Home Quasi-plurisubharmonic envelopes 3: Solving Monge–Ampère equations on hermitian manifolds
Article
Licensed
Unlicensed Requires Authentication

Quasi-plurisubharmonic envelopes 3: Solving Monge–Ampère equations on hermitian manifolds

  • Vincent Guedj and Chinh H. Lu
Published/Copyright: June 9, 2023

Abstract

We develop a new approach to L -a priori estimates for degenerate complex Monge–Ampère equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel [Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds, preprint (2021), https://arxiv.org/abs/2106.04273], we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in Kähler geometry. In [Quasi-plurisubharmonic envelopes 2: Bounds on Monge–Ampère volumes, Algebr. Geom. 9 (2022), 6, 688–713], we have studied the behavior of Monge–Ampère volumes on hermitian manifolds. We extend here the techniques of the former to the hermitian setting and use the bounds established in the latter, producing new relative a priori estimates, as well as several existence results for degenerate complex Monge–Ampère equations on compact hermitian manifolds.

Award Identifier / Grant number: ANR-11-LABX-0040

Funding statement: This work has benefited from state aid managed by the ANR under the “PIA” program bearing the reference ANR-11-LABX-0040 (research project HERMETIC). The authors are also partially supported by the ANR project PARAPLUI.

Acknowledgements

We thank D. Angella and V. Tosatti for useful discussions, as well as T. D. Tô and C.-M. Pan for a careful reading of a first draft. We also thank the referee for useful comments which help improve the presentation.

References

[1] D. Angella, V. Guedj and C. H. Lu, Plurisigned Hermitian metrics, Trans. Amer. Math. Soc. (2022), 10.1090/tran/8916. 10.1090/tran/8916Search in Google Scholar

[2] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1–2, 1–40. 10.1007/BF02392348Search in Google Scholar

[3] R. J. Berman, S. Boucksom, P. Eyssidieux, V. Guedj and A. Zeriahi, Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties, J. reine angew. Math. 751 (2019), 27–89. 10.1515/crelle-2016-0033Search in Google Scholar

[4] R. J. Berman, S. Boucksom and M. Jonsson, A variational approach to the Yau–Tian–Donaldson conjecture, J. Amer. Math. Soc. 34 (2021), no. 3, 605–652. 10.1090/jams/964Search in Google Scholar

[5] Z. Błocki, On uniform estimate in Calabi–Yau theorem, Sci. China Ser. A 48 (2005), 244–247. 10.1007/BF02884710Search in Google Scholar

[6] Z. Błocki, On the uniform estimate in the Calabi–Yau theorem, II, Sci. China Math. 54 (2011), no. 7, 1375–1377. 10.1007/s11425-011-4197-6Search in Google Scholar

[7] S. Boucksom, P. Eyssidieux, V. Guedj and A. Zeriahi, Monge–Ampère equations in big cohomology classes, Acta Math. 205 (2010), no. 2, 199–262. 10.1007/s11511-010-0054-7Search in Google Scholar

[8] X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics (I)—A priori estimates, J. Amer. Math. Soc. 34 (2021), no. 4, 909–936. 10.1090/jams/967Search in Google Scholar

[9] X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics (II)—Existence results, J. Amer. Math. Soc. 34 (2021), no. 4, 937–1009. 10.1090/jams/966Search in Google Scholar

[10] P. Cherrier, Équations de Monge–Ampère sur les variétés hermitiennes compactes, Bull. Sci. Math. (2) 111 (1987), no. 4, 343–385. Search in Google Scholar

[11] J.-P. Demailly, Regularization of closed positive currents of type ( 1 , 1 ) by the flow of a Chern connection, Contributions to complex analysis and analytic geometry, Aspects Math. E26, Friedrich Vieweg, Braunschweig (1994), 105–126. 10.1007/978-3-663-14196-9_4Search in Google Scholar

[12] J.-P. Demailly, Analytic methods in algebraic geometry, Surv. Mod. Math. 1, International Press, Somerville 2012. Search in Google Scholar

[13] J.-P. Demailly, On the cohomology of pseudoeffective line bundles, Complex geometry and dynamics, Abel Symp. 10, Springer, Cham (2015), 51–99. 10.1007/978-3-319-20337-9_4Search in Google Scholar

