Constant scalar curvature kähler metrics: uniqueness, stability
Séminaire Bourbaki (2004-2005)
- Volume: 47, page 1-32
- ISSN: 0303-1179
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topBiquard, Olivier. "Métriques kählériennes à courbure scalaire constante : unicité, stabilité." Séminaire Bourbaki 47 (2004-2005): 1-32. <http://eudml.org/doc/252160>.
@article{Biquard2004-2005,
abstract = {Un des problèmes les plus intéressants de la géométrie différentielle complexe consiste à comprendre les classes de Kähler de variétés complexes admettant des métriques à courbure scalaire constante. La question de l’unicité a été récemment résolue par Donaldson, Mabuchi, Chen–Tian. Des liens forts avec la stabilité algébrique des variétés ont été mis en évidence. L’exposé s’attachera à exposer les idées nouvelles qui ont mené à ces résultats.},
author = {Biquard, Olivier},
journal = {Séminaire Bourbaki},
keywords = {Kähler manifold; extremal metric; stability},
language = {fre},
pages = {1-32},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Métriques kählériennes à courbure scalaire constante : unicité, stabilité},
url = {http://eudml.org/doc/252160},
volume = {47},
year = {2004-2005},
}
TY - JOUR
AU - Biquard, Olivier
TI - Métriques kählériennes à courbure scalaire constante : unicité, stabilité
JO - Séminaire Bourbaki
PY - 2004-2005
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 47
SP - 1
EP - 32
AB - Un des problèmes les plus intéressants de la géométrie différentielle complexe consiste à comprendre les classes de Kähler de variétés complexes admettant des métriques à courbure scalaire constante. La question de l’unicité a été récemment résolue par Donaldson, Mabuchi, Chen–Tian. Des liens forts avec la stabilité algébrique des variétés ont été mis en évidence. L’exposé s’attachera à exposer les idées nouvelles qui ont mené à ces résultats.
LA - fre
KW - Kähler manifold; extremal metric; stability
UR - http://eudml.org/doc/252160
ER -
References
top- [1] M. Abreu – “Kähler geometry of toric varieties and extremal metrics”, Internat. J. Math. 9 (1998), no. 6, p. 641–651. Zbl0932.53043MR1644291
- [2] T. Aubin – “Équations du type Monge-Ampère sur les variétés kähleriennes compactes”, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 3, p. Aiii, A119–A121. Zbl0333.53040MR433520
- [3] S. Bando & T. Mabuchi – “Uniqueness of Einstein Kähler metrics modulo connected group actions”, in Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, p. 11–40. Zbl0641.53065MR946233
- [4] A.L. Besse – Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10, Springer-Verlag, Berlin, 1987. Zbl0613.53001MR867684
- [5] J.-P. Bourguignon – “Métriques d’Einstein-Kähler sur les variétés de Fano : obstructions et existence (d’après Y. Matsushima, A. Futaki, S.T. Yau, A. Nadel et G. Tian)”, in Séminaire Bourbaki, 1996/97, Astérisque, vol. 245, 1997, Exp. no 830, p. 277–305. Zbl0935.32019MR1627115
- [6] L. Boutet de Monvel & J. Sjöstrand – “Sur la singularité des noyaux de Bergman et de Szegö”, in Journées Équations aux Dérivées Partielles (Rennes, 1975), Astérisque, vol. 34-35, Paris, 1976, p. 123–164. Zbl0344.32010MR590106
- [7] D. Burns & P. de Bartolomeis – “Stability of vector bundles and extremal metrics”, Invent. Math. 92 (1988), no. 2, p. 403–407. Zbl0645.53037MR936089
- [8] E. Calabi – “Extremal Kähler metrics”, in Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, p. 259–290. Zbl0487.53057MR645743
- [9] —, “Extremal Kähler metrics. II”, in Differential geometry and complex analysis, Springer, Berlin, 1985, p. 95–114. Zbl0574.58006
- [10] E. Calabi & X.X. Chen – “The space of Kähler metrics. II”, J. Differential Geom. 61 (2002), no. 2, p. 173–193. Zbl1067.58010MR1969662
- [11] D.M.J. Calderbank & M.A. Singer – “Einstein metrics and complex singularities”, Invent. Math. 156 (2004), no. 2, p. 405–443. Zbl1061.53026MR2052611
- [12] D. Catlin – “The Bergman kernel and a theorem of Tian”, in Analysis and geometry in several complex variables (Katata, 1997), Trends Math., Birkhäuser Boston, Boston, MA, 1999, p. 1–23. Zbl0941.32002MR1699887
- [13] X.X. Chen – “The space of Kähler metrics”, J. Differential Geom. 56 (2000), no. 2, p. 189–234. Zbl1041.58003MR1863016
- [14] X.X. Chen & G. Tian – “Geometry of Kähler metrics and holomorphic foliation by discs”, arXiv : math.DG/0409433.
