Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2017: 1.49

Online
ISSN
1435-5345
See all formats and pricing
More options …
Volume 2019, Issue 751

Issues

Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties

Robert J. Berman / Sebastien Boucksom / Philippe Eyssidieux / Vincent Guedj / Ahmed Zeriahi
Published Online: 2016-09-14 | DOI: https://doi.org/10.1515/crelle-2016-0033

Abstract

We prove the existence and uniqueness of Kähler–Einstein metrics on -Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler–Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler–Ricci flow provides weak convergence independently of Perelman’s celebrated estimates.

References

  • [1]

    C. Arezzo, A. Ghigi and G. P. Pirola, Symmetries, quotients and Kähler–Einstein metrics, J. reine angew. Math. 591 (2006), 177–200. Google Scholar

  • [2]

    T. Aubin, Equation de type Monge–Ampère sur les variétés kählériennes compactes, Bull. Sci. Math. 102 (1978), 63–95. Google Scholar

  • [3]

    S. Bando and T. Mabuchi, Uniqueness of Einstein–Kähler metrics modulo connected group actions, Algebraic geometry (Sendai 1985), Adv. Stud. Pure Math. 10, Kinokuniya, Tokyo (1987), 11–40. CrossrefGoogle Scholar

  • [4]

    E. Bedford and B. A. Taylor, Fine topology, Šilov boundary, and (ddc)n, J. Funct. Anal. 72 (1987), no. 2, 225–251. Google Scholar

  • [5]

    R. Berman, A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics, Adv. Math. 248 (2013), 1254–1297. CrossrefGoogle Scholar

  • [6]

    R. Berman and S. Boucksom, Growth of balls of holomorphic sections and energy at equilibrium, Invent. Math. 181 (2010), no. 2, 337–394.CrossrefGoogle Scholar

  • [7]

    R. Berman, S. Boucksom, V. Guedj and A. Zeriahi, A variational approach to complex Monge–Ampère equations, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 179–245. CrossrefGoogle Scholar

  • [8]

    B. Berndtsson, Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, 1633–1662. CrossrefGoogle Scholar

  • [9]

    B. Berndtsson, A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry, Invent. Math 200 (2015), no. 1, 149–200. CrossrefGoogle Scholar

  • [10]

    Z. Błocki, The Calabi–Yau theorem, Complex Monge–Ampère equations and geodesics in the space of Kähler metrics, Lecture Notes in Math. 2038, Springer, Berlin (2012), 201–227. Google Scholar

  • [11]

    Z. Błocki and S. Kołodziej, On regularization of plurisubharmonic functions on manifolds, Proc. Amer. Math. Soc. 135 (2007), no. 7, 2089–2093. CrossrefGoogle Scholar

  • [12]

    S. Boucksom, On the volume of a line bundle, Internat. J. Math. 13 (2002), no. 10, 1043–1063. CrossrefGoogle Scholar

  • [13]

    S. Boucksom, P. Eyssidieux, V. Guedj and A. Zeriahi, Monge–Ampère equations in big cohomology classes, Acta Math. 205 (2010), 199–262. CrossrefGoogle Scholar

  • [14]

    S. Boucksom, C. Favre and M. Jonsson, Singular semipositive metrics in non-Archimedean geometry, J. Algebraic Geom. 25 (2016), 77–139. Google Scholar

  • [15]

    S. Boucksom and V. Guedj, Kähler–Ricci flows on singular varieties, lecture notes (2012), www.math.univ-toulouse.fr/~guedj/fichierspdf/NotesBoucksomGuedj.pdf.

  • [16]

    N. Bourbaki, Eléments de mathématiques. Topologie générale. Chapitre 9, Springer, Berlin 2007. Google Scholar

  • [17]

    M. Brion, Introduction to actions of algebraic groups, Lecture Notes (2009), www-fourier.ujf-grenoble.fr/~mbrion/notes.html.

  • [18]

    F. Campana, H. Guenancia and M. Paun, Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 6, 879–916. CrossrefGoogle Scholar

  • [19]

    J. Cheeger, T. Colding and G. Tian, On the singularities of spaces with bounded Ricci curvature, Geom. Funct. Anal. 12 (2002), 873–914. CrossrefGoogle Scholar

  • [20]

    X. X. Chen, On the lower bound of the Mabuchi energy and its application, Int. Math. Res. Not. IMRN 2000 (2000), no. 12, 607–623. CrossrefGoogle Scholar

  • [21]

    X. X. Chen, S. Donaldson and S. Sun, Kähler–Einstein metrics and stability, Int. Math. Res. Not. IMRN 2014 (2014), no. 8, 2119–2125. CrossrefGoogle Scholar

  • [22]

    X. X. Chen, S. Donaldson and S. Sun, Kähler–Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof, J. Amer. Math. Soc. 28 (2015), no. 1, 235–278. Google Scholar

  • [23]

    D. Coman, V. Guedj and A. Zeriahi, Extension of plurisubharmonic functions with growth control, J. reine angew. Math. 676 (2013), 33–49. Google Scholar

  • [24]

    T. Darvas, The Mabuchi geometry of finite energy classes, preprint (2014), https://arxiv.org/abs/1409.2072.

  • [25]

    T. Darvas and Y. Rubinstein, Tian’s properness conjectures and Finsler geometry of the space of Kahler metrics, preprint (2015), http://arxiv.org/abs/1506.07129.

