon type criterion for supercritical deformed Hermitian-Yang-Mills equation.” arXiv:2105.10725.

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

  • T. Collins and G. Székelyhidi “Convergence of the J-flow on toric manifolds.” J. Differential. Geom, 107(1) (2017): 47-81.

  • T. Collins, D. Xie, and S.T. Yau. “The deformed Hermitian-Yang-Mills equation in geometry and physics.” Geometry and Physics: Volume 1: A Festschrift in Honour of Nigel Hitchin 1 (2018): 69.

  • T. Collins and S.T. Yau. “Moment Maps, Nonlinear PDE and Stability in Mirror Symmetry, I: Geodesics.” Annals of PDE, 7(1) (2021): 1-73.

  • J-P. Demailly and M. Paun. “ Numerical characterization of the Kähler cone of a compact Kähler manifold.” Ann. Math, (2004): 1247-1274.

  • R. Dervan and J. Keller. “A finite dimensional approach to Donaldson’s J-flow.” Comm. Anal. Geom. 27(5) (2019): 1025-1085.

  • S. K. Donaldson. “Moment maps and diffeomorphisms.” Asian J. Math. 3 (1999): 1-15.

  • Z. Dyrefelt. “Optimal lower bounds for Donaldson’s J-functional.” Advances in Math. 374(2020): 107271.

    Article  MathSciNet  Google Scholar 

  • H. Fang, M. Lai, and X. Ma. “On a class of fully nonlinear flows in Kähler geometry.” J. für die reine ang. Math. 653 (2011): 189-220.

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

  • X. Han and X. Jin. “A rigidity theorem for deformed Hermitian-Yang-Mills equation.” Calc. Var. Part. Diff. Eq. 60(1) (2021): 1-16.

  • X. Han and X. Jin. “Stability of line bundle mean curvature flow.” arXiv:2001.07406 (2020).

  • X. Han and H. Yamamoto. “An \(\varepsilon \)-regularity theorem for line bundle mean curvature flow.” arXiv:1904.02391.

  • Y. Hashimoto and Julien Keller. “About J-flow, J-balanced metrics, uniform J-stability and K-stability.” arXiv:1705.02000 (2017).

  • H. Hironaka. “Resolution of singularities of an algebraic variety over a field of characteristic zero. I” Ann. Math.2, 79(1) (1964): 109-203.

  • H. Hironaka. “Resolution of singularities of an algebraic variety over a field of characteristic zero. II" Ann. Math. 2, 79(1) (1964): 205-326.

  • A. Jacob. “Weak Geodesics for the deformed Hermitian-Yang-Mills equation.” arXiv:1906.07128.

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

  • K. Kawai and H. Yamamoto. “Deformation theory of deformed Hermitian Yang-Mills connections and deformed Donaldson-Thomas connections.” arXiv:2004.00532 (2020).

  • J. Kollár. Lectures on resolution of singularities (AM-166). Annals of Mathematics Studies, Princeton University Press (2007).

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

  • V. Pingali. “A note on the deformed Hermitian-Yang-Mills PDE.” Comp. Var. Ellip. Eq. 64.3 (2019): 503-518.

  • V. Pingali. “On a generalised Monge-Ampère equation.” J. Part. Diff. Eq., 27(4) (2014): 333-346.

  • V. Pingali. “The deformed Hermitian Yang-Mills equation on three-folds.” Analysis and PDE (In press) arXiv:1910.01870 (2019).

  • D. Phong and J. Sturm. “The Dirichlet problem for degenerate complex Monge-Ampere equations.” arXiv preprint arXiv:0904.1898 (2009).

  • R. Richberg. “Stetige streng pseudokonvexe Funktionen.” Math. Ann. 175 (1968): 257-286.

  • Y-T Siu. Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics: delivered at the German Mathematical Society Seminar in Düsseldorf in June, (1986) (Vol. 8). Birkhäuser.

  • E. Schlitzer and J. Stoppa. “Deformed Hermitian Yang-Mills connections, extended gauge group and scalar curvature.” arXiv:1911.10852 (2019).

  • J. Song and B. Weinkove. “On the convergence and singularities of the J-flow with applications to the Mabuchi energy.” Comm. Pure Appl. Math. 61(2) (2008): 210-229

  • J. Song and B. Weinkove. “The degenerate J-flow and the Mabuchi energy on minimal surfaces of general type.” Univ. Iagellonicae Acta Math. 50 : 89-106.

  • J. Song. “Nakai-Moishezon criterions for complex Hessian equations.” arXiv:2012.07956.

  • W. Sun. “Generalized complex Monge-Ampère type equations on closed Hermitian manifolds.” arXiv:1412.8192.

  • G. Székelyhidi. “Fully non-linear elliptic equations on compact Hermitian manifolds.” J. Differential Geom. 109(2) (2018): 337-378.

  • R. Takahashi. “Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow.” arXiv:2002.05132 (2020).

  • R. Takahashi. “Collapsing of the line bundle mean curvature flow on Kähler surfaces.” Calc. Var. Part. Diff. Eq. 60(1) (2021): 1-18

  • V. Tosatti and B. Weinkove. “Estimates for the complex Monge-Ampére equation on Hermitian and balanced manifolds.” Asian J. Math. 14(1) (2010): 19-40

  • V. Tosatti and B. Weinkove. “The complex Monge-Ampére equation on compact Hermitian manifolds.” J. Am. Math. Soc. 23(4) (2010): 1187-1195

  • B. Weinkove. “Convergence of the J-flow on Kähler surfaces.” Comm. Anal. Geom. 12(4) (2004): 949-965.

  • B. Weinkove. “On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy.” J. Differential Geom. 73(2) (2006): 351-358

  • J. Xiao. “Positivity in convergence of the inverse \({n-1} \)-flow.” arXiv:1610.09584 (2016).

  • J. Xiao. “A remark on the convergence of the inverse -flow.” Comp. Rend. Math. 354(4) (2016): 395-399.

  • H. Yamamoto. “Special Lagrangian and deformed Hermitian Yang-Mills on tropical manifolds.” Math. Zeit. 290(3-4) (2018): 1023-1040

  • Y. Yao. “The J-flow on toric manifolds.” Acta Math. Sinica, English Series, 31(10) (2015): 1582-1592

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

  • K. Zheng. “I-properness of Mabuchi’s K-energy.” Calc. Variations Par. Diff. Eq, 54(3) (2015): 2807-2830.

Download references

Acknowledgements

This is work partially supported by grant F.510/25/CAS-II/2018(SAP-I) from UGC (Govt. of India). The first author (V. Datar) is grateful to the Infosys Foundation for their support through the Young Investigator Award. The second author (V. Pingali) is partially supported by a MATRICS grant MTR/2020/000100 from SERB (Govt. of India) We are deeply indebted to Gao Chen for many clarifications regarding his paper, and for his comments on all drafts of this work. We thank Jian Song for his comments on the first draft of the paper, and for many useful discussions on the issue of relaxing uniform positivity in the Theorem 1.3. We also thank Indranil Biswas, Apoorva Khare, Kapil Paranjape, Venkatesh Rajendran for help with regard to some algebraic aspects of the work. Finally, we thank the anonymous reviewer for detailed and useful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vamsi Pritham Pingali.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Datar, V.V., Pingali, V.P. A numerical criterion for generalised Monge-Ampère equations on projective manifolds. Geom. Funct. Anal. 31, 767–814 (2021). https://doi.org/10.1007/s00039-021-00577-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-021-00577-1

Navigation