A variational approach to the Yau–Tian–Donaldson conjecture
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- by Robert J. Berman, Sébastien Boucksom and Mattias Jonsson
- J. Amer. Math. Soc. 34 (2021), 605-652
- DOI: https://doi.org/10.1090/jams/964
- Published electronically: April 1, 2021
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Abstract:
We give a variational proof of a version of the Yau–Tian–Donaldson conjecture for twisted Kähler–Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve a continuity method or the Cheeger–Colding–Tian theory, and uses instead pluripotential theory and valuations. Along the way, we study the relationship between geodesic rays and non-Archimedean metrics.References
- Thierry Aubin, Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull. Sci. Math. (2) 102 (1978), no. 1, 63–95 (French, with English summary). MR 494932
- Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709, DOI 10.1090/surv/033
- Robert J. Berman, A thermodynamical formalism for Monge-Ampère equations, Moser-Trudinger inequalities and Kähler-Einstein metrics, Adv. Math. 248 (2013), 1254–1297. MR 3107540, DOI 10.1016/j.aim.2013.08.024
- Robert J. Berman, K-polystability of ${\Bbb Q}$-Fano varieties admitting Kähler-Einstein metrics, Invent. Math. 203 (2016), no. 3, 973–1025. MR 3461370, DOI 10.1007/s00222-015-0607-7
- Robert J. Berman, From Monge-Ampère equations to envelopes and geodesic rays in the zero temperature limit, Math. Z. 291 (2019), no. 1-2, 365–394. MR 3936074, DOI 10.1007/s00209-018-2087-0
- Robert J. Berman and Bo Berndtsson, Convexity of the $K$-energy on the space of Kähler metrics and uniqueness of extremal metrics, J. Amer. Math. Soc. 30 (2017), no. 4, 1165–1196. MR 3671939, DOI 10.1090/jams/880
- Robert J. Berman, Sébastien Boucksom, Vincent Guedj, and Ahmed Zeriahi, A variational approach to complex Monge-Ampère equations, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 179–245. MR 3090260, DOI 10.1007/s10240-012-0046-6
- Robert J. Berman, Sebastien Boucksom, Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi, Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties, J. Reine Angew. Math. 751 (2019), 27–89. MR 3956691, DOI 10.1515/crelle-2016-0033
- R. J. Berman, S. Boucksom, M. Jonsson. A variational approach to the Yau–Tian–Donaldson conjecture. \text{arXiv:1509.04561}, 2015.
- Robert J. Berman, Tamás Darvas, and Chinh H. Lu, Convexity of the extended K-energy and the large time behavior of the weak Calabi flow, Geom. Topol. 21 (2017), no. 5, 2945–2988. MR 3687111, DOI 10.2140/gt.2017.21.2945
- Robert J. Berman and Henri Guenancia, Kähler-Einstein metrics on stable varieties and log canonical pairs, Geom. Funct. Anal. 24 (2014), no. 6, 1683–1730. MR 3283927, DOI 10.1007/s00039-014-0301-8
- Bo Berndtsson, Curvature of vector bundles associated to holomorphic fibrations, Ann. of Math. (2) 169 (2009), no. 2, 531–560. MR 2480611, DOI 10.4007/annals.2009.169.531
- Bo Berndtsson, A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry, Invent. Math. 200 (2015), no. 1, 149–200. MR 3323577, DOI 10.1007/s00222-014-0532-1
- Bo Berndtsson, The openness conjecture and complex Brunn-Minkowski inequalities, Complex geometry and dynamics, Abel Symp., vol. 10, Springer, Cham, 2015, pp. 29–44. MR 3587460
- Zbigniew Błocki, A gradient estimate in the Calabi-Yau theorem, Math. Ann. 344 (2009), no. 2, 317–327. MR 2495772, DOI 10.1007/s00208-008-0307-3
- Zbigniew Błocki, On geodesics in the space of Kähler metrics, Advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 21, Int. Press, Somerville, MA, 2012, pp. 3–19. MR 3077245
- Zbigniew Błocki and Sławomir Kołodziej, On regularization of plurisubharmonic functions on manifolds, Proc. Amer. Math. Soc. 135 (2007), no. 7, 2089–2093. MR 2299485, DOI 10.1090/S0002-9939-07-08858-2
- Harold Blum and Mattias Jonsson, Thresholds, valuations, and K-stability, Adv. Math. 365 (2020), 107062, 57. MR 4067358, DOI 10.1016/j.aim.2020.107062
- H. Blum and Y. Liu, Openness of uniform K-stability in families of $\mathbb {Q}$-Fano varieties, arXiv:1808.09070.
