Advertisement

Cohomology of Nilmanifolds

  • Daniele Angella
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2095)

Abstract

Nilmanifolds and solvmanifolds appear as “toy-examples” in non-Kähler geometry: indeed, on the one hand, non-tori nilmanifolds admit no Kähler structure, (Benson and Gordon, Topology 27(4):513–518, 1988; Lupton and Oprea, J. Pure Appl. Algebra 91(1–3):193–207, 1994), and, more in general, solvmanifolds admitting a Kähler structure are characterized, (Hasegawa, Proc. Am. Math. Soc. 106(1):65–71, 1989); on the other hand, the geometry and cohomology of solvmanifolds can be often reduced to study left-invariant geometry.In Sect. 3.1, it is shown that, for certain classes of complex structures on nilmanifolds (that is, compact quotients of connected simply-connected nilpotent Lie groups by co-compact discrete subgroups), the de Rham, Dolbeault, Bott-Chern, and Aeppli cohomologies are completely determined by the associated Lie algebra endowed with the induced linear complex structure, Theorem 3.6, giving a sort of result à la Nomizu for the Bott-Chern cohomology. This will allow us to explicitly study the Bott-Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations, in Sect. 3.2, and of the complex structures on six-dimensional nilmanifolds in M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa’s classification, (Ceballos et al., Classification of complex structures on 6-dimensional nilpotent Lie algebras, arXiv:1111.5873v3 [math.DG], 2011), in Sect. 3.3. Finally, in Appendix: Cohomology of Solvmanifolds, we recall some facts concerning cohomologies of solvmanifolds.

