no. 6
Stability under deformations of Hermite-Einstein almost Kähler metrics
[Stabilité sous déformations des métriques presque-kählériennes de Hermite-Einstein]
Lejmi, Mehdi1
1 Université Libre de Bruxelles CP218 Département de Mathématiques Boulevard du Triomphe Bruxelles 1050, (Belgique)
Annales de l'Institut Fourier, Tome 64 (2014) no. 6, pp. 2251-2263.

Sur une variété symplectique compacte de dimension 4, nous considérons une famille lisse de structures presque-complexes compatibles tel qu’en temps zéro, la métrique induite est presque-kählérienne de Hermite-Einstein avec une courbure scalaire hermitienne nulle ou négative. Nous prouvons, sous une certaine hypothèse, l’existence d’une famille lisse de structures presque-complexes, difféomorphe à chaque temps à la structure initiale et induisant une métrique à courbure scalaire hermitienne constante.

On a 4-dimensional compact symplectic manifold, we consider a smooth family of compatible almost-complex structures such that at time zero the induced metric is Hermite-Einstein almost-Kähler metric with zero or negative Hermitian scalar curvature. We prove, under certain hypothesis, the existence of a smooth family of compatible almost-complex structures, diffeomorphic at each time to the initial one, and inducing constant Hermitian scalar curvature metrics.

DOI : 10.5802/aif.2911
Classification : 53C55, 53C15, 53D20
Keywords: Almost-Kähler geometry, extremal almost-Kähler metrics, constant Hermitian scalar curvature almost-Kähler metrics
Mot clés : Géométrie presque-kählérienne, métrique presque-kählériennes extrémales, métriques presque-kählériennes à courbure scalaire hermitienne constante

Lejmi, Mehdi 1

1 Université Libre de Bruxelles CP218 Département de Mathématiques Boulevard du Triomphe Bruxelles 1050, (Belgique) 
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Lejmi, Mehdi. Stability under deformations of Hermite-Einstein almost Kähler metrics. Annales de l'Institut Fourier, Tome 64 (2014) no. 6, pp. 2251-2263. doi : 10.5802/aif.2911. https://aif.centre-mersenne.org/articles/10.5802/aif.2911/

[1] Apostolov, Vestislav; Calderbank, David M. J.; Gauduchon, Paul; Tønnesen-Friedman, Christina W. Extremal Kähler metrics on projective bundles over a curve, Adv. Math., Volume 227 (2011) no. 6, pp. 2385-2424 | DOI | MR | Zbl

[2] Apostolov, Vestislav; Drăghici, Tedi The curvature and the integrability of almost-Kähler manifolds: a survey, Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001) (Fields Inst. Commun.), Volume 35, Amer. Math. Soc., Providence, RI, 2003, pp. 25-53 | MR | Zbl

[3] Calabi, Eugenio Extremal Kähler metrics, Seminar on Differential Geometry (Ann. of Math. Stud.), Volume 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 259-290 | MR | Zbl

[4] Deligne, Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis Real homotopy theory of Kähler manifolds, Invent. Math., Volume 29 (1975) no. 3, pp. 245-274 | DOI | MR | Zbl

[5] Donaldson, S. K. Remarks on gauge theory, complex geometry and 4-manifold topology, Fields Medallists’ lectures (World Sci. Ser. 20th Century Math.), Volume 5, World Sci. Publ., River Edge, NJ, 1997, pp. 384-403 | DOI | MR

[6] Drăghici, Tedi Lecture notes and private communications

[7] Drăghici, Tedi; Li, Tian-Jun; Zhang, Weiyi Symplectic forms and cohomology decomposition of almost complex four-manifolds, Int. Math. Res. Not. IMRN (2010) no. 1, pp. 1-17 | DOI | MR | Zbl

[8] Fujiki, Akira Moduli space of polarized algebraic manifolds and Kähler metrics [translation of Sûgaku 42 (1990), no. 3, 231–243; MR1073369 (92b:32032)], Sugaku Expositions, Volume 5 (1992) no. 2, pp. 173-191 (Sugaku Expositions) | MR | Zbl

[9] Fujiki, Akira; Schumacher, Georg The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics, Publ. Res. Inst. Math. Sci., Volume 26 (1990) no. 1, pp. 101-183 | DOI | MR | Zbl