- Harold Blum and Chenyang Xu, Uniqueness of K-polystable degenerations of Fano varieties, Ann. of Math. (2) 190 (2019), no. 2, 609–656. MR 3997130, DOI 10.4007/annals.2019.190.2.4
- S. Boucksom, Cônes positifs des variétés complexes compactes, Ph.D. thesis, 2002. https://tel.archives-ouvertes.fr/tel-00002268.
- Sébastien Boucksom, Variational and non-archimedean aspects of the Yau-Tian-Donaldson conjecture, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 591–617. MR 3966781
- Sébastien Boucksom and Huayi Chen, Okounkov bodies of filtered linear series, Compos. Math. 147 (2011), no. 4, 1205–1229. MR 2822867, DOI 10.1112/S0010437X11005355
- Sébastien Boucksom, Charles Favre, and Mattias Jonsson, Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 449–494. MR 2426355, DOI 10.2977/prims/1210167334
- Sébastien Boucksom, Charles Favre, and Mattias Jonsson, Solution to a non-Archimedean Monge-Ampère equation, J. Amer. Math. Soc. 28 (2015), no. 3, 617–667. MR 3327532, DOI 10.1090/S0894-0347-2014-00806-7
- Sébastien Boucksom, Charles Favre, and Mattias Jonsson, Singular semipositive metrics in non-Archimedean geometry, J. Algebraic Geom. 25 (2016), no. 1, 77–139. MR 3419957, DOI 10.1090/jag/656
- Sébastien Boucksom, Tomoyuki Hisamoto, and Mattias Jonsson, Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 743–841 (English, with English and French summaries). MR 3669511, DOI 10.5802/aif.3096
- Sébastien Boucksom, Tomoyuki Hisamoto, and Mattias Jonsson, Uniform K-stability and asymptotics of energy functionals in Kähler geometry, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 9, 2905–2944. MR 3985614, DOI 10.4171/JEMS/894
- S. Boucksom and M. Jonsson, Singular semipositive metrics on line bundles on varieties over trivially valued fields, arXiv:1801.08229v1, 2018.
- S. Boucksom and M. Jonsson, A non-Archimedean approach to K-stability, arXiv:1805.11160v1, 2018.
- Sébastien Boucksom, Alex Küronya, Catriona Maclean, and Tomasz Szemberg, Vanishing sequences and Okounkov bodies, Math. Ann. 361 (2015), no. 3-4, 811–834. MR 3319549, DOI 10.1007/s00208-014-1081-z
- Jacob Cable, Greatest lower bounds on Ricci curvature for Fano $T$-manifolds of complexity one, Bull. Lond. Math. Soc. 51 (2019), no. 1, 34–42. MR 3919559, DOI 10.1112/blms.12206
- Lennart Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225986
- Urban Cegrell, Pluricomplex energy, Acta Math. 180 (1998), no. 2, 187–217. MR 1638768, DOI 10.1007/BF02392899
- Ivan A. Cheltsov, Yanir A. Rubinstein, and Kewei Zhang, Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces, Selecta Math. (N.S.) 25 (2019), no. 2, Paper No. 34, 36. MR 3945265, DOI 10.1007/s00029-019-0473-z
- Ivan Cheltsov and Kewei Zhang, Delta invariants of smooth cubic surfaces, Eur. J. Math. 5 (2019), no. 3, 729–762. MR 3993261, DOI 10.1007/s40879-019-00357-0
- Xiuxiong Chen, The space of Kähler metrics, J. Differential Geom. 56 (2000), no. 2, 189–234. MR 1863016
- Xiuxiong Chen, On the lower bound of the Mabuchi energy and its application, Internat. Math. Res. Notices 12 (2000), 607–623. MR 1772078, DOI 10.1155/S1073792800000337
- X. X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics (I)—a priori estimates, arXiv:1712.06697, 2017.
- X. X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics (II)—existence results, arXiv:1801.000656, 2018.
- X. X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics (III)—general automorphism group, arXiv:1801.05907, 2018.
