Extremal Kähler metrics on ruled manifolds and stability

Apostolov, Vestislav ; Calderbank, David M. J. ; Gauduchon, Paul ; Tønnesen-Friedman, Christina W.

Géométrie différentielle, physique mathématique, mathématiques et société (II) - Volume en l'honneur de Jean-Pierre Bourguignon, Astérisque, no. 322 (2008), pp. 93-150

MR Zbl

@incollection{AST_2008__322__93_0,
     author = {Apostolov, Vestislav and Calderbank, David M. J. and Gauduchon, Paul and T{\o}nnesen-Friedman, Christina W.},
     title = {Extremal {K\"ahler} metrics on ruled manifolds and stability},
     booktitle = {G\'eom\'etrie diff\'erentielle, physique math\'ematique, math\'ematiques et soci\'et\'e (II) - Volume en l'honneur de Jean-Pierre Bourguignon},
     editor = {Hijazi Oussama},
     series = {Ast\'erisque},
     pages = {93--150},
     year = {2008},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {322},
     mrnumber = {2521655},
     zbl = {1204.53028},
     language = {en},
     url = {https://www.numdam.org/item/AST_2008__322__93_0/}
}

TY  - CHAP
AU  - Apostolov, Vestislav
AU  - Calderbank, David M. J.
AU  - Gauduchon, Paul
AU  - Tønnesen-Friedman, Christina W.
TI  - Extremal Kähler metrics on ruled manifolds and stability
BT  - Géométrie différentielle, physique mathématique, mathématiques et société (II) - Volume en l'honneur de Jean-Pierre Bourguignon
AU  - Collectif
ED  - Hijazi Oussama
T3  - Astérisque
PY  - 2008
SP  - 93
EP  - 150
IS  - 322
PB  - Société mathématique de France
UR  - https://www.numdam.org/item/AST_2008__322__93_0/
LA  - en
ID  - AST_2008__322__93_0
ER  -

%0 Book Section
%A Apostolov, Vestislav
%A Calderbank, David M. J.
%A Gauduchon, Paul
%A Tønnesen-Friedman, Christina W.
%T Extremal Kähler metrics on ruled manifolds and stability
%B Géométrie différentielle, physique mathématique, mathématiques et société (II) - Volume en l'honneur de Jean-Pierre Bourguignon
%A Collectif
%E Hijazi Oussama
%S Astérisque
%D 2008
%P 93-150
%N 322
%I Société mathématique de France
%U https://www.numdam.org/item/AST_2008__322__93_0/
%G en
%F AST_2008__322__93_0

Apostolov, Vestislav; Calderbank, David M. J.; Gauduchon, Paul; Tønnesen-Friedman, Christina W. Extremal Kähler metrics on ruled manifolds and stability, dans Géométrie différentielle, physique mathématique, mathématiques et société (II) - Volume en l'honneur de Jean-Pierre Bourguignon, Astérisque, no. 322 (2008), pp. 93-150. https://www.numdam.org/item/AST_2008__322__93_0/

Bibliographie
Cité par

[1] V. Apostolov, D. M. J. Calderbank & P. Gauduchon - "Hamiltonian $2$ -forms in Kähler geometry. I. General theory", J. Differential Geom. 73 (2006), p. 359-412. | MR | Zbl | DOI

[2] V. Apostolov, D. M. J. Calderbank, P. Gauduchon & C. W. Tønnesen-Friedman - "Hamiltonian $2$ -forms in Kähler geometry. II. Global classification", J. Differential Geom. 68 (2004), p. 277-345. | MR | Zbl | DOI

[3] V. Apostolov, D. M. J. Calderbank, P. Gauduchon & C. W. Tønnesen-Friedman, "Hamiltonian $2$ -forms in Kähler geometry. III. Extremal metrics and stability", Invent. Math. 173 (2008), p. 547-601. | MR | Zbl | DOI

[4] V. Apostolov, D. M. J. Calderbank, P. Gauduchon & C. W. Tønnesen-Friedman, "Hamiltonian $2$ -forms in Kähler geometry. IV. Weakly Bochner-flat Kähler manifolds", Comm. Anal. Geom. 16 (2008), p. 91-126. | MR | Zbl | DOI

[5] V. Apostolov & C. W. Tønnesen-Friedman - "A remark on Kähler metrics of constant scalar curvature on ruled complex surfaces", Bull. London Math. Soc. 38 (2006), p. 494-500. | MR | Zbl | DOI

[6] N. Bourbaki - Éléments de Mathématique, Algèbre, Chapitre 7, Masson, Paris, 1981. | Zbl

[7] J. P. Bourguignon - "Invariants intégraux fonctionnels pour des équations aux dérivées partielles d'origine géométrique", in Partial differential equations, Part 1, 2 (Warsaw, 1990), Banach Center Publ., 27, Part 1, vol. 2, Polish Acad. Sci., 1992, p. 65-73. | MR | EuDML | Zbl

[8] E. Calabi - "Extremal Kähler metrics", in Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, 1982, p. 259-290. | MR | Zbl

[9] E. Calabi, "Extremal Kähler metrics. II", in Differential geometry and complex analysis, Springer, 1985, p. 95-114. | MR | Zbl

[10] X. Chen - "The space of Kähler metrics", J. Differential Geom. 56 (2000), p. 189-234. | MR | Zbl | DOI

[11] X. Chen & G. Tian - "Partial regularity for homogeneous complex Monge-Ampere equations", C. R. Math. Acad. Sci. Paris 340 (2005), p. 337-340. | MR | Zbl | DOI

