On the de Rham cohomology of solvmanifolds
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 4, pp. 801-818.

Using results by D. Witte [35] on the superigidity of lattices in solvable Lie groups we get a new proof of a recent remarkable result obtained by D. Guan [15] on the de Rham cohomology of a compact solvmanifold, i.e., of a quotient of a connected and simply connected solvable Lie group $G$ by a lattice $\Gamma$. This result can be applied to compute the Betti numbers of a compact solvmanifold $G/\Gamma$ even in the case that the solvable Lie group $G$ and the lattice $\Gamma$ do not satisfy the Mostow condition.

Publié le :
Classification : 53C30,  22E25,  22E40
@article{ASNSP_2011_5_10_4_801_0,
author = {Console, Sergio and Fino, Anna},
title = {On the de Rham cohomology of solvmanifolds},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {801--818},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 10},
number = {4},
year = {2011},
zbl = {1242.53055},
mrnumber = {2932894},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2011_5_10_4_801_0/}
}
Console, Sergio; Fino, Anna. On the de Rham cohomology of solvmanifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 4, pp. 801-818. http://www.numdam.org/item/ASNSP_2011_5_10_4_801_0/

[1] L. Auslander, An exposition of the structure of solvmanidolds I and II, Bull. Amer. Math. Soc. 79 (1973), 227–261. | MR 486307 | Zbl 0265.22016

[2] L. Auslander and J. Brezin, Almost algebraic Lie algebras, J. Algebra 8 (1968), 295–313. | MR 224745 | Zbl 0197.03002

[3] L. Auslander and R. Tolimieri, Splitting theorems and the structure of solvmanifolds, Ann. of Math. (2) 92 (1970), 164–173. | MR 276995 | Zbl 0229.22020

[4] O. Baues and V. Cortes, Aspherical Kähler manifolds with solvable fundamental group, Geom. Dedicata 122 (2006), 215–229. | MR 2295551 | Zbl 1128.53043

[5] C. Bock, On low-dimensional solvmanifolds, preprint arXiv:0903.2926 (2009). | MR 3480018

[6] A. Borel, “Linear Algebraic Groups”, Second edition, Graduate Texts in Mathematics, Vol. 126, Springer, New York, 1991. | MR 1102012 | Zbl 0726.20030

[7] S. Console and A. Fino, Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001), 111–124. | MR 1835667 | Zbl 1028.58024

[8] A. Cordero, M. Fernández, A. Gray and L. Ugarte, Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology, Trans. Amer. Math. Soc. 352 (2000), 5405–5433. | MR 1665327 | Zbl 0965.32026

[9] P. de Bartolomeis and A. Tomassini, On solvable generalized Calabi-Yau manifolds, Ann. Inst. Fourier (Grenoble) 56 (2006), 1281–1296. | EuDML 10177 | Numdam | MR 2273857 | Zbl 1127.53065

[10] K. Dekimpe, Semi-simple splitting for solvable Lie groups and polynomial structures, Forum Math. 12 (2000), 77–96. | MR 1736093 | Zbl 0946.22009

[11] V.V. Gorbatsevich, Splittings of Lie groups and their application to the study of homogeneous spaces, Math. USSR-Izv. 15 (1980), 441–467. | Zbl 0453.22006

[12] V.V. Gorbatsevich, Plesicompact homogeneous spaces, Siber. Math. J. 30 (1989), 217–226. | MR 997468 | Zbl 0705.22005

[13] V.V. Gorbatsevich, Symplectic structures and cohomologies on some solvmanifolds, Siber. Math. J. 44 (2003), 260–274. | EuDML 51132 | MR 1981370 | Zbl 1054.53084

[14] W. Greub, S. Halperin and R. Vanstone, “Connections, Curvature and Cohomology”, Vol. I, Academic Press, New York and London, 1973. | MR 336651 | Zbl 0322.58001

[15] D. Guan, Modification and the cohomology groups of compact solvmanifolds, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 74–81. | MR 2358304 | Zbl 1134.53024

