Stability under deformations of Hermite-Einstein almost Kähler metrics
Annales de l'Institut Fourier, Volume 64 (2014) no. 6, pp. 2251-2263.

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.

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.

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, Volume 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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