- X.X. Chen, S. K. Donaldson and S. Sun. Kähler–Einstein metrics on Fano manifolds, I-III. J. Amer. Math. Soc. 28 (2015), 183–197, 199–234, 235–278.
- XiuXiong Chen, Long Li, and Mihai Păuni, Approximation of weak geodesics and subharmonicity of Mabuchi energy, Ann. Fac. Sci. Toulouse Math. (6) 25 (2016), no. 5, 935–957 (English, with English and French summaries). MR 3582114, DOI 10.5802/afst.1516
- Xiuxiong Chen, Song Sun, and Bing Wang, Kähler-Ricci flow, Kähler-Einstein metric, and K-stability, Geom. Topol. 22 (2018), no. 6, 3145–3173. MR 3858762, DOI 10.2140/gt.2018.22.3145
- Jianchun Chu, Valentino Tosatti, and Ben Weinkove, $C^{1,1}$ regularity for degenerate complex Monge-Ampère equations and geodesic rays, Comm. Partial Differential Equations 43 (2018), no. 2, 292–312. MR 3777876, DOI 10.1080/03605302.2018.1446167
- G. Codogni and Z . Patakfalvi, Positivity of the CM line bundle for families of K-stable klt Fanos, arXiv:1806.07180, 2018.
- Tamás Darvas, The Mabuchi geometry of finite energy classes, Adv. Math. 285 (2015), 182–219. MR 3406499, DOI 10.1016/j.aim.2015.08.005
- Tamás Darvas, Weak geodesic rays in the space of Kähler potentials and the class $\mathcal {E}(X,\omega )$, J. Inst. Math. Jussieu 16 (2017), no. 4, 837–858. MR 3680345, DOI 10.1017/S1474748015000316
- Tamás Darvas, The Mabuchi completion of the space of Kähler potentials, Amer. J. Math. 139 (2017), no. 5, 1275–1313. MR 3702499, DOI 10.1353/ajm.2017.0032
- Tamás Darvas and Weiyong He, Geodesic rays and Kähler-Ricci trajectories on Fano manifolds, Trans. Amer. Math. Soc. 369 (2017), no. 7, 5069–5085. MR 3632560, DOI 10.1090/tran/6878
- Tamás Darvas and Yanir A. Rubinstein, Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics, J. Amer. Math. Soc. 30 (2017), no. 2, 347–387. MR 3600039, DOI 10.1090/jams/873
- Ved Datar and Gábor Székelyhidi, Kähler-Einstein metrics along the smooth continuity method, Geom. Funct. Anal. 26 (2016), no. 4, 975–1010. MR 3558304, DOI 10.1007/s00039-016-0377-4
- Jean-Pierre Demailly, A numerical criterion for very ample line bundles, J. Differential Geom. 37 (1993), no. 2, 323–374. MR 1205448
- Jean-Pierre Demailly, Lawrence Ein, and Robert Lazarsfeld, A subadditivity property of multiplier ideals, Michigan Math. J. 48 (2000), 137–156. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786484, DOI 10.1307/mmj/1030132712
- Jean-Pierre Demailly and János Kollár, Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 4, 525–556 (English, with English and French summaries). MR 1852009, DOI 10.1016/S0012-9593(01)01069-2
- Ruadhaí Dervan, Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. IMRN 15 (2016), 4728–4783. MR 3564626, DOI 10.1093/imrn/rnv291
- Wei Yue Ding, Remarks on the existence problem of positive Kähler-Einstein metrics, Math. Ann. 282 (1988), no. 3, 463–471. MR 967024, DOI 10.1007/BF01460045
- Charles Favre and Mattias Jonsson, Valuative analysis of planar plurisubharmonic functions, Invent. Math. 162 (2005), no. 2, 271–311. MR 2199007, DOI 10.1007/s00222-005-0443-2
- Charles Favre and Mattias Jonsson, Valuations and multiplier ideals, J. Amer. Math. Soc. 18 (2005), no. 3, 655–684. MR 2138140, DOI 10.1090/S0894-0347-05-00481-9
- Kento Fujita, A valuative criterion for uniform K-stability of $\Bbb Q$-Fano varieties, J. Reine Angew. Math. 751 (2019), 309–338. MR 3956698, DOI 10.1515/crelle-2016-0055
- Kento Fujita and Yuji Odaka, On the K-stability of Fano varieties and anticanonical divisors, Tohoku Math. J. (2) 70 (2018), no. 4, 511–521. MR 3896135, DOI 10.2748/tmj/1546570823
- Qi’an Guan and Xiangyu Zhou, A proof of Demailly’s strong openness conjecture, Ann. of Math. (2) 182 (2015), no. 2, 605–616. MR 3418526, DOI 10.4007/annals.2015.182.2.5
- Vincent Guedj and Ahmed Zeriahi, Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005), no. 4, 607–639. MR 2203165, DOI 10.1007/BF02922247
- Vincent Guedj and Ahmed Zeriahi, The weighted Monge-Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal. 250 (2007), no. 2, 442–482. MR 2352488, DOI 10.1016/j.jfa.2007.04.018
- Henri Guenancia and Mihai Păun, Conic singularities metrics with prescribed Ricci curvature: general cone angles along normal crossing divisors, J. Differential Geom. 103 (2016), no. 1, 15–57. MR 3488129
- T. Hisamoto, Orthogonal projection of a test configuration to vector fields, arXiv:1610.07158, 2016.