[14] J.-P. Demailly, S. A. Dinew, V. Guedj, H. H. Pham, S. Kołodziej and A. Zeriahi, Hölder continuous solutions to Monge–Ampère equations, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 4, 619–647. 10.4171/JEMS/442Search in Google Scholar

[15] J.-P. Demailly and N. Pali, Degenerate complex Monge–Ampère equations over compact Kähler manifolds, Internat. J. Math. 21 (2010), no. 3, 357–405. 10.1142/S0129167X10006070Search in Google Scholar

[16] J.-P. Demailly and M. Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2) 159 (2004), no. 3, 1247–1274. 10.4007/annals.2004.159.1247Search in Google Scholar

[17] E. Di Nezza and C. H. Lu, Generalized Monge–Ampère capacities, Int. Math. Res. Not. IMRN 2015 (2015), no. 16, 7287–7322. 10.1093/imrn/rnu166Search in Google Scholar

[18] E. Di Nezza and C. H. Lu, Complex Monge–Ampère equations on quasi-projective varieties, J. reine angew. Math. 727 (2017), 145–167. 10.1515/crelle-2014-0090Search in Google Scholar

[19] S. Dinew, Pluripotential theory on compact Hermitian manifolds, Ann. Fac. Sci. Toulouse Math. (6) 25 (2016), no. 1, 91–139. 10.5802/afst.1488Search in Google Scholar

[20] S. Dinew and S. Kołodziej, Pluripotential estimates on compact Hermitian manifolds, Advances in geometric analysis, Adv. Lect. Math. (ALM) 21, International Press, Somerville (2012), 69–86. Search in Google Scholar

[21] S. Donaldson, Some recent developments in Kähler geometry and exceptional holonomy, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenary lectures, World Scientific, Hackensack (2018), 425–451. 10.1142/9789813272880_0019Search in Google Scholar

[22] P. Eyssidieux, V. Guedj and A. Zeriahi, A priori L -estimates for degenerate complex Monge–Ampère equations, Int. Math. Res. Not. IMRN 2008 (2008), Article ID rnn 070. Search in Google Scholar

[23] P. Eyssidieux, V. Guedj and A. Zeriahi, Singular Kähler–Einstein metrics, J. Amer. Math. Soc. 22 (2009), no. 3, 607–639. 10.1090/S0894-0347-09-00629-8Search in Google Scholar

[24] A. Fino and A. Tomassini, Blow-ups and resolutions of strong Kähler with torsion metrics, Adv. Math. 221 (2009), no. 3, 914–935. 10.1016/j.aim.2009.02.001Search in Google Scholar

[25] J. Fu, J. Li and S.-T. Yau, Balanced metrics on non-Kähler Calabi–Yau threefolds, J. Differential Geom. 90 (2012), no. 1, 81–129. 10.4310/jdg/1335209490Search in Google Scholar

[26] B. Guan and Q. Li, Complex Monge–Ampère equations and totally real submanifolds, Adv. Math. 225 (2010), no. 3, 1185–1223. 10.1016/j.aim.2010.03.019Search in Google Scholar

[27] V. Guedj and C. H. Lu, Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds, preprint (2021), https://arxiv.org/abs/2106.04273. Search in Google Scholar

[28] V. Guedj and C. H. Lu, Quasi-plurisubharmonic envelopes 2: Bounds on Monge–Ampère volumes, Algebr. Geom. 9 (2022), no. 6, 688–713. 10.14231/AG-2022-021Search in Google Scholar

[29] V. Guedj, C. H. Lu and A. Zeriahi, Stability of solutions to complex Monge–Ampère flows, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 7, 2819–2836. 10.5802/aif.3227Search in Google Scholar

[30] V. Guedj and A. Zeriahi, Degenerate complex Monge–Ampère equations, EMS Tracts Math. 26, European Mathematical Society, Zürich 2017. 10.4171/167Search in Google Scholar

[31] B. Guo, D. H. Phong and F. Tong, On L estimates for complex Monge–Ampère equations, Ann. of Math. (2) 198 (2023), no. 1, 393–418. 10.4007/annals.2023.198.1.4Search in Google Scholar