[16] D. Guan, Classification of compact complex homogeneous manifolds with pseudo-Khlerian structures, J. Algebra 324 (2010), 2010–2024. | MR 2678834 | Zbl 1205.53055

[17] D. Guan, Classification of compact homogeneous manifolds with pseudo-Kählerian structures, C. R. Math. Acad. Sci. Soc. R. Can. 31 (2009), 20–23. | MR 2519018 | Zbl 1177.53028

[18] K. Hasegawa, Complex and Kähler structures on compact solvmanifolds, J. Symplectic Geom. 3 (2005), 749–767. | MR 2235860 | Zbl 1120.53043

[19] A. Hattori, Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 289–331. | MR 124918 | Zbl 0099.18003

[20] A. Malcev, On solvable Lie algebras, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 9 (1945), 329–356. | MR 22217

[21] Y. I. Merzilyakov, “Rational Groups”, Nauka, Moscow, 1987. | MR 895823 | Zbl 0647.20042

[22] J. Milnor, Curvature of left invariant metrics on Lie groups, Adv. Math. 21 (1976), 293–329. | MR 425012 | Zbl 0341.53030

[23] M. V. Milovanov, Description of solvable Lie groups with a given uniform subgroup, Mat. Sb. (N.S.) 113(155) (1980), 98–117, 175. | EuDML 71216 | MR 590540 | Zbl 0496.22013

[24] D. V. Millionschikov, Multivalued functionals, one-forms and deformed de Rham complex, e-print math.AT/0512572 (2005).

[25] G. Mostow, Factor spaces of solvable spaces, Ann. of Math. (2) 60 (1954), 1–27. | MR 61611 | Zbl 0057.26103

[26] G. Mostow, Cohomology of topological groups and solvmanifolds, Ann. of Math. (2) 73 (1961), 20–48. | MR 125179 | Zbl 0103.26501

[27] K. Nomizu, On the cohomology of homogeneous spaces of nilpotent Lie Groups, Ann. of Math. (2) 59 (1954), 531–538. | MR 64057 | Zbl 0058.02202

[28] A. L. Onishchik and E. B. Vinberg, “Lie Groups and Lie Algebras II. Discrete Subgroups of Lie Groups and Cohomologies of Lie Groups and Lie Algebras”, Encyclopaedia of Mathematical Sciences, Vol. 21, Springer-Verlag, Berlin, 2000. | MR 1756407 | Zbl 0932.00011

[29] J. Oprea and A. Tralle, “Symplectic Manifolds with no Kähler Structure”, Lecture Notes in Mathematics, Vol. 1661, Springer, Berlin, 1997. | MR 1465676 | Zbl 0891.53001

[30] M. S. Raghunathan, “Discrete subgroups of Lie groups”, Springer, Berlin, 1972. | MR 507234 | Zbl 0254.22005

[31] S. Rollenske, Lie-algebra Dolbeault-cohomology and small deformations of nilmanifolds, J. London Math. Soc. (2) 79 (2009), 346–362. | MR 2496518 | Zbl 1194.32006

[32] A. N. Starkov, “Algebraic Groups and Homogeneous Spaces of Finite Volume”, Translated from the 1999 Russian original by the author, Translations of Mathematical Monographs, Vol. 190, American Mathematical Society, Providence, RI, 2000. | MR 1746847 | Zbl 1143.37300

[33] T. Yamada, A pseudo-Kähler structure on a nontoral compact complex parallelizable solvmanifold, Geom. Dedicata 112 (2005), 115–122. | MR 2163892 | Zbl 1083.53034

[34] D. Witte, Zero-entropy affine maps on homogeneous spaces, Amer. J. Math. 109 (1987), 927–961. | MR 910358 | Zbl 0653.22005

[35] D. Witte, Superrigidity of lattices in solvable Lie groups, Invent. Math. 122 (1995), 147–193. | EuDML 144318 | MR 1354957 | Zbl 0844.22015