References

  1. [AB90]
    L. Alessandrini, G. Bassanelli, Small deformations of a class of compact non-Kähler manifolds. Proc. Am. Math. Soc. 109(4), 1059–1062 (1990)MathSciNetzbMATHGoogle Scholar
  2. [ABDM11]
    A. Andrada, M.L. Barberis, I.G. Dotti Miatello, Classification of abelian complex structures on 6-dimensional Lie algebras. J. Lond. Math. Soc. (2) 83(1), 232–255 (2011)Google Scholar
  3. [AFR12]
    D. Angella, M.G. Franzini, F.A. Rossi, Degree of non-kählerianity for 6-dimensional nilmanifolds, arXiv:1210.0406 [math.DG], 2012Google Scholar
  4. [AGS97]
    E. Abbena, S. Garbiero, S. Salamon, Hermitian geometry on the Iwasawa manifold. Boll. Un. Mat. Ital. B (7) 11(2, Suppl.), 231–249 (1997)Google Scholar
  5. [AI01]
    B. Alexandrov, S. Ivanov, Vanishing theorems on Hermitian manifolds. Differ. Geom. Appl. 14(3), 251–265 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [AK12]
    D. Angella, H. Kasuya, Bott-chern cohomology of solvmanifolds, arXiv: 1212.5708v3 [math.DG], 2012Google Scholar
  7. [AK13a]
    D. Angella, H. Kasuya, Cohomologies of deformations of solvmanifolds and closedness of some properties, arXiv:1305.6709v1 [math.CV], 2013Google Scholar
  8. [AK13b]
    D. Angella, H. Kasuya, Symplectic Bott-Chern cohomology of solvmanifolds. arXiv:1308.4258v1 [math.SG] (2013)Google Scholar
  9. [Ang11]
    D. Angella, The cohomologies of the Iwasawa manifold and of its small deformations. J. Geom. Anal. 23(3), 1355–1378 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [AT11]
    D. Angella, A. Tomassini, On cohomological decomposition of almost-complex manifolds and deformations. J. Symplectic Geom. 9(3), 403–428 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Bas99]
    G. Bassanelli, Area-minimizing Riemann surfaces on the Iwasawa manifold. J. Geom. Anal. 9(2), 179–201 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [BDMM95]
    M.L. Barberis, I.G. Dotti Miatello, R.J. Miatello, On certain locally homogeneous Clifford manifolds. Ann. Global Anal. Geom. 13(3), 289–301 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [Bel00]
    F.A. Belgun, On the metric structure of non-Kähler complex surfaces. Math. Ann. 317(1), 1–40 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Bis89]
    J.-M. Bismut, A local index theorem for non-Kähler manifolds. Math. Ann. 284(4), 681–699 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [Boc09]
    C. Bock, On low-dimensional solvmanifolds, arXiv:0903.2926v4 [math.DG], 2009Google Scholar
  16. [CF01]
    S. Console, A. Fino, Dolbeault cohomology of compact nilmanifolds. Transform. Groups 6(2), 111–124 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [CF11]
    S. Console, A. Fino, On the de Rham cohomology of solvmanifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) X(4), 801–818 (2011)Google Scholar
  18. [CFGU00]
    L.A. Cordero, M. Fernández, A. Gray, L. Ugarte, Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology. Trans. Am. Math. Soc. 352(12), 5405–5433 (2000)CrossRefzbMATHGoogle Scholar
  19. [CFK13]
    S. Console, A.M. Fino, H. Kasuya, Modifications and cohomologies of solvmanifolds, arXiv:1301.6042v1 [math.DG], 2013Google Scholar
  20. [CM12]
    S. Console, M. Macrì, Lattices, cohomology and models of six dimensional almost abelian solvmanifolds, arXiv:1206.5977v1 [math.DG], 2012Google Scholar
  21. [Con06]
    S. Console, Dolbeault cohomology and deformations of nilmanifolds. Rev. Un. Mat. Argentina 47(1), 51–60 (2006)MathSciNetzbMATHGoogle Scholar
  22. [COUV11]
    M. Ceballos, A. Otal, L. Ugarte, R. Villacampa, Classification of complex structures on 6-dimensional nilpotent Lie algebras, arXiv:1111.5873v3 [math.DG], 2011Google Scholar
  23. [dAFdLM92]
    L.C. de Andrés, M. Fernández, M. de León, J.J. Mencía, Some six-dimensional compact symplectic and complex solvmanifolds. Rend. Mat. Appl. (7) 12(1), 59–67 (1992)Google Scholar
  24. [dBT06]
    P. de Bartolomeis, A. Tomassini, On solvable generalized Calabi-Yau manifolds. Ann. Inst. Fourier (Grenoble) 56 (5), 1281–1296 (2006)Google Scholar
  25. [Dek00]
    K. Dekimpe, Semi-simple splittings for solvable Lie groups and polynomial structures. Forum Math. 12(1), 77–96 (2000)MathSciNetzbMATHGoogle Scholar
  26. [DGMS75]
    P. Deligne, Ph.A. Griffiths, J. Morgan, D.P. Sullivan, Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [DM05]
    S.G. Dani, M.G. Mainkar, Anosov automorphisms on compact nilmanifolds associated with graphs. Trans. Am. Math. Soc. 357(6), 2235–2251 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [DtER03]
    N. Dungey, A.F.M. ter Elst, D.W. Robinson, Analysis on Lie Groups with Polynomial Growth. Progress in Mathematics, vol. 214 (Birkhäuser Boston, Boston, 2003)Google Scholar