- T. Hisamoto, Stability and coercivity for toric polarizations, arXiv:1610.07998, 2016.
- Mattias Jonsson and Mircea Mustaţă, Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 6, 2145–2209 (2013) (English, with English and French summaries). MR 3060755, DOI 10.5802/aif.2746
- Mattias Jonsson and Mircea Mustaţă, An algebraic approach to the openness conjecture of Demailly and Kollár, J. Inst. Math. Jussieu 13 (2014), no. 1, 119–144. MR 3134017, DOI 10.1017/S1474748013000091
- Sławomir Kołodziej, The complex Monge-Ampère equation, Acta Math. 180 (1998), no. 1, 69–117. MR 1618325, DOI 10.1007/BF02392879
- Chi Li, Greatest lower bounds on Ricci curvature for toric Fano manifolds, Adv. Math. 226 (2011), no. 6, 4921–4932. MR 2775890, DOI 10.1016/j.aim.2010.12.023
- Chi Li, Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds, J. Reine Angew. Math. 733 (2017), 55–85. MR 3731324, DOI 10.1515/crelle-2014-0156
- C. Li, G-uniform stability and Kähler–Einstein metrics on singular Fano varieties, arXiv:1907.09399.
- Chi Li and Chenyang Xu, Special test configuration and K-stability of Fano varieties, Ann. of Math. (2) 180 (2014), no. 1, 197–232. MR 3194814, DOI 10.4007/annals.2014.180.1.4
- C. Li. Geodesic rays and stability in the cscK problem. arXiv:2001.01366, 2020.
- Chi Li and Song Sun, Conical Kähler-Einstein metrics revisited, Comm. Math. Phys. 331 (2014), no. 3, 927–973. MR 3248054, DOI 10.1007/s00220-014-2123-9
- C. Li, G. Tian and F. Wang. On Yau–Tian–Donaldson conjecture for singular Fano varieties. arXiv:1711.09530, 2017.
- C. Li, G. Tian, and F. Wang, The uniform version of Yau–Tian–Donaldson conjecture for singular Fano varieties, arXiv:1903.01215, 2019.