[32] A. Hanani, Équations du type de Monge–Ampère sur les variétés hermitiennes compactes, J. Funct. Anal. 137 (1996), no. 1, 49–75. 10.1006/jfan.1996.0040Search in Google Scholar

[33] S. Kołodziej, The complex Monge–Ampère equation, Acta Math. 180 (1998), no. 1, 69–117. 10.1007/BF02392879Search in Google Scholar

[34] S. Kołodziej and N. C. Nguyen, Weak solutions to the complex Monge–Ampère equation on Hermitian manifolds, Analysis, complex geometry, and mathematical physics: In honor of Duong H. Phong, Contemp. Math. 644, American Mathematical Society, Providence (2015), 141–158. 10.1090/conm/644/12775Search in Google Scholar

[35] S. Kołodziej and N. C. Nguyen, Stability and regularity of solutions of the Monge–Ampère equation on Hermitian manifolds, Adv. Math. 346 (2019), 264–304. 10.1016/j.aim.2019.02.004Search in Google Scholar

[36] S. Kołodziej and N. C. Nguyen, Continuous solutions to Monge-Ampère equations on Hermitian manifolds for measures dominated by capacity, Calc. Var. Partial Differential Equations 60 (2021), no. 3, Paper No. 93. 10.1007/s00526-021-01944-4Search in Google Scholar

[37] C. H. Lu, T.-T. Phung and T.-D. Tô, Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds, Ann. Inst. Fourier (Grenoble) 71 (2021), no. 5, 2019–2045. 10.5802/aif.3436Search in Google Scholar

[38] N. C. Nguyen, The complex Monge–Ampère type equation on compact Hermitian manifolds and applications, Adv. Math. 286 (2016), 240–285. 10.1016/j.aim.2015.09.009Search in Google Scholar

[39] D. Popovici, Sufficient bigness criterion for differences of two nef classes, Math. Ann. 364 (2016), no. 1–2, 649–655. 10.1007/s00208-015-1230-zSearch in Google Scholar

[40] H. Skoda, Sous-ensembles analytiques d’ordre fini ou infini dans C n , Bull. Soc. Math. France 100 (1972), 353–408. 10.24033/bsmf.1743Search in Google Scholar

[41] G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differential Geom. 109 (2018), no. 2, 337–378. 10.4310/jdg/1527040875Search in Google Scholar

[42] G. Székelyhidi, V. Tosatti and B. Weinkove, Gauduchon metrics with prescribed volume form, Acta Math. 219 (2017), no. 1, 181–211. 10.4310/ACTA.2017.v219.n1.a6Search in Google Scholar

[43] T. D. Tô, Regularizing properties of complex Monge–Ampère flows II: Hermitian manifolds, Math. Ann. 372 (2018), no. 1–2, 699–741. 10.1007/s00208-017-1574-7Search in Google Scholar

[44] V. Tosatti and B. Weinkove, Estimates for the complex Monge–Ampère equation on Hermitian and balanced manifolds, Asian J. Math. 14 (2010), no. 1, 19–40. 10.4310/AJM.2010.v14.n1.a3Search in Google Scholar

[45] V. Tosatti and B. Weinkove, The complex Monge–Ampère equation on compact Hermitian manifolds, J. Amer. Math. Soc. 23 (2010), no. 4, 1187–1195. 10.1090/S0894-0347-2010-00673-XSearch in Google Scholar

[46] V. Tosatti and B. Weinkove, On the evolution of a Hermitian metric by its Chern–Ricci form, J. Differential Geom. 99 (2015), no. 1, 125–163. 10.4310/jdg/1418345539Search in Google Scholar

[47] V. Tosatti and B. Weinkove, The Aleksandrov–Bakelman–Pucci estimate and the Calabi–Yau equation, Nonlinear analysis in geometry and applied mathematics. Part 2, Harv. Univ. Cent. Math. Sci. Appl. Ser. Math. 2, International Press, Somerville (2018), 147–158. Search in Google Scholar

[48] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411. 10.1002/cpa.3160310304Search in Google Scholar

Received: 2022-07-12
Revised: 2023-04-16
Published Online: 2023-06-09
Published in Print: 2023-07-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 12.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2023-0030/html
Scroll to top button