  29. [ES93]
    M.G. Eastwood, M.A. Singer, The Fröhlicher spectral sequence on a twistor space. J. Differ. Geom. 38(3), 653–669 (1993)MathSciNetzbMATHGoogle Scholar
  30. [FG86]
    M. Fernández, A. Gray, The Iwasawa manifold, in Differential Geometry, Peñíscola 1985. Lecture Notes in Mathematics, vol. 1209 (Springer, Berlin, 1986), pp. 157–159Google Scholar
  31. [FG04]
    A. Fino, G. Grantcharov, Properties of manifolds with skew-symmetric torsion and special holonomy. Adv. Math. 189(2), 439–450 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. [FMS03]
    M. Fernández, V. Muñoz, J.A. Santisteban, Cohomologically Kähler manifolds with no Kähler metrics. Int. J. Math. Math. Sci. 2003(52), 3315–3325 (2003)CrossRefzbMATHGoogle Scholar
  33. [FPS04]
    A. Fino, M. Parton, S. Salamon, Families of strong KT structures in six dimensions. Comment. Math. Helv. 79(2), 317–340 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [Fra11]
    M.G. Franzini, Deformazioni di varietà bilanciate e loro proprietà coomologiche. Tesi di Laurea Magistrale in Matematica, Università degli Studi di Parma, 2011Google Scholar
  35. [Gua07]
    D. Guan, Modification and the cohomology groups of compact solvmanifolds. Electron. Res. Announc. Am. Math. Soc. 13, 74–81 (2007)CrossRefzbMATHGoogle Scholar
  36. [Has06]
    K. Hasegawa, A note on compact solvmanifolds with Kähler structures. Osaka J. Math. 43(1), 131–135 (2006)MathSciNetzbMATHGoogle Scholar
  37. [Has10]
    K. Hasegawa, Small deformations and non-left-invariant complex structures on six-dimensional compact solvmanifolds. Differ. Geom. Appl. 28(2), 220–227 (2010)CrossRefzbMATHGoogle Scholar
  38. [Hat60]
    A. Hattori, Spectral sequence in the de Rham cohomology of fibre bundles. J. Fac. Sci. Univ. Tokyo Sect. I 8(2), 289–331 (1960)MathSciNetzbMATHGoogle Scholar
  39. [Hir95]
    F. Hirzebruch, Topological Methods in Algebraic Geometry. Classics in Mathematics (Springer, Berlin, 1995). Translated from the German and Appendix One by R.L.E. Schwarzenberger, With a preface to the third English edition by the author and Schwarzenberger, Appendix Two by A. Borel, Reprint of the 1978 editionGoogle Scholar
  40. [Iit72]
    S. Iitaka, Genus and classification of algebraic varieties. I. Sûgaku 24(1), 14–27 (1972)MathSciNetGoogle Scholar
  41. [Kas11]
    H. Kasuya, Hodge symmetry and decomposition on non-Kähler solvmanifolds, arXiv:1109.5929v4 [math.DG], 2011Google Scholar
  42. [Kas12a]
    H. Kasuya, De Rham and Dolbeault cohomology of solvmanifolds with local systems, arXiv:1207.3988v3 [math.DG], 2012Google Scholar
  43. [Kas12b]
    H. Kasuya, Degenerations of the Frölicher spectral sequences of solvmanifolds, arXiv:1210.2661v2 [math.DG], 2012Google Scholar
  44. [Kas12c]
    H. Kasuya, Differential Gerstenhaber algebras and generalized deformations of solvmanifolds, arXiv:1211.4188v2 [math.DG], 2012Google Scholar
  45. [Kas12d]
    H. Kasuya, Geometrical formality of solvmanifolds and solvable Lie type geometries, in RIMS Kokyuroku Bessatsu B 39. Geometry of Transformation Groups and Combinatorics (2012), pp. 21–34Google Scholar
  46. [Kas13a]
    H. Kasuya, Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems. J. Differ. Geom. 93 (2), 269–297 (2013)MathSciNetzbMATHGoogle Scholar
  47. [Kas13b]
    H. Kasuya, Techniques of computations of Dolbeault cohomology of solvmanifolds. Math. Z. 273(1–2), 437–447 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  48. [Kir08]
    A. Kirillov Jr., An Introduction to Lie Groups and Lie Algebras. Cambridge Studies in Advanced Mathematics, vol. 113 (Cambridge University Press, Cambridge, 2008)Google Scholar
  49. [KS60]
    K. Kodaira, D.C. Spencer, On deformations of complex analytic structures. III. Stability theorems for complex structures. Ann. Math. (2) 71(1), 43–76 (1960)Google Scholar
  50. [KS04]
    G. Ketsetzis, S. Salamon, Complex structures on the Iwasawa manifold. Adv. Geom. 4(2), 165–179 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  51. [Ler50a]
    J. Leray, L’anneau spectral et l’anneau filtré d’homologie d’un espace localement compact et d’une application continue. J. Math. Pures Appl. (9) 29, 1–80, 81–139 (1950)Google Scholar
  52. [Ler50b]
    J. Leray, L’homologie d’un espace fibré dont la fibre est connexe. J. Math. Pures Appl. (9) 29, 169–213 (1950)Google Scholar
  53. [LUV12]
    A. Latorre, L. Ugarte, R. Villacampa, On the Bott-Chern cohomology and balanced Hermitian nilmanifolds, arXiv:1210.0395v1 [math.DG], 2012Google Scholar
  54. [Mac13]
    M. Macrì, Cohomological properties of unimodular six dimensional solvable Lie algebras. Differ. Geom. Appl. 31(1), 112–129 (2013)CrossRefzbMATHGoogle Scholar
  55. [Mat51]