- [15] X. Dai, K. Liu & X. Ma – “On the asymptotic expansion of Bergman kernel”, J. Differential Geom. 72 (2006), no. 1, p. 1–41. Zbl1099.32003MR2215454
- [16] S.K. Donaldson – “Infinite determinants, stable bundles and curvature”, Duke Math. J. 54 (1987), no. 1, p. 231–247. Zbl0627.53052MR885784
- [17] —, “Remarks on gauge theory, complex geometry and -manifold topology”, in Fields Medallists’ lectures, World Sci. Ser. 20th Century Math., vol. 5, World Sci. Publishing, River Edge, NJ, 1997, p. 384–403. MR1622931
- [18] —, “Symmetric spaces, Kähler geometry and Hamiltonian dynamics”, in Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc., Providence, RI, 1999, p. 13–33. Zbl0972.53025
- [19] —, “Scalar curvature and projective embeddings. I”, J. Differential Geom. 59 (2001), no. 3, p. 479–522. Zbl1052.32017MR1916953
- [20] —, “Holomorphic discs and the complex Monge-Ampère equation”, J. Symplectic Geom. 1 (2002), no. 2, p. 171–196. Zbl1035.53102MR1959581
- [21] —, “Scalar curvature and stability of toric varieties”, J. Differential Geom. 62 (2002), no. 2, p. 289–349. Zbl1074.53059MR1988506
- [22] —, “Interior estimates for solutions of Abreu’s equation”, Collect. Math. 56 (2005), no. 2, p. 103–142. Zbl1085.53063MR2154300
- [23] —, “Scalar curvature and projective embeddings, II”, Q. J. Math. 56 (2005), no. 3, p. 345–356. Zbl1159.32012MR2161248
- [24] S.K. Donaldson & P. B. Kronheimer – The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1990, Oxford Science Publications. Zbl0904.57001MR1079726
- [25] A. Futaki – “An obstruction to the existence of Einstein Kähler metrics”, Invent. Math. 73 (1983), no. 3, p. 437–443. Zbl0506.53030MR718940
- [26] P. Gauduchon – Calabi’s extremal Kähler metrics : an elementary introduction. Zbl0487.53057
- [27] D. Gieseker – “Global moduli for surfaces of general type”, Invent. Math. 43 (1977), no. 3, p. 233–282. Zbl0389.14006MR498596
- [28] D. Guan – “On modified Mabuchi functional and Mabuchi moduli space of Kähler metrics on toric bundles”, Math. Res. Lett. 6 (1999), no. 5-6, p. 547–555. Zbl0968.53050MR1739213
- [29] G. Kempf & L. Ness – “The length of vectors in representation spaces”, in Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lect. Notes in Math., vol. 732, Springer, Berlin, 1979, p. 233–243. Zbl0407.22012MR555701
- [30] J. Kim, C. LeBrun & M. Pontecorvo – “Scalar-flat Kähler surfaces of all genera”, J. Reine Angew. Math.486 (1997), p. 69–95. Zbl0876.53044MR1450751
- [31] C. LeBrun – “Scalar-flat Kähler metrics on blown-up ruled surfaces.”, J. Reine Angew. Math.420 (1991), p. 161–177. Zbl0727.53067MR1124569
- [32] —, “Polarized 4-manifolds, extremal Kähler metrics, and Seiberg-Witten theory”, Math. Res. Lett. 2 (1995), no. 5, p. 653–662. Zbl0874.53051MR1359969
- [33] C. LeBrun & M. Singer – “Existence and deformation theory for scalar-flat Kähler metrics on compact complex surfaces”, Invent. Math. 112 (1993), no. 2, p. 273–313. Zbl0793.53067MR1213104
- [34] A. Lichnerowicz – “Sur les transformations analytiques des variétés kählériennes compactes”, C. R. Acad. Sci. Paris244 (1957), p. 3011–3013. Zbl0080.37501MR94479
- [35] Z. Lu – “On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch”, Amer. J. Math. 122 (2000), no. 2, p. 235–273. Zbl0972.53042MR1749048
- [36] H. Luo – “Geometric criterion for Gieseker-Mumford stability of polarized manifolds”, J. Differential Geom. 49 (1998), no. 3, p. 577–599. Zbl1006.32022MR1669716
- [37] T. Mabuchi – “-energy maps integrating Futaki invariants”, Tohoku Math. J. (2) 38 (1986), no. 4, p. 575–593. Zbl0619.53040MR867064
- [38] —, “Some symplectic geometry on compact Kähler manifolds. I”, Osaka J. Math. 24 (1987), no. 2, p. 227–252. Zbl0645.53038MR909015
- [39] —, “Stability of extremal Kähler metrics”, Osaka J. Math. 41 (2004), no. 3.
