Métriques kählériennes à courbure scalaire constante : unicité, stabilité
Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 938, pp. 1-31.

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.

One of the most interesting problems in complex differential geometry is to understand the Kähler classes of complex compact manifolds which admit constant scalar curvature metrics. The uniqueness question has been recently solved in the works of Donaldson, Mabuchi, Chen and Tian, and strong relations appeared between existence and the stability of algebraic varieties. The seminar explains some of the new ideas leading to these results.

Classification : 32Q15, 53D20, 53C55
Mot clés : variété kählérienne, métrique extrémale, stabilité
Keywords: Kähler manifold, extremal metric, stability
@incollection{SB_2004-2005__47__1_0,
     author = {Biquard, Olivier},
     title = {M\'etriques k\"ahl\'eriennes \`a courbure scalaire constante : unicit\'e, stabilit\'e},
     booktitle = {S\'eminaire Bourbaki : volume 2004/2005, expos\'es 938-951},
     series = {Ast\'erisque},
     note = {talk:938},
     pages = {1--31},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {307},
     year = {2006},
     mrnumber = {2296414},
     zbl = {1142.32010},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2004-2005__47__1_0/}
}
TY  - CHAP
AU  - Biquard, Olivier
TI  - Métriques kählériennes à courbure scalaire constante : unicité, stabilité
BT  - Séminaire Bourbaki : volume 2004/2005, exposés 938-951
AU  - Collectif
T3  - Astérisque
N1  - talk:938
PY  - 2006
SP  - 1
EP  - 31
IS  - 307
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/SB_2004-2005__47__1_0/
LA  - fr
ID  - SB_2004-2005__47__1_0
ER  - 
%0 Book Section
%A Biquard, Olivier
%T Métriques kählériennes à courbure scalaire constante : unicité, stabilité
%B Séminaire Bourbaki : volume 2004/2005, exposés 938-951
%A Collectif
%S Astérisque
%Z talk:938
%D 2006
%P 1-31
%N 307
%I Société mathématique de France
%U http://www.numdam.org/item/SB_2004-2005__47__1_0/
%G fr
%F SB_2004-2005__47__1_0
Biquard, Olivier. Métriques kählériennes à courbure scalaire constante : unicité, stabilité, dans Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 938, pp. 1-31. http://www.numdam.org/item/SB_2004-2005__47__1_0/

[1] M. Abreu - “Kähler geometry of toric varieties and extremal metrics”, Internat. J. Math. 9 (1998), no. 6, p. 641-651. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[4] A. L. Besse - Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10, Springer-Verlag, Berlin, 1987. | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

[7] D. Burns & P. De Bartolomeis - “Stability of vector bundles and extremal metrics”, Invent. Math. 92 (1988), no. 2, p. 403-407. | EuDML | MR | Zbl

[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. | MR | Zbl

[9] -, “Extremal Kähler metrics. II”, in Differential geometry and complex analysis, Springer, Berlin, 1985, p. 95-114. | Zbl

[10] E. Calabi & X. X. Chen - “The space of Kähler metrics. II”, J. Differential Geom. 61 (2002), no. 2, p. 173-193. | MR | Zbl

[11] D. M. J. Calderbank & M. A. Singer - “Einstein metrics and complex singularities”, Invent. Math. 156 (2004), no. 2, p. 405-443. | MR | Zbl

[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. | MR | Zbl

[13] X. X. Chen - “The space of Kähler metrics”, J. Differential Geom. 56 (2000), no. 2, p. 189-234. | MR | Zbl

[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. | MR | Zbl

[16] S. K. Donaldson - “Infinite determinants, stable bundles and curvature”, Duke Math. J. 54 (1987), no. 1, p. 231-247. | MR | Zbl

[17] -, “Remarks on gauge theory, complex geometry and 4-manifold topology”, in Fields Medallists' lectures, World Sci. Ser. 20th Century Math., vol. 5, World Sci. Publishing, River Edge, NJ, 1997, p. 384-403. | MR

[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. | Zbl

[19] -, “Scalar curvature and projective embeddings. I”, J. Differential Geom. 59 (2001), no. 3, p. 479-522. | MR | Zbl

[20] -, “Holomorphic discs and the complex Monge-Ampère equation”, J. Symplectic Geom. 1 (2002), no. 2, p. 171-196. | MR | Zbl

[21] -, “Scalar curvature and stability of toric varieties”, J. Differential Geom. 62 (2002), no. 2, p. 289-349. | MR | Zbl

[22] -, “Interior estimates for solutions of Abreu's equation”, Collect. Math. 56 (2005), no. 2, p. 103-142. | EuDML | MR | Zbl

[23] -, “Scalar curvature and projective embeddings, II”, Q. J. Math. 56 (2005), no. 3, p. 345-356. | MR | Zbl

[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. | MR | Zbl

[25] A. Futaki - “An obstruction to the existence of Einstein Kähler metrics”, Invent. Math. 73 (1983), no. 3, p. 437-443. | EuDML | MR | Zbl

[26] P. Gauduchon - Calabi's extremal Kähler metrics : an elementary introduction. | Zbl

