Quasi-Kähler Chern-flat manifolds and complex 2-step nilpotent Lie algebras
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 1, pp. 41-60.

The study of quasi-Kähler Chern-flat almost Hermitian manifolds is strictly related to the study of anti-bi-invariant almost complex Lie algebras. In the present paper we show that quasi-Kähler Chern-flat almost Hermitian structures on compact manifolds are in correspondence to complex parallelisable Hermitian structures satisfying the second Gray identity. From an algebraic point of view this correspondence reads as a natural correspondence between anti-bi-invariant almost complex structures on Lie algebras and bi-invariant complex structures. Some natural algebraic problems are approached and some exotic examples are carefully described.

Publié le :
Classification : 53C15, 53C55
Di Scala, Antonio J. 1 ; Lauret, Jorge 2 ; Vezzoni, Luigi 3

1 Dipartimento di Scienze Matematiche Politecnico di Torino Corso Duca degli Abruzzi, 24 10129 Torino, Italia
2 FaMAF and CIEM Universidad Nacional de Cordoba Ciudad Universitaria 5000 Córdoba, Argentina
3 Dipartimento di Matematica Università di Torino Via Carlo Alberto, 10 10123 Torino, Italia
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     title = {Quasi-K\"ahler {Chern-flat} manifolds and complex $2$-step nilpotent {Lie} algebras},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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Di Scala, Antonio J.; Lauret, Jorge; Vezzoni, Luigi. Quasi-Kähler Chern-flat manifolds and complex $2$-step nilpotent Lie algebras. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 1, pp. 41-60. http://archive.numdam.org/item/ASNSP_2012_5_11_1_41_0/

[1] E. Abbena, S. Garbiero and S. Salamon, Almost Hermitian geometry on six dimensional nilmanifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 30 (2001), 147–170. | EuDML | Numdam | MR | Zbl

[2] E. Abbena and A. Grassi, Hermitian left invariant metrics on complex Lie groups and cosymplectic Hermitian manifolds, Boll. Un. Mat. Ital. A (6) 5 (1986), 371–379. | MR | Zbl

[3] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. | MR | Zbl

[4] S.-S. Chern, Characteristic classes of Hermitian manifolds, Ann. of Math. 47 (1946), 85–121. | MR | Zbl

[5] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T, Ivey, D. Knopf, P. Lu, F. Luo and L. Ni, “The Ricci Flow: Techniques and Applications, Part I: Geometric Aspects”, AMS Math. Surv. Mon., Vol. 135, 2007, Amer. Math. Soc., Providence. | MR | Zbl

[6] A. J. Di Scala and L. Vezzoni, Quasi-Kähler manifolds with trivial Chern Holonomy, Math. Z. (2008), to appear. | MR | Zbl

[7] I. Dolgachev, “Lectures on Invariant Theory”, Lect. Notes Ser. London Math. Soc., Vol. 296, 2003, Cambridge University Press. | MR | Zbl

[8] C. Ehresmann and P. Libermann, Sur les structures presque hermitiennes isotropes, C. R. Acad. Sci. Paris 232 (1951), 1281–1283. | MR | Zbl

[9] M. Gauger, On the classification of metabelian Lie algebras, Trans. Amer. Math. Soc. 179 (1973), 293–329. | MR | Zbl

[10] L. Galitski and D. Timashev, On classification of metabelian Lie algebras, J. Lie Theory 9 (1999), 125–156. | EuDML | MR | Zbl

[11] A. Gray, Curvature identities for Hermitian and almost Hermitian manifolds, Tôhoku Math. J. (2) 28 (1976), 601–612. | MR | Zbl

[12] J. Lauret, A canonical compatible metric for geometric structures on nilmanifolds, Ann. Global Anal. Geom. 30 (2006), 107–138. | MR | Zbl

[13] J. Lauret, Minimal metrics on nilmanifolds, Diff. Geom. and its Appl., Proc. Conf. Prague September 2004 (2005), 77–94 (arXiv: math.DG/0411257). | MR | Zbl

[14] J. Lauret, Rational forms of nilpotent Lie algebras, Monatsh. Math. 155 (2008), 15–30. | MR | Zbl

[15] J. Lauret, Einstein solvmanifolds and nilsolitons, Contemp. Math. 491 (2009), 1–35. | MR | Zbl

[16] C. Seeley, 7-dimensional nilpotent Lie algebras, Trans. Amer. Math. Soc., 335 (1993), 479–496. | MR | Zbl

[17] F. Tricerri and L. Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), 365–397. | MR | Zbl

[18] H.-C. Wang, Complex parallisable manifolds, Proc. Amer. Math. Soc. 5 (1954), 771–776. | MR | Zbl

[19] C. E. Will, A curve of nilpotent Lie algebras which are not Einstein nilradicals, Monatsh. Math. 159 (2010), 425–437. | MR | Zbl