Analysis & PDE
- Anal. PDE
- Volume 6, Number 1 (2013), 131-180.
Relative Kähler–Ricci flows and their quantization
Full-text: Open access
Abstract
Let be a holomorphic fibration and let be a relatively ample line bundle over . We define relative Kähler–Ricci flows on the space of all Hermitian metrics on with relatively positive curvature and study their convergence properties. Mainly three different settings are investigated: the case when the fibers are Calabi–Yau manifolds and the case when is the relative (anti)canonical line bundle. The main theme studied is whether “positivity in families” is preserved under the flows and its relation to the variation of the moduli of the complex structures of the fibers. The “quantization” of this setting is also studied, where the role of the Kähler–Ricci flow is played by Donaldson’s iteration on the space of all Hermitian metrics on the finite rank vector bundle . Applications to the construction of canonical metrics on the relative canonical bundles of canonically polarized families and Weil–Petersson geometry are given. Some of the main results are a parabolic analogue of a recent elliptic equation of Schumacher and the convergence towards the Kähler–Ricci flow of Donaldson’s iteration in a certain double scaling limit.
Article information
Source
Anal. PDE, Volume 6, Number 1 (2013), 131-180.
Dates
Received: 21 August 2011
Revised: 15 November 2011
Accepted: 20 December 2011
First available in Project Euclid: 20 December 2017
Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731292
Digital Object Identifier
doi:10.2140/apde.2013.6.131
Mathematical Reviews number (MathSciNet)
MR3068542
Zentralblatt MATH identifier
1282.14069
Subjects
Primary: 14J32: Calabi-Yau manifolds 32G05: Deformations of complex structures [See also 13D10, 16S80, 58H10, 58H15] 32Q20: Kähler-Einstein manifolds [See also 53Cxx] 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]
Keywords
Kähler–Ricci flow positivity Kähler–Einstein metric balanced metric Weil–Petersson metric
Citation
Berman, Robert. Relative Kähler–Ricci flows and their quantization. Anal. PDE 6 (2013), no. 1, 131--180. doi:10.2140/apde.2013.6.131. https://projecteuclid.org/euclid.apde/1513731292
References
- R. J. Berman, “Determinantal point processes and fermions on complex manifolds: Bulk universality”, preprint, 2008. arXiv: 0811.3341
- R. Berman and S. Boucksom, “Growth of balls of holomorphic sections and energy at equilibrium”, Invent. Math. 181:2 (2010), 337–394. Mathematical Reviews (MathSciNet): MR2011h:32021
Zentralblatt MATH: 1208.32020
Digital Object Identifier: doi:10.1007/s00222-010-0248-9 - R. Berman and J.-P. Demailly, “Regularity of plurisubharmonic upper envelopes in big cohomology classes”, pp. 39–66 in Perspectives in analysis, geometry, and topology, Progress in Mathematics 296, Birkhäuser, New York, 2012.
- R. Berman, S. Boucksom, V. Guedj, and A. Zeriahi, “A variational approach to complex Monge–Ampère equations”, preprint, 2009. arXiv: 0907.4490
- B. Berndtsson, “Curvature of vector bundles associated to holomorphic fibrations”, Ann. of Math. $(2)$ 169:2 (2009), 531–560. Mathematical Reviews (MathSciNet): MR2010c:32036
Zentralblatt MATH: 1195.32012
Digital Object Identifier: doi:10.4007/annals.2009.169.531 - B. Berndtsson, “Positivity of direct image bundles and convexity on the space of Kähler metrics”, J. Differential Geom. 81:3 (2009), 457–482. Mathematical Reviews (MathSciNet): MR2010e:32020
Zentralblatt MATH: 1187.53076
Digital Object Identifier: doi:10.4310/jdg/1236604342
Project Euclid: euclid.jdg/1236604342 - B. Berndtsson, “Remarks on a theorem by H. Tsuji”, pp. 1107–1110 in Multiplier Ideal Sheaves in Algebraic and Complex Geometry (Oberwolfach, 2009), edited by S. Kebekus et al., Mathematisches Forschungsinstitut Oberwolfach 6, European Mathematical Society, Zürich, 2009.
