×

zbMATH — the first resource for mathematics

On de Rham and Dolbeault cohomology of solvmanifolds. (English) Zbl 1352.22007
For a simply connected non-nilpotent solvable Lie group \(G\) with a lattice \(\Gamma\), the de Rham and Dolbeault cohomologies of \(G/\Gamma\) are not in general isomorphic to the cohomologies of its Lie algebra \(\mathfrak g\). This article constructs, up to a finite group, a new Lie algebra \(\widetilde{\mathfrak g}\) whose cohomology is isomorphic to the de Rham cohomology of \(G/\Gamma\). It also gives a Dolbeault version of such technique when \(G/\Gamma\) is complex.

MSC:
22E25 Nilpotent and solvable Lie groups
14F40 de Rham cohomology and algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Auslander, L, An exposition of the structure of solvmanidolds I and II, Bull. Am. Math. Soc., 79, 227-261, (1973) · Zbl 0265.22016
[2] Auslander, L; Tolimieri, R, Splitting theorems and the structure of solvmanifolds, Ann. Math., 92, 164-173, (1970) · Zbl 0229.22020
[3] Barberis, ML; Dotti, IG; Miatello, RJ, On certain locally homogeneous Clifford manifolds, Ann. Glob. Anal. Geom., 13, 513-518, (1995) · Zbl 0832.53039
[4] Belgun, F, On the metric structure of non-Kähler complex surfaces, Math. Ann., 317, 1-40, (2000) · Zbl 0988.32017
[5] A. Borel, Linear Algebraic Groups, 2nd ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991. · Zbl 0726.20030
[6] Console, S; Fino, A, Dolbeault cohomology of compact nilmanifolds, Transform. Groups, 6, 111-124, (2001) · Zbl 1028.58024
[7] S. Console, A. Fino, On the de Rham cohomology of solvmanifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), no. 4, 801-818. · Zbl 1242.53055
[8] Console, S; Fino, A; Poon, YS, Stability of abelian complex structures, Internat. J. Math., 17, 401-416, (2006) · Zbl 1096.32009
[9] Console, S; Ovando, GP; Subils, M, Solvable models for Kodaira surfaces, Mediterr. J. Math., 12, 187-204, (2015) · Zbl 1316.53082
[10] Cordero, L; Fernandez, M; Gray, A; Ugarte, L, Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology, Trans. Amer. Math. Soc., 352, 5405-5433, (2000) · Zbl 0965.32026
[11] Bartolomeis, P; Tomassini, A, On solvable generalized Calabi-Yau manifolds, Ann. Inst. Fourier, 56, 1281-1296, (2006) · Zbl 1127.53065
[12] Dekimpe, K, Solvable Lie algebras, Lie groups and polynomial structures, Compositio Math., 121, 183-204, (2000) · Zbl 1002.22003
[13] Dekimpe, K, Semi-simple splitting for solvable Lie groups and polynomial structures, Forum Math., 12, 77-96, (2000) · Zbl 0946.22009
[14] N. Dungey, A. F. M. ter Elst, D. W. Robinson, Analysis on Lie Groups with Polynomial Growth, Progress in Mathematics, Vol. 214, Birkäuser Boston, Boston, MA, 2003. · Zbl 1041.43003
[15] H. R. Fischer, F. L. Williams, The Borel spectral sequence: some remarks and applications, in: Differential Geometry, Calculus of Variations, and Their Applications, Lecture Notes in Pure and Appl. Math., Vol. 100, Dekker, New York, 1985, pp. 255-266. · Zbl 0965.32026
[16] Guan, D, Modification and the cohomology groups of compact solvmanifolds, Electron. Res. Announc. Amer. Math. Soc., 13, 74-81, (2007) · Zbl 1134.53024
[17] Hattori, A, Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I, 8, 289-331, (1960) · Zbl 0099.18003
[18] Kasuya, H, Techniques of computations of Dolbeault cohomology, Math. Z., 273, 437-447, (2013) · Zbl 1261.22009
[19] Kasuya, H, Minimal models, formality and hard Lefschetz properties of solvmanifolds with local systems, J. Diff. Geom., 93, 269-298, (2013) · Zbl 1373.53069
[20] H. Kasuya, de Rham and Dolbeault Cohomology of solvmanifolds with local systems , Math. Res. Lett. 21 (2014), no. 4, 781-805. · Zbl 1314.17010
[21] A. И. Maльцeв, Oб oднoм клacce oднopoдныx пpocтpaнcтв, Изв. AH CCCCP. Cep. мaт. 13 (1949), no. 1, 9-32. Engl. transl.: A. I. Malcev, On a class of homogeneous spaces, Amer. Math. Soc. Translation 1951 (1951), no. 39, 33 pp. · Zbl 0361.22005
[22] G. Mostow, Cohomology of topological groups and solvmanifolds, Ann. of Math. (2) 73 (1961), 20-48. · Zbl 0103.26501
[23] K. Nomizu, On the cohomology of homogeneous spaces of nilpotent Lie Groups, Ann. of Math. (2) 59 (1954), 531-538. · Zbl 0058.02202
[24] M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer-Verlag, Berlin, 1972. Russian transl.: M. Paгунaтaн, Диcкpeтныe пoдгpуппыe гpупп Ли, Mиp, M., 1977. · Zbl 0254.22005
[25] S. Rollenske, Geometry of nilmanifolds with left-invariant complex structure and deformations in the large, Proc. Lond. Math. Soc. (3) 99 (2009), no. 2, 425-460. · Zbl 1175.32006
[26] Sakane, Y, On compact complex parallelisable solvmanifolds, Osaka J. Math., 13, 187-212, (1976) · Zbl 0361.22005
[27] N. Steenrod, The Topology of Fibre Bundles, Princeton University Press, 1951. Russian transl.: H. Cтинpoд, Toпoлoгия кocыx, пpoизвeдeний, ИЛ, M., 1953. · Zbl 0054.07103
[28] Yamada, T, A pseudo-Kähler structure on a nontoral compact complex parallelizable solvmanifold, Geom. Dedicata, 112, 115-122, (2005) · Zbl 1083.53034
[29] Witte, D, Zero-entropy affine maps on homogeneous spaces, Am. J. Math., 109, 927-961, (1987) · Zbl 0653.22005
[30] Witte, D, Superrigidity of lattices in solvable Lie groups, Invent. Math., 122, 147-193, (1995) · Zbl 0844.22015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.