[27] D. Gieseker - “Global moduli for surfaces of general type”, Invent. Math. 43 (1977), no. 3, p. 233-282. | EuDML | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[30] J. Kim, C. Lebrun & M. Pontecorvo - “Scalar-flat Kähler surfaces of all genera”, J. Reine Angew. Math. 486 (1997), p. 69-95. | EuDML | MR | Zbl

[31] C. Lebrun - “Scalar-flat Kähler metrics on blown-up ruled surfaces.”, J. Reine Angew. Math. 420 (1991), p. 161-177. | EuDML | MR | Zbl

[32] -, “Polarized 4-manifolds, extremal Kähler metrics, and Seiberg-Witten theory”, Math. Res. Lett. 2 (1995), no. 5, p. 653-662. | MR | Zbl

[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. | EuDML | MR | Zbl

[34] A. Lichnerowicz - “Sur les transformations analytiques des variétés kählériennes compactes”, C. R. Acad. Sci. Paris 244 (1957), p. 3011-3013. | MR | Zbl

[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. | MR | Zbl

[36] H. Luo - “Geometric criterion for Gieseker-Mumford stability of polarized manifolds”, J. Differential Geom. 49 (1998), no. 3, p. 577-599. | MR | Zbl

[37] T. Mabuchi - K-energy maps integrating Futaki invariants”, Tohoku Math. J. (2) 38 (1986), no. 4, p. 575-593. | MR | Zbl

[38] -, “Some symplectic geometry on compact Kähler manifolds. I”, Osaka J. Math. 24 (1987), no. 2, p. 227-252. | MR | Zbl

[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. | MR | Zbl

[41] -, “An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, I”, Invent. Math. 159 (2005), no. 2, p. 225-243. | MR | Zbl

[42] -, “An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, II”, J. Differential Geom. (à paraître). | Zbl

[43] Y. Matsushima - “Sur la structure du groupe d'homéomorphismes analytiques d'une certaine variété kählérienne”, Nagoya Math. J. 11 (1957), p. 145-150. | DOI | MR | Zbl

[44] D. Mumford - “Stability of projective varieties”, Enseignement Math. (2) 23 (1977), no. 1-2, p. 39-110. | MR | Zbl

[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. | DOI | MR | Zbl

[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. | MR | Zbl

[48] -, “Scalar curvature, moment maps, and the Deligne pairing”, Amer. J. Math. 126 (2004), no. 3, p. 693-712. | MR | Zbl

[49] Y. Rollin & M. Singer - “Non-minimal scalar-flat Kähler surfaces and parabolic stability”, Invent. Math. 162 (2005), no. 2, p. 235-270. | MR | Zbl

[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. | MR | Zbl

[52] -, “A study of the Hilbert-Mumford criterion for the stability of projective varieties”, arXiv : math.AG/0412519, . | Zbl

[53] W.-D. Ruan - “Canonical coordinates and Bergmann metrics”, Comm. Anal. Geom. 6 (1998), no. 3, p. 589-631. | MR | Zbl

[54] S. Semmes - “Complex Monge-Ampère and symplectic manifolds”, Amer. J. Math. 114 (1992), no. 3, p. 495-550. | MR | Zbl

[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. | MR | Zbl

[56] G. Tian - “On Kähler-Einstein metrics on certain Kähler manifolds with C 1 (M)>0, Invent. Math. 89 (1987), no. 2, p. 225-246. | EuDML | MR | Zbl

[57] -, “On a set of polarized Kähler metrics on algebraic manifolds”, J. Differential Geom. 32 (1990), no. 1, p. 99-130. | MR | Zbl

[58] -, “On Calabi's conjecture for complex surfaces with positive first Chern class”, Invent. Math. 101 (1990), no. 1, p. 101-172. | EuDML | MR | Zbl

[59] -, “The K-energy on hypersurfaces and stability”, Comm. Anal. Geom. 2 (1994), no. 2, p. 239-265. | MR | Zbl

[60] -, “Kähler-Einstein metrics with positive scalar curvature”, Invent. Math. 130 (1997), no. 1, p. 1-37. | MR | Zbl

[61] -, “Bott-Chern forms and geometric stability”, Discrete Contin. Dynam. Systems 6 (2000), no. 1, p. 211-220. | MR | Zbl

[62] -, Canonical metrics in Kähler geometry, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2000, Notes taken by Meike Akveld. | MR | Zbl

[63] -, “Extremal metrics and geometric stability”, Houston J. Math. 28 (2002), no. 2, p. 411-432, Special issue for S. S. Chern. | MR | Zbl

[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). | MR | Zbl

[65] E. Viehweg - Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 30, Springer-Verlag, Berlin, 1995. | DOI | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[69] S. Zelditch - “Szegö kernels and a theorem of Tian”, Internat. Math. Res. Notices (1998), no. 6, p. 317-331. | MR | Zbl

[70] S. Zhang - “Heights and reductions of semi-stable varieties”, Compositio Math. 104 (1996), no. 1, p. 77-105. | EuDML | Numdam | MR | Zbl