- B. Berndtsson, “Strict and nonstrict positivity of direct image bundles”, Math. Z. 269:3-4 (2011), 1201–1218. Zentralblatt MATH: 05996694
- B. Berndtsson and M. Păun, “A Bergman kernel proof of the Kawamata subadjunction theorem”, preprint, 2008. arXiv: 0804.3884
- B. Berndtsson and M. Păun, “Bergman kernels and the pseudoeffectivity of relative canonical bundles”, Duke Math. J. 145:2 (2008), 341–378. Mathematical Reviews (MathSciNet): MR2009k:32020
Zentralblatt MATH: 1181.32025
Digital Object Identifier: doi:10.1215/00127094-2008-054
Project Euclid: euclid.dmj/1224508840 - S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi, “Monge–Ampère equations in big cohomology classes”, Acta Math. 205:2 (2010), 199–262. Mathematical Reviews (MathSciNet): MR2011k:32049
Zentralblatt MATH: 1213.32025
Digital Object Identifier: doi:10.1007/s11511-010-0054-7
Project Euclid: euclid.acta/1485892500 - J.-P. Bourguignon, P. Li, and S.-T. Yau, “Upper bound for the first eigenvalue of algebraic submanifolds”, Comment. Math. Helv. 69:2 (1994), 199–207. Mathematical Reviews (MathSciNet): MR95j:58168
Zentralblatt MATH: 0814.53040
Digital Object Identifier: doi:10.1007/BF02564482 - H. D. Cao, “Deformation of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds”, Invent. Math. 81:2 (1985), 359–372. Mathematical Reviews (MathSciNet): MR87d:58051
Zentralblatt MATH: 0574.53042
Digital Object Identifier: doi:10.1007/BF01389058 - X. Chen, “The space of Kähler metrics”, J. Differential Geom. 56:2 (2000), 189–234. Mathematical Reviews (MathSciNet): MR2003b:32031
Zentralblatt MATH: 1041.58003
Digital Object Identifier: doi:10.4310/jdg/1090347643
Project Euclid: euclid.jdg/1090347643 - X. X. Chen and G. Tian, “Ricci flow on Kähler–Einstein surfaces”, Invent. Math. 147:3 (2002), 487–544. Mathematical Reviews (MathSciNet): MR2003c:53095
Zentralblatt MATH: 1047.53043
Digital Object Identifier: doi:10.1007/s002220100181 - P. Deligne, “Le déterminant de la cohomologie”, pp. 93–177 in Current trends in arithmetical algebraic geometry (Arcata, CA, 1985), edited by K. A. Ribet, Contemporary Mathematics 67, American Mathematical Society, Providence, RI, 1987.
- S. K. Donaldson, “Scalar curvature and projective embeddings, I”, J. Differential Geom. 59:3 (2001), 479–522. Mathematical Reviews (MathSciNet): MR2003j:32030
Zentralblatt MATH: 1052.32017
Digital Object Identifier: doi:10.4310/jdg/1090349449
Project Euclid: euclid.jdg/1090349449 - S. K. Donaldson, “Scalar curvature and projective embeddings, II”, Q. J. Math. 56:3 (2005), 345–356. Mathematical Reviews (MathSciNet): MR2006f:32033
Zentralblatt MATH: 1159.32012
Digital Object Identifier: doi:10.1093/qmath/hah044 - S. K. Donaldson, “Some numerical results in complex differential geometry”, Pure Appl. Math. Q. 5:2 (2009), 571–618. Mathematical Reviews (MathSciNet): MR2010d:32020
Zentralblatt MATH: 1178.32018
Digital Object Identifier: doi:10.4310/PAMQ.2009.v5.n2.a2 - P. Eyssidieux, V. Guedj, and A. Zeriahi, “Singular Kähler–Einstein metrics”, J. Amer. Math. Soc. 22:3 (2009), 607–639. Mathematical Reviews (MathSciNet): MR2010k:32031
Zentralblatt MATH: 1215.32017
Digital Object Identifier: doi:10.1090/S0894-0347-09-00629-8 - J. Fine, “Calabi flow and projective embeddings”, J. Differential Geom. 84:3 (2010), 489–523. Mathematical Reviews (MathSciNet): MR2012a:32023
Zentralblatt MATH: 1202.32018
Digital Object Identifier: doi:10.4310/jdg/1279114299
Project Euclid: euclid.jdg/1279114299 - A. Fujiki and G. Schumacher, “The moduli space of extremal compact Kähler manifolds and generalized Weil–Petersson metrics”, Publ. Res. Inst. Math. Sci. 26:1 (1990), 101–183. Mathematical Reviews (MathSciNet): MR92e:32012
Zentralblatt MATH: 0714.32007