[10] Gauduchon, Paul Calabi’s extremal Kähler metrics: An elementary introduction (In preparation)

[11] Gauduchon, Paul Hermitian connections and Dirac operators, Boll. Un. Mat. Ital. B (7), Volume 11 (1997) no. 2, suppl., pp. 257-288 | MR | Zbl

[12] Kodaira, Kunihiko Complex manifolds and deformation of complex structures, Classics in Mathematics, Springer-Verlag, Berlin, 2005, pp. x+465 (Translated from the 1981 Japanese original by Kazuo Akao) | MR | Zbl

[13] LeBrun, Claude; Simanca, Santiago R. On the Kähler classes of extremal metrics, Geometry and global analysis (Sendai, 1993), Tohoku Univ., Sendai, 1993, pp. 255-271 | MR | Zbl

[14] LeBrun, Claude; Simanca, Santiago R. Extremal Kähler metrics and complex deformation theory, Geom. Funct. Anal., Volume 4 (1994) no. 3, pp. 298-336 | DOI | MR | Zbl

[15] LeBrun, Claude; Simanca, Santiago R. On Kähler surfaces of constant positive scalar curvature, J. Geom. Anal., Volume 5 (1995) no. 1, pp. 115-127 | DOI | MR | Zbl

[16] Lejmi, Mehdi Extremal almost-Kähler metrics, Internat. J. Math. (2010) no. 12, pp. 1639-1662 | DOI | MR | Zbl

[17] Lejmi, Mehdi Stability under deformations of extremal almost-Kähler metrics in dimension 4, Math. Res. Lett., Volume 17 (2010) no. 4, pp. 601-612 | DOI | MR | Zbl

[18] Li, Tian-Jun Symplectic Calabi-Yau surfaces, Handbook of geometric analysis, No. 3 (Adv. Lect. Math. (ALM)), Volume 14, Int. Press, Somerville, MA, 2010, pp. 231-356 | MR | Zbl

[19] Li, Tian-Jun; Tomassini, Adriano Almost Kähler structures on four dimensional unimodular Lie algebras, J. Geom. Phys., Volume 62 (2012) no. 7, pp. 1714-1731 | DOI | MR | Zbl

[20] Libermann, Paulette Sur les connexions hermitiennes, C. R. Acad. Sci. Paris, Volume 239 (1954), pp. 1579-1581 | MR | Zbl

[21] Merkulov, S. A. Formality of canonical symplectic complexes and Frobenius manifolds, Internat. Math. Res. Notices (1998) no. 14, pp. 727-733 | DOI | MR | Zbl

[22] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34, Springer-Verlag, Berlin, 1994, pp. xiv+292 | DOI | MR | Zbl

[23] Rollin, Yann; Simanca, Santiago R.; Tipler, Carl Deformation of extremal metrics, complex manifolds and the relative Futaki invariant, Math. Z., Volume 273 (2013) no. 1-2, pp. 547-568 | DOI | MR | Zbl

[24] Rollin, Yann; Tipler, Carl Deformations of extremal toric manifolds (preprint 2013, math.DG/1201.4137) | MR

[25] Székelyhidi, Gábor The Kähler-Ricci flow and K-polystability, Amer. J. Math., Volume 132 (2010) no. 4, pp. 1077-1090 | DOI | MR | Zbl

[26] Tan, Q.; Wang, H.; Zhang, Y.; Zhu, P. Symplectic cohomology and the stability of J-anti-invariant cohomology (preprint 2013, math.DG/1307.1513)

[27] Tian, G. K-stability and Kähler-Einstein metrics (preprint 2013, math.DG/1211.4669)

[28] Vezzoni, Luigi A note on canonical Ricci forms on 2-step nilmanifolds, Proc. Amer. Math. Soc., Volume 141 (2013) no. 1, pp. 325-333 | DOI | MR | Zbl

[29] Weinkove, Ben The Calabi-Yau equation on almost-Kähler four-manifolds, J. Differential Geom., Volume 76 (2007) no. 2, pp. 317-349 http://projecteuclid.org/euclid.jdg/1180135681 | MR | Zbl

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