    Y. Matsushima, On the discrete subgroups and homogeneous spaces of nilpotent Lie groups. Nagoya Math. J. 2, 95–110 (1951)MathSciNetzbMATHGoogle Scholar
  56. [McC01]
    J. McCleary, A User’s Guide to Spectral Sequences, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 58 (Cambridge University Press, Cambridge, 2001)Google Scholar
  57. [Mic82]
    M.L. Michelsohn, On the existence of special metrics in complex geometry. Acta Math. 149(3–4), 261–295 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  58. [Mil76]
    J. Milnor, Curvatures of left invariant metrics on Lie groups. Adv. Math. 21(3), 293–329 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  59. [MK06]
    J. Morrow, K. Kodaira, Complex Manifolds (AMS Chelsea Publishing, Providence, 2006). Reprint of the 1971 edition with errataGoogle Scholar
  60. [Mos54]
    G.D. Mostow, Factor spaces of solvable groups. Ann. Math. (2) 60(1), 1–27 (1954)Google Scholar
  61. [Nak75]
    I. Nakamura, Complex parallelisable manifolds and their small deformations. J. Differ. Geom. 10, 85–112 (1975)zbMATHGoogle Scholar
  62. [NN57]
    A. Newlander, L. Nirenberg, Complex analytic coordinates in almost complex manifolds. Ann. Math. (2) 65(3), 391–404 (1957)Google Scholar
  63. [Nom54]
    K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. Math. (2) 59, 531–538 (1954)Google Scholar
  64. [PT09]
    H. Pouseele, P. Tirao, Compact symplectic nilmanifolds associated with graphs. J. Pure Appl. Algebra 213(9), 1788–1794 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  65. [Rag72]
    M.S. Raghunathan, Discrete Subgroups of Lie Groups (Springer, New York, 1972). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68Google Scholar
  66. [Rol07]
    S. Rollenske, Nilmanifolds: complex structures, geometry and deformations, Ph.D, Thesis, Universität Bayreuth, 2007, http://opus.ub.uni-bayreuth.de/opus4-ubbayreuth/frontdoor/index/index/docId/280,
  67. [Rol09a]
    S. Rollenske, Geometry of nilmanifolds with left-invariant complex structure and deformations in the large. Proc. Lond. Math. Soc. (3) 99(2), 425–460 (2009)Google Scholar
  68. [Rol09b]
    S. Rollenske, Lie-algebra Dolbeault-cohomology and small deformations of nilmanifolds. J. Lond. Math. Soc. (2) 79(2), 346–362 (2009)Google Scholar
  69. [Rol11a]
    S. Rollenske, Dolbeault cohomology of nilmanifolds with left-invariant complex structure, in Complex and Differential Geometry, ed. by W. Ebeling, K. Hulek, K. Smoczyk. Springer Proceedings in Mathematics, vol. 8 (Springer, Berlin, 2011), pp. 369–392Google Scholar
  70. [Rol11b]
    S. Rollenske, The Kuranishi space of complex parallelisable nilmanifolds. J. Eur. Math. Soc. (JEMS) 13(3), 513–531 (2011)Google Scholar
  71. [Sak76]
    Y. Sakane, On compact complex parallelisable solvmanifolds. Osaka J. Math. 13(1), 187–212 (1976)MathSciNetzbMATHGoogle Scholar
  72. [Sch07]
    M. Schweitzer, Autour de la cohomologie de Bott-Chern, arXiv:0709.3528 [math.AG], 2007Google Scholar
  73. [Ser51]
    J.-P. Serre, Homologie singulière des espaces fibrés. Applications. Ann. Math. (2) 54, 425–505 (1951)Google Scholar
  74. [TO97]
    A. Tralle, J. Oprea, Symplectic Manifolds with No Kähler Structure. Lecture Notes in Mathematics, vol. 1661 (Springer, Berlin, 1997)Google Scholar
  75. [TY11]
    L.-S. Tseng, S.-T. Yau, Generalized cohomologies and supersymmetry, arXiv:1111.6968v1 [hep-th], 2011Google Scholar
  76. [Uen75]
    K. Ueno, Classification Theory of Algebraic Varieties and Compact Complex Spaces. Lecture Notes in Mathematics, vol. 439 (Springer, Berlin, 1975). Notes written in collaboration with P. Cherenack.Google Scholar
  77. [Uga07]
    L. Ugarte, Hermitian structures on six-dimensional nilmanifolds. Transform. Groups 12(1), 175–202 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  78. [UV09]
    L. Ugarte, R. Villacampa, Non-nilpotent complex geometry of nilmanifolds and heterotic supersymmetry, arXiv:0912.5110v2 [math.DG], 2009Google Scholar
  79. [Wan54]
    H.-C. Wang, Complex parallisable manifolds. Proc. Am. Math. Soc. 5(5), 771–776 (1954)CrossRefzbMATHGoogle Scholar
  80. [War83]
    F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, vol. 94 (Springer, New York, 1983). Corrected reprint of the 1971 editionGoogle Scholar
  81. [Yam05]
    T. Yamada, A pseudo-Kähler structure on a nontoral compact complex parallelizable solvmanifold. Geom. Dedicata 112, 115–122 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  82. [Ye08]
    X. Ye, The jumping phenomenon of Hodge numbers. Pac. J. Math. 235(2), 379–398 (2008)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Daniele Angella
    • 1
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

Personalised recommendations