Stability under deformations of Hermite-Einstein almost Kähler metrics
Annales de l'Institut Fourier, Volume 64 (2014) no. 6, p. 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 : https://doi.org/10.5802/aif.2911
Classification:  53C55,  53C15,  53D20
Keywords: Almost-Kähler geometry, extremal almost-Kähler metrics, constant Hermitian scalar curvature almost-Kähler metrics
@article{AIF_2014__64_6_2251_0,
     author = {Lejmi, Mehdi},
     title = {Stability under deformations of Hermite-Einstein almost K\"ahler metrics},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {6},
     year = {2014},
     pages = {2251-2263},
     doi = {10.5802/aif.2911},
     mrnumber = {3331165},
     zbl = {06387338},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2014__64_6_2251_0}
}
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. http://www.numdam.org/item/AIF_2014__64_6_2251_0/

[1] Apostolov, Vestislav; Calderbank, David M. J.; Gauduchon, Paul; Tønnesen-Friedman, Christina W. Extremal Kähler metrics on projective bundles over a curve, Adv. Math., Tome 227 (2011) no. 6, pp. 2385-2424 | Article | MR 2807093 | Zbl 1232.32011

[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), Amer. Math. Soc., Providence, RI (Fields Inst. Commun.) Tome 35 (2003), pp. 25-53 | MR 1969266 | Zbl 1050.53031

[3] Calabi, Eugenio Extremal Kähler metrics, Seminar on Differential Geometry, Princeton Univ. Press, Princeton, N.J. (Ann. of Math. Stud.) Tome 102 (1982), pp. 259-290 | MR 645743 | Zbl 0574.58006

[4] Deligne, Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis Real homotopy theory of Kähler manifolds, Invent. Math., Tome 29 (1975) no. 3, pp. 245-274 | Article | MR 382702 | Zbl 0312.55011

[5] Donaldson, S. K. Remarks on gauge theory, complex geometry and 4-manifold topology, Fields Medallists’ lectures, World Sci. Publ., River Edge, NJ (World Sci. Ser. 20th Century Math.) Tome 5 (1997), pp. 384-403 | Article | MR 1622931

[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 | Article | MR 2576281 | Zbl 1190.32021

[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, Tome 5 (1992) no. 2, pp. 173-191 (Sugaku Expositions) | MR 1073369 | Zbl 0796.32009

[9] Fujiki, Akira; Schumacher, Georg The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics, Publ. Res. Inst. Math. Sci., Tome 26 (1990) no. 1, pp. 101-183 | Article | MR 1053910 | Zbl 0714.32007

[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), Tome 11 (1997) no. 2, suppl., pp. 257-288 | MR 1456265 | Zbl 0876.53015

[12] Kodaira, Kunihiko Complex manifolds and deformation of complex structures, Springer-Verlag, Berlin, Classics in Mathematics (2005), pp. x+465 (Translated from the 1981 Japanese original by Kazuo Akao) | MR 2109686 | Zbl 1058.32007

[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 1361191 | Zbl 0921.53032

[14] Lebrun, Claude; Simanca, Santiago R. Extremal Kähler metrics and complex deformation theory, Geom. Funct. Anal., Tome 4 (1994) no. 3, pp. 298-336 | Article | MR 1274118 | Zbl 0801.53050

[15] Lebrun, Claude; Simanca, Santiago R. On Kähler surfaces of constant positive scalar curvature, J. Geom. Anal., Tome 5 (1995) no. 1, pp. 115-127 | Article | MR 1315659 | Zbl 0815.53075

[16] Lejmi, Mehdi Extremal almost-Kähler metrics, Internat. J. Math. (2010) no. 12, pp. 1639-1662 | Article | MR 2747965 | Zbl 1253.53067

[17] Lejmi, Mehdi Stability under deformations of extremal almost-Kähler metrics in dimension 4, Math. Res. Lett., Tome 17 (2010) no. 4, pp. 601-612 | Article | MR 2661166 | Zbl 1223.53057

[18] Li, Tian-Jun Symplectic Calabi-Yau surfaces, Handbook of geometric analysis, No. 3, Int. Press, Somerville, MA (Adv. Lect. Math. (ALM)) Tome 14 (2010), pp. 231-356 | MR 2743450 | Zbl 1216.32016

[19] Li, Tian-Jun; Tomassini, Adriano Almost Kähler structures on four dimensional unimodular Lie algebras, J. Geom. Phys., Tome 62 (2012) no. 7, pp. 1714-1731 | Article | MR 2922031 | Zbl 1257.53104

[20] Libermann, Paulette Sur les connexions hermitiennes, C. R. Acad. Sci. Paris, Tome 239 (1954), pp. 1579-1581 | MR 66733 | Zbl 0057.14203

[21] Merkulov, S. A. Formality of canonical symplectic complexes and Frobenius manifolds, Internat. Math. Res. Notices (1998) no. 14, pp. 727-733 | Article | MR 1637093 | Zbl 0931.58002

[22] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], Tome 34 (1994), pp. xiv+292 | Article | MR 1304906 | Zbl 0797.14004

[23] Rollin, Yann; Simanca, Santiago R.; Tipler, Carl Deformation of extremal metrics, complex manifolds and the relative Futaki invariant, Math. Z., Tome 273 (2013) no. 1-2, pp. 547-568 | Article | MR 3010175 | Zbl 1271.32026

[24] Rollin, Yann; Tipler, Carl Deformations of extremal toric manifolds (preprint 2013, math.DG/1201.4137) | MR 3261726

[25] Székelyhidi, Gábor The Kähler-Ricci flow and K-polystability, Amer. J. Math., Tome 132 (2010) no. 4, pp. 1077-1090 | Article | MR 2663648 | Zbl 1206.53075

[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., Tome 141 (2013) no. 1, pp. 325-333 | Article | MR 2988734 | Zbl 1272.53020

[29] Weinkove, Ben The Calabi-Yau equation on almost-Kähler four-manifolds, J. Differential Geom., Tome 76 (2007) no. 2, pp. 317-349 http://projecteuclid.org/euclid.jdg/1180135681 | MR 2330417 | Zbl 1123.32015