[12] X. Chen & G. Tian, "Uniqueness of extremal Kähler metrics", C. R. Math. Acad. Sci. Paris 340 (2005), p. 287-290. | MR | Zbl | DOI

[13] X. Chen & G. Tian, "Geometry of Kähler metrics and foliations by holomorphic discs", Publ. Math. Inst. Hautes Études Sci. 107 (2008), p. 1-107. | MR | Zbl | EuDML | Numdam | DOI

[14] S. K. Donaldson - "Symmetric spaces, Kähler geometry and Hamiltonian dynamics", in Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transi. Ser. 2, vol. 196, Amer. Math. Soc., 1999, p. 13-33. | MR | Zbl

[15] S. K. Donaldson, "Scalar curvature and stability of toric varieties", J. Differential Geom. 62 (2002), p. 289-349. | MR | Zbl | DOI

[16] W. Fulton - Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 2, Springer, 1998. | Zbl

[17] A. Futaki - "An obstruction to the existence of Einstein Kähler metrics", Invent. Math. 73 (1983), p. 437-443. | MR | Zbl | EuDML | DOI

[18] A. Futaki & T. Mabuchi - "Bilinear forms and extremal Kähler vector fields associated with Kähler classes", Math. Ann. 301 (1995), p. 199-210. | MR | Zbl | EuDML | DOI

[19] P. Gauduchon - "Calabi extremal Kähler metrics: An elementary introduction", in preparation.

[20] P. Griffiths & J. Harris - Principles of algebraic geometry, Wiley-Interscience, John Wiley & Sons, 1978, Pure and Applied Mathematics. | MR | Zbl

[21] D. Guan - "Existence of extremal metrics on compact almost homogeneous Kähler manifolds with two ends", Trans. Amer. Math. Soc. 347 (1995), p. 2255-2262. | MR | Zbl

[22] D. Guan, "On modified Mabuchi functional and Mabuchi moduli space of Kähler metrics on toric bundles", Math. Res. Lett. 6 (1999), p. 547-555. | MR | Zbl | DOI

[23] F. Hirzebruch - "Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten", Math. Ann. 124 (1951), p. 77-86. | MR | Zbl | EuDML | DOI

[24] F. Hirzebruch, Topological methods in algebraic geometry, Classics in Mathematics, Springer, 1995. | MR | Zbl

[25] A. D. Hwang - "On existence of Kähler metrics with constant scalar curvature", Osaka J. Math. 31 (1994), p. 561-595. | MR | Zbl

[26] A. D. Hwang & M. A. Singer - "A momentum construction for circle-invariant Kähler metrics", Trans. Amer. Math. Soc. 354 (2002), p. 2285-2325. | MR | Zbl | DOI

[27] S. Kobayashi - Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 70, Springer, 1972. | MR | Zbl

[28] S. Lang - Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. | MR | Zbl

[29] C. R. Lebrun & S. R. Simanca - "On the Kähler classes of extremal metrics", in Geometry and Global Analysis, First MSJ, Intern. Res. Inst. Sendai, Japan, Eds. Kotake, Nishikawa and Schoen, 1993. | MR | Zbl

[30] C. R. Lebrun & S. R. Simanca, "Extremal Kähler metrics and complex deformation theory", Geom. Funct. Anal. 4 (1994), p. 298-336. | MR | Zbl | EuDML | DOI

[31] L. Lempert - "Symmetries and other transformations of the complex Monge-Ampère equation", Duke Math. J. 52 (1985), p. 869-885. | MR | Zbl | DOI

[32] T. Mabuchi - "Some symplectic geometry on compact Kähler manifolds. I", Osaka J. Math. 24 (1987), p. 227-252. | MR | Zbl

[33] R. D. Macpherson - "Chern classes for singular algebraic varieties", Ann. of Math. 100 (1974), p. 423-432. | MR | Zbl | DOI

[34] J. Ross & R. Thomas - "An obstruction to the existence of constant scalar curvature Kähler metrics", J. Differential Geom. 72 (2006), p. 429-466. | MR | Zbl | DOI

[35] J. Ross & R. Thomas, "A study of the Hilbert-Mumford criterion for the stability of projective varieties", J. Algebraic Geom. 16 (2007), p. 201-255. | MR | Zbl | DOI

[36] S. Semmes - "Complex Monge-Ampère and symplectic manifolds", Amer. J. Math. 114 (1992), p. 495-550. | MR | Zbl | DOI

[37] S. R. Simanca - "A note on extremal metrics of nonconstant scalar curvature", Israel J. Math. 78 (1992), p. 85-93. | MR | Zbl | DOI

[38] S. R. Simanca, "A $K$ -energy characterization of extremal Kähler metrics", Proc. Amer. Math. Soc. 128 (2000), p. 1531-1535. | MR | Zbl | DOI

[39] G. Székelyhidi - "Extremal metrics and $K$ -stability", Bull. Lond. Math. Soc. 39 (2007), p. 76-84. | MR | Zbl | DOI

[40] G. Székelyhidi, "The Calabi functional on a ruled surface", preprint arXiv:math.DG/0703562. | MR | Zbl | Numdam | DOI

[41] G. Tian - "Kähler-Einstein metrics with positive scalar curvature", Invent. Math. 130 (1997), p. 1-37. | MR | Zbl | DOI

[42] C. W. Tønnesen-Friedman - "Extremal Kähler metrics on minimal ruled surfaces", J. reine angew. Math. 502 (1998), p. 175-197. | MR | Zbl