- David McKinnon and Mike Roth, Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties, Invent. Math. 200 (2015), no. 2, 513–583. MR 3338009, DOI 10.1007/s00222-014-0540-1
- Alan Michael Nadel, Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Ann. of Math. (2) 132 (1990), no. 3, 549–596. MR 1078269, DOI 10.2307/1971429
- Yuji Odaka, The GIT stability of polarized varieties via discrepancy, Ann. of Math. (2) 177 (2013), no. 2, 645–661. MR 3010808, DOI 10.4007/annals.2013.177.2.6
- Jihun Park and Joonyeong Won, K-stability of smooth del Pezzo surfaces, Math. Ann. 372 (2018), no. 3-4, 1239–1276. MR 3880298, DOI 10.1007/s00208-017-1602-7
- D. H. Phong, Julius Ross, and Jacob Sturm, Deligne pairings and the Knudsen-Mumford expansion, J. Differential Geom. 78 (2008), no. 3, 475–496. MR 2396251, DOI 10.4310/jdg/1207834553
- D. H. Phong and Jacob Sturm, The Monge-Ampère operator and geodesics in the space of Kähler potentials, Invent. Math. 166 (2006), no. 1, 125–149. MR 2242635, DOI 10.1007/s00222-006-0512-1
- Julius Ross and David Witt Nyström, Analytic test configurations and geodesic rays, J. Symplectic Geom. 12 (2014), no. 1, 125–169. MR 3194078, DOI 10.4310/JSG.2014.v12.n1.a5
- Yanir A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Adv. Math. 218 (2008), no. 5, 1526–1565. MR 2419932, DOI 10.1016/j.aim.2008.03.017
- Yanir A. Rubinstein, On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow, Trans. Amer. Math. Soc. 361 (2009), no. 11, 5839–5850. MR 2529916, DOI 10.1090/S0002-9947-09-04675-3
- Zakarias Sjöström Dyrefelt, K-semistability of cscK manifolds with transcendental cohomology class, J. Geom. Anal. 28 (2018), no. 4, 2927–2960. MR 3881961, DOI 10.1007/s12220-017-9942-9
- Jian Song and Gang Tian, Canonical measures and Kähler-Ricci flow, J. Amer. Math. Soc. 25 (2012), no. 2, 303–353. MR 2869020, DOI 10.1090/S0894-0347-2011-00717-0
- Jian Song and Xiaowei Wang, The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality, Geom. Topol. 20 (2016), no. 1, 49–102. MR 3470713, DOI 10.2140/gt.2016.20.49
- Gábor Székelyhidi, Greatest lower bounds on the Ricci curvature of Fano manifolds, Compos. Math. 147 (2011), no. 1, 319–331. MR 2771134, DOI 10.1112/S0010437X10004938
- Gábor Székelyhidi, The partial $C^0$-estimate along the continuity method, J. Amer. Math. Soc. 29 (2016), no. 2, 537–560. MR 3454382, DOI 10.1090/jams/833
- Gang Tian, On stability of the tangent bundles of Fano varieties, Internat. J. Math. 3 (1992), no. 3, 401–413. MR 1163733, DOI 10.1142/S0129167X92000175
- Gang Tian, Canonical metrics in Kähler geometry, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2000. Notes taken by Meike Akveld. MR 1787650, DOI 10.1007/978-3-0348-8389-4
- Gang Tian, K-stability and Kähler-Einstein metrics, Comm. Pure Appl. Math. 68 (2015), no. 7, 1085–1156. MR 3352459, DOI 10.1002/cpa.21578
- Valentino Tosatti, Adiabatic limits of Ricci-flat Kähler metrics, J. Differential Geom. 84 (2010), no. 2, 427–453. MR 2652468
- H. Tsuji, Canonical measures and the dynamical systems of Bergman kernels, arXiv:0805.1829, 2008.
- Shing Tung 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. MR 480350, DOI 10.1002/cpa.3160310304
Bibliographic Information
- Robert J. Berman
- Affiliation: Mathematical Sciences, Chalmers University of Technology; and University of Gothenburg, SE-412 96 Göteborg, Sweden
- MR Author ID: 743613
- Email: robertb@chalmers.se
- Sébastien Boucksom
- Affiliation: CNRS-CMLS, École Polytechnique, F-91128 Palaiseau Cedex, France
- MR Author ID: 688226
- Email: sebastien.boucksom@polytechnique.edu
- Mattias Jonsson
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
- MR Author ID: 631360
- Email: mattiasj@umich.edu
- Received by editor(s): January 24, 2019
- Received by editor(s) in revised form: June 2, 2020
- Published electronically: April 1, 2021
- Additional Notes: The first author was partially supported by the Swedish Research Council, the European Research Council, the Knut and Alice Wallenberg foundation, and the Göran Gustafsson foundation.
The second author was partially supported by the ANR projects GRACK, MACK and POSITIVE.
The third author was partially supported by NSF grants DMS-1600011 and DMS-1900025, the Knut and Alice Wallenberg foundation, and the United States—Israel Binational Science Foundation. - © Copyright 2021 American Mathematical Society
- Journal: J. Amer. Math. Soc. 34 (2021), 605-652
- MSC (2020): Primary 32Q20, 32Q26
- DOI: https://doi.org/10.1090/jams/964
- MathSciNet review: 4334189