  • [26]

    V. Datar and G. Székelyhidi, Kähler–Einstein metrics along the smooth continuity method, Geom. Funct. Anal. (2016), 10.1007/s00039-016-0377-4. Google Scholar

  • [27]

    J. P. Demailly, Mesures de Monge–Ampère et caractérisation géométrique des variétés algébriques affines, Mém. Soc. Math. Fr. (N.S.) 19 (1985), 1–124. Google Scholar

  • [28]

    J. P. Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), no. 3, 361–409. Google Scholar

  • [29]

    J. P. Demailly, Théorie de Hodge L2 et théorèmes d’annulation, Introduction à la théorie de Hodge, Panor. Synth‘eses 3, Société Mathématique de France, Paris (1996), 3–111. Google Scholar

  • [30]

    A. Dembo and O. Zeitouni, Large deviations techniques and applications. Corrected reprint of the second (1998) edition, Stoch. Model. Appl. Probab. 38, Springer, Berlin 2010. Google Scholar

  • [31]

    W.-Y. Ding, Remarks on the existence problem of positive Kähler–Einstein metrics, Math. Ann. 282 (1988), 463–471. CrossrefGoogle Scholar

  • [32]

    W.-Y. Ding and G. Tian, Kähler–Einstein metrics and the generalized Futaki invariant, Invent. Math. 110 (1992), no. 2, 315–335. CrossrefGoogle Scholar

  • [33]

    S. K. Donaldson and S. Sun, Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, Acta Math. 213 (2014), no. 1, 63–106. CrossrefGoogle Scholar

  • [34]

    P. Eyssidieux, V. Guedj and A. Zeriahi, Singular Kähler–Einstein metrics, J. Amer. Math. Soc. 22 (2009), 607–639. CrossrefGoogle Scholar

  • [35]

    P. Eyssidieux, V. Guedj and A. Zeriahi, Viscosity solutions to degenerate Complex Monge–Ampère equations, Comm. Pure Appl. Math. 64 (2011), 1059–1094. CrossrefGoogle Scholar

  • [36]

    J. E. Fornaess and R. Narasimhan, The Levi problem on complex spaces with singularities, Math. Ann. 248 (1980), no. 1, 47–72. CrossrefGoogle Scholar

  • [37]

    A. Ghigi and J. Kollár, Kähler–Einstein metrics on orbifolds and Einstein metrics on spheres, Comment. Math. Helv. 82 (2007), 877–902. Google Scholar

  • [38]

    H. Grauert and R. Remmert, Plurisubharmonische Funktionen in komplexen Räumen, Math. Z. 65 (1956), 175–194. CrossrefGoogle Scholar

  • [39]

    V. Guedj and A. Zeriahi, The weighted Monge–Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal. 250 (2007), 442–482. CrossrefGoogle Scholar

  • [40]

    V. Guedj and A. Zeriahi, Stability of solutions to complex Monge-Ampère equations in big cohomology classes, Math. Res. Lett. 19 (2012), no. 5, 1025–1042. CrossrefGoogle Scholar

  • [41]

    S. Izumi, Linear complementary inequalities for orders of germs of analytic functions, Invent. Math. 65 (1982), no. 3, 459–471. CrossrefGoogle Scholar

  • [42]

    T. D. Jeffres, R. Mazzeo and Y. A. Rubinstein, Kähler–Einstein metrics with edge singularities, Ann. of Math. (2) 183 (2016), 95–176. Google Scholar

  • [43]

    J. Keller, Ricci iterations on Kähler classes, J. Inst. Math. Jussieu 8 (2009), no. 4, 743–768. CrossrefGoogle Scholar

  • [44]

    J. Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin 1996. Google Scholar

  • [45]

    J. Kollár, Singularities of pairs, Algebraic geometry (Santa Cruz 1995), Proc. Sympos. Pure Math. 62. Part 1, American Mathematical Society, Providence (1997), 221–287. Google Scholar

  • [46]

    Y. Lee, Chow stability criterion in terms of log canonical threshold, J. Korean Math. Soc. 45 (2008), no. 2, 467–477. CrossrefGoogle Scholar

  • [47]

    C. Li and C. Y. Xu, Special test configurations and K-stability of Fano varities, Ann. of Math. (2) 180 (2014), no. 1, 197–232. CrossrefGoogle Scholar

  • [48]

    H. Li, On the lower bound of the K-energy and F-functional, Osaka J. Math. 45 (2008), no. 1, 253–264. Google Scholar

  • [49]

    Y. Matsushima, Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variétié kählérienne, Nagoya Math. J. 11 (1957), 145–150. CrossrefGoogle Scholar