Digital Object Identifier: doi:10.2977/prims/1195171664 - B. R. Greene, A. Shapere, C. Vafa, and S.-T. Yau, “Stringy cosmic strings and noncompact Calabi–Yau manifolds”, Nuclear Phys. B 337:1 (1990), 1–36. Mathematical Reviews (MathSciNet): MR91g:83057
Zentralblatt MATH: 0744.53045
Digital Object Identifier: doi:10.1016/0550-3213(90)90248-C - M. Gross and P. M. H. Wilson, “Large complex structure limits of $K3$ surfaces”, J. Differential Geom. 55:3 (2000), 475–546. Mathematical Reviews (MathSciNet): MR2003a:32042
Zentralblatt MATH: 1027.32021
Digital Object Identifier: doi:10.4310/jdg/1090341262
Project Euclid: euclid.jdg/1090341262 - V. Guedj and A. Zeriahi, “Intrinsic capacities on compact Kähler manifolds”, J. Geom. Anal. 15:4 (2005), 607–639. Mathematical Reviews (MathSciNet): MR2006j:32041
Zentralblatt MATH: 1087.32020
Digital Object Identifier: doi:10.1007/BF02922247 - R. S. Hamilton, “Three-manifolds with positive Ricci curvature”, J. Differential Geom. 17:2 (1982), 255–306. Mathematical Reviews (MathSciNet): MR84a:53050
Zentralblatt MATH: 0504.53034
Digital Object Identifier: doi:10.4310/jdg/1214436922
Project Euclid: euclid.jdg/1214436922 - Y. Kawamata, “Kodaira dimension of algebraic fiber spaces over curves”, Invent. Math. 66:1 (1982), 57–71. Mathematical Reviews (MathSciNet): MR83h:14025
Zentralblatt MATH: 0461.14004
Digital Object Identifier: doi:10.1007/BF01404756 - J. Keller, “Ricci iterations on Kähler classes”, J. Inst. Math. Jussieu 8:4 (2009), 743–768. Mathematical Reviews (MathSciNet): MR2010h:32026
Zentralblatt MATH: 1178.32020
Digital Object Identifier: doi:10.1017/S1474748009000103 - J. Keller and S. Lukic, “Numerical Weil–Petersson metrics on moduli spaces of Calabi–Yau manifolds”, preprint, 2009. arXiv: :0907.1387
- N. Koiso, “Einstein metrics and complex structures”, Invent. Math. 73:1 (1983), 71–106. Mathematical Reviews (MathSciNet): MR85d:58018
Zentralblatt MATH: 0515.53040
Digital Object Identifier: doi:10.1007/BF01393826 - R. Lazarsfeld, Positivity in algebraic geometry, I: Classical setting: Line bundles and linear series, Ergeb. Math. Grenzgeb. (3) 48, Springer, Berlin, 2004.
- K. Liu, X. Sun, and S.-T. Yau, “Good geometry on the curve moduli”, Publ. Res. Inst. Math. Sci. 44:2 (2008), 699–724. Mathematical Reviews (MathSciNet): MR2009h:32020
Zentralblatt MATH: 1219.14012
Digital Object Identifier: doi:10.2977/prims/1210167341 - T. Mabuchi, “$K$-energy maps integrating Futaki invariants”, Tohoku Math. J. $(2)$ 38:4 (1986), 575–593. Mathematical Reviews (MathSciNet): MR88b:53060
Zentralblatt MATH: 0619.53040
Digital Object Identifier: doi:10.2748/tmj/1178228410
Project Euclid: euclid.tmj/1178228410 - M. S. Narasimhan and R. R. Simha, “Manifolds with ample canonical class”, Invent. Math. 5 (1968), 120–128. Mathematical Reviews (MathSciNet): MR38:5253
Zentralblatt MATH: 0159.37902
Digital Object Identifier: doi:10.1007/BF01425543 - D. H. Phong and J. Sturm, “Scalar curvature, moment maps, and the Deligne pairing”, Amer. J. Math. 126:3 (2004), 693–712. Mathematical Reviews (MathSciNet): MR2005b:53137
Zentralblatt MATH: 1077.53068
Digital Object Identifier: doi:10.1353/ajm.2004.0019 - D. H. Phong, N. Sesum, and J. Sturm, “Multiplier ideal sheaves and the Kähler–Ricci flow”, Comm. Anal. Geom. 15:3 (2007), 613–632. Mathematical Reviews (MathSciNet): MR2009a:32037
Zentralblatt MATH: 1143.53064
Digital Object Identifier: doi:10.4310/CAG.2007.v15.n3.a7 - D. H. Phong, J. Song, J. Sturm, and B. Weinkove, “The Moser–Trudinger inequality on Kähler–Einstein manifolds”, Amer. J. Math. 130:4 (2008), 1067–1085. Mathematical Reviews (MathSciNet): MR2009e:32027
Zentralblatt MATH: 1158.58005
Digital Object Identifier: doi:10.1353/ajm.0.0013 - M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, NJ, 1967.