- [40] —, “Uniqueness of extremal Kähler metrics for an integral Kähler class”, Internat. J. Math. 15 (2004), no. 6, p. 531–546. Zbl1058.32017MR2078878
- [41] —, “An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, I”, Invent. Math. 159 (2005), no. 2, p. 225–243. Zbl1118.53047MR2116275
- [42] —, “An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, II”, J. Differential Geom. (à paraître). Zbl1209.53032
- [43] Y. Matsushima – “Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne”, Nagoya Math. J.11 (1957), p. 145–150. Zbl0091.34803MR94478
- [44] D. Mumford – “Stability of projective varieties”, Enseignement Math. (2) 23 (1977), no. 1-2, p. 39–110. Zbl0363.14003MR450272
- [45] D. Mumford, J. Fogarty & F. Kirwan – Geometric invariant theory, 3e ’ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34, Springer-Verlag, Berlin, 1994. Zbl0797.14004MR1304906
- [46] S.T. Paul & G. Tian – “Algebraic and analytic K-stability”, arXiv : math.DG/0405530.
- [47] D.H. Phong & J. Sturm – “Stability, energy functionals, and Kähler-Einstein metrics”, Comm. Anal. Geom. 11 (2003), no. 3, p. 565–597. Zbl1098.32012MR2015757
- [48] —, “Scalar curvature, moment maps, and the Deligne pairing”, Amer. J. Math. 126 (2004), no. 3, p. 693–712. Zbl1077.53068MR2058389
- [49] Y. Rollin & M. Singer – “Non-minimal scalar-flat Kähler surfaces and parabolic stability”, Invent. Math. 162 (2005), no. 2, p. 235–270. Zbl1083.32021MR2199006
- [50] J. Ross – “Instability of polarised algebraic varieties”, PhD thesis, Imperial College, 2003.
- [51] J. Ross & R. Thomas – “An obstruction to the existence of constant scalar curvature Kähler metrics”, J. Differential Geom. 72 (2006), no. 3, p. 429–466. Zbl1125.53057MR2219940
- [52] —, “A study of the Hilbert-Mumford criterion for the stability of projective varieties”, arXiv : math.AG/0412519, . Zbl1200.14095
- [53] W.-D. Ruan – “Canonical coordinates and Bergmann metrics”, Comm. Anal. Geom. 6 (1998), no. 3, p. 589–631. Zbl0917.53026MR1638878
- [54] S. Semmes – “Complex Monge-Ampère and symplectic manifolds”, Amer. J. Math. 114 (1992), no. 3, p. 495–550. Zbl0790.32017MR1165352
- [55] C.T. Simpson – “Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization”, J. Amer. Math. Soc. 1 (1988), no. 4, p. 867–918. Zbl0669.58008MR944577
- [56] G. Tian – “On Kähler-Einstein metrics on certain Kähler manifolds with ”, Invent. Math. 89 (1987), no. 2, p. 225–246. Zbl0599.53046MR894378
- [57] —, “On a set of polarized Kähler metrics on algebraic manifolds”, J. Differential Geom. 32 (1990), no. 1, p. 99–130. Zbl0706.53036MR1064867
- [58] —, “On Calabi’s conjecture for complex surfaces with positive first Chern class”, Invent. Math. 101 (1990), no. 1, p. 101–172. Zbl0716.32019MR1055713
- [59] —, “The -energy on hypersurfaces and stability”, Comm. Anal. Geom. 2 (1994), no. 2, p. 239–265. Zbl0846.32019MR1312688
- [60] —, “Kähler-Einstein metrics with positive scalar curvature”, Invent. Math. 130 (1997), no. 1, p. 1–37. Zbl0892.53027MR1471884
- [61] —, “Bott-Chern forms and geometric stability”, Discrete Contin. Dynam. Systems 6 (2000), no. 1, p. 211–220. Zbl1022.32009MR1739924
- [62] —, Canonical metrics in Kähler geometry, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2000, Notes taken by Meike Akveld. Zbl0978.53002MR1787650
- [63] —, “Extremal metrics and geometric stability”, Houston J. Math. 28 (2002), no. 2, p. 411–432, Special issue for S. S. Chern. Zbl1026.53023MR1898198
- [64] K. Uhlenbeck & S.-T. Yau – “On the existence of Hermitian-Yang-Mills connections in stable vector bundles”, Comm. Pure Appl. Math. 39 (1986), no. S, suppl., p. S257–S293, Frontiers of the mathematical sciences : 1985 (New York, 1985). Zbl0615.58045MR861491
- [65] E. Viehweg – Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 30, Springer-Verlag, Berlin, 1995. Zbl0844.14004MR1368632
- [66] S.-T. Yau – “Calabi’s conjecture and some new results in algebraic geometry”, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, p. 1798–1799. Zbl0355.32028MR451180
- [67] —, “On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I”, Comm. Pure Appl. Math. 31 (1978), no. 3, p. 339–411. Zbl0369.53059MR480350
- [68] —, “Open problems in geometry”, in Differential geometry : partial differential equations on manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, p. 1–28. Zbl0801.53001MR1216573
- [69] S. Zelditch – “Szegö kernels and a theorem of Tian”, Internat. Math. Res. Notices (1998), no. 6, p. 317–331. Zbl0922.58082MR1616718
- [70] S. Zhang – “Heights and reductions of semi-stable varieties”, Compositio Math. 104 (1996), no. 1, p. 77–105. Zbl0924.11055MR1420712
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