  • [50]

    M. Păun, Regularity properties of the degenerate Monge–Ampère equations on compact Kähler manifolds, Chin. Ann. Math. Ser. B 29 (2008), no. 6, 623–630. CrossrefGoogle Scholar

  • [51]

    D. H. Phong, N. Sesum and J. Sturm, Multiplier ideal sheaves and the Kähler–Ricci flow, Comm. Anal. Geom. 15 (2007), no. 3, 613–632. CrossrefGoogle Scholar

  • [52]

    D. H. Phong, J. Song, J. Sturm and B. Weinkove, The Kähler–Ricci flow with positive bisectional curvature, Invent. Math. 173 (2008), no. 3, 651–665. CrossrefGoogle Scholar

  • [53]

    D. H. Phong, J. Song, J. Sturm and B. Weinkove, The Moser–Trudinger inequality on Kähler–Einstein manifolds, Amer. J. Math. 130 (2008), no. 4, 1067–1085. CrossrefGoogle Scholar

  • [54]

    D. H. Phong and J. Sturm, Lectures on stability and constant scalar curvature, Handbook of geometric analysis. No. 3, Adv. Lect. Math. (ALM) 14, International Press, Somerville (2010), 357–436. Google Scholar

  • [55]

    Y. 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. CrossrefGoogle Scholar

  • [56]

    N. Sesum and G. Tian, Bounding scalar curvature and diameter along the Kähler–Ricci flow (after Perelman), J. Inst. Math. Jussieu 7 (2008), no. 3, 575–587. Google Scholar

  • [57]

    Y. T. Siu, Lectures on Hermitian–Einstein metrics for stable bundles and Kähler–Einstein metrics, DMV Seminar 8, Birkhäuser, Basel 1987. Google Scholar

  • [58]

    H. Skoda, Sous-ensembles analytiques d’ordre fini ou infini dans n, Bull. Soc. Math. France 100 (1972), 353–408. Google Scholar

  • [59]

    J. Song and G. Tian, The Kähler–Ricci flow through singularities, preprint (2009), https://arxiv.org/abs/0909.4898.

  • [60]

    J. Song and G. Tian, Canonical measures and Kähler–Ricci flow, J. Amer. Math. Soc. 25 (2012), no. 2, 303–353. CrossrefGoogle Scholar

  • [61]

    G. Tian, On Kähler–Einstein metrics on certain Kähler manifolds with c1(M)>0, Invent. Math. 89 (1987), no. 2, 225–246. Google Scholar

  • [62]

    G. Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), 239–265. Google Scholar

  • [63]

    G. Tian, Canonical metrics in Kähler geometry, Lect. Math. ETH Zürich, Birkhäuser, Basel 2000. Google Scholar

  • [64]

    G. Tian, Existence of Einstein metrics on Fano manifolds, Metric and differential geometry (Tianjin and Beijing 2009), Progr. Math. 297, Springer, Berlin (2012), 119–159. Google Scholar

  • [65]

    G. Tian, K-stability implies CM-stability, preprint (2014), http://arxiv.org/abs/1409.7836.

  • [66]

    G. Tian, K-stability and Kähler–Einstein metrics, Comm. Pure Appl. Math. 68 (2015), no. 7, 1085–1156. CrossrefGoogle Scholar

  • [67]

    G. Tian and S. T. Yau, Kähler–Einstein metrics on complex surfaces with c1>0, Comm. Math. Phys. 112 (1987), no. 1, 175–203. Google Scholar

  • [68]

    G. Tian and X. Zhu, Convergence of Kähler–Ricci flow, J. Amer. Math. Soc. 20 (2007), no. 3, 675–699. CrossrefGoogle Scholar

  • [69]

    H. Tsuji, Existence and degeneration of Kähler–Einstein metrics on minimal algebraic varieties of general type, Math. Ann. 281 (1988), no. 1, 123–133. CrossrefGoogle Scholar

  • [70]

    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. CrossrefGoogle Scholar

  • [71]

    L. Yi, A Bando–Mabuchi uniqueness theorem, preprint (2013), https://arxiv.org/abs/1301.2847.

  • [72]

    A. Zeriahi, Volume and capacity of sublevel sets of a Lelong class of psh functions, Indiana Univ. Math. J. 50 (2001), no. 1, 671–703. CrossrefGoogle Scholar

About the article

Received: 2014-08-03

Revised: 2016-05-04

Published Online: 2016-09-14

Published in Print: 2019-06-01


Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: MACK

The authors are partially supported by the French ANR project MACK.


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 751, Pages 27–89, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2016-0033.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Tamás Darvas and Yanir A. Rubinstein
Analysis & PDE, 2019, Volume 12, Number 3, Page 721
[2]
Phylippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi
Communications in Mathematical Physics, 2018
[3]
Jakob Hultgren and D Witt Nyström
International Mathematics Research Notices, 2018

Comments (0)

Please log in or register to comment.
Log in