- Y. A. Rubinstein, “Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics”, Adv. Math. 218:5 (2008), 1526–1565. Mathematical Reviews (MathSciNet): MR2009m:32045
Zentralblatt MATH: 1143.53065
Digital Object Identifier: doi:10.1016/j.aim.2008.03.017 - Y. Sano, “Numerical algorithm for finding balanced metrics”, Osaka J. Math. 43:3 (2006), 679–688. http://msp.org/idx/mr/2008i:32031MR 2008i:32031 Zentralblatt MATH: 1108.32014
URL: Link to item
Mathematical Reviews (MathSciNet): MR2283416
Project Euclid: euclid.ojm/1159190008 - G. Schumacher, “Positivity of relative canonical bundles for families of canonically polarized manifolds”, preprint, 2008. arXiv: 0808.3259
- Y. T. Siu, “Curvature of the Weil–Petersson metric in the moduli space of compact Kähler–Einstein manifolds of negative first Chern class”, pp. 261–298 in Contributions to several complex variables (Notre Dame, IN, 1984), edited by A. Howard and P.-M. Wong, Aspects Math. E9, Vieweg, Braunschweig, 1986.
- Y.-T. Siu, “Invariance of plurigenera”, Invent. Math. 134:3 (1998), 661–673. Mathematical Reviews (MathSciNet): MR99i:32035
Zentralblatt MATH: 0955.32017
Digital Object Identifier: doi:10.1007/s002220050276 - J. Song and G. Tian, “The Kahler–Ricci flow through singularities”, preprint, 2009. arXiv: 0909.4898
- J. Song and G. Tian, “Canonical measures and Kähler–Ricci flow”, J. Amer. Math. Soc. 25:2 (2012), 303–353. Mathematical Reviews (MathSciNet): MR2869020
Zentralblatt MATH: 1239.53086
Digital Object Identifier: doi:10.1090/S0894-0347-2011-00717-0 - J. Song and B. Weinkove, “Constructions of Kähler–Einstein metrics with negative scalar curvature”, Math. Ann. 347:1 (2010), 59–79. Mathematical Reviews (MathSciNet): MR2011b:32037
Zentralblatt MATH: 1205.53077
Digital Object Identifier: doi:10.1007/s00208-009-0427-4 - G. Tian, “Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric”, pp. 629–646 in Mathematical aspects of string theory (San Diego, CA, 1986), edited by S.-T. Yau, Adv. Ser. Math. Phys. 1, World Science Publishing, Singapore, 1987.
- G. Tian, Canonical metrics in Kähler geometry, Birkhäuser, Basel, 2000.
- G. Tian and X. Zhu, “Convergence of Kähler–Ricci flow”, J. Amer. Math. Soc. 20:3 (2007), 675–699. Mathematical Reviews (MathSciNet): MR2007k:53107
Zentralblatt MATH: 1185.53078
Digital Object Identifier: doi:10.1090/S0894-0347-06-00552-2 - A. N. Todorov, “The Weil–Petersson geometry of the moduli space of ${\rm SU}(n\geq 3)$ (Calabi–Yau) manifolds, I”, Comm. Math. Phys. 126:2 (1989), 325–346. Mathematical Reviews (MathSciNet): MR91f:32022
Zentralblatt MATH: 0688.53030
Digital Object Identifier: doi:10.1007/BF02125128
Project Euclid: euclid.cmp/1104179854 - H. Tsuji, “Dynamical construction of Kähler–Einstein metrics”, preprint, 2006. arXiv: math/0606626
- H. Tsuji, “Canonical singular Hermitian metrics on relative canonical bundles”, Amer. J. Math. 133:6 (2011), 1469–1501. Mathematical Reviews (MathSciNet): MR2863368
Zentralblatt MATH: 05994418
Digital Object Identifier: doi:10.1353/ajm.2011.0047 - C. Voisin, Hodge theory and complex algebraic geometry, vol. 1, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2007.
- X. Wang, “Canonical metrics on stable vector bundles”, Comm. Anal. Geom. 13:2 (2005), 253–285. http://msp.org/idx/mr/2006b:32031MR 2006b:32031 1108.32014 Zentralblatt MATH: 1093.32008
URL: Link to item
Mathematical Reviews (MathSciNet): MR2154820
Digital Object Identifier: doi:10.4310/CAG.2005.v13.n2.a1 - 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. Mathematical Reviews (MathSciNet): MR81d:53045
Zentralblatt MATH: 0369.53059
Digital Object Identifier: doi:10.1002/cpa.3160310304 - S.-T. Yau, “Nonlinear analysis in geometry”, Enseign. Math. $(2)$ 33:1-2 (1987), 109–158.
- S. Zelditch, “Book review of Holomorphic Morse inequalities and Bergman kernels by Xiaonan Ma and George Marinescu; MR2339952”, Bull. Amer. Math. Soc. $($N.S.$)$ 46:2 (2009), 349–361. Mathematical Reviews (MathSciNet): MR2729895
Digital Object Identifier: doi:10.1090/S0273-0979-08-01224-X
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