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Cohomology of Nilmanifolds

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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.

Keywords

  • Nilmanifolds
  • Bott-Chern Cohomology
  • Iwasawa Manifold
  • Solvmanifolds
  • Aeppli Cohomology

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Fig. 3.1
Fig. 3.2
Fig. 3.3
Fig. 3.4
Fig. 3.5

Notes

  1. 1.

    We recall that, on six-dimensional nilmanifolds endowed with left-invariant complex structures, the existence of pluriclosed metrics and the existence of balanced metrics are complementary properties: indeed, by [FPS04, Proposition 1.4], see also [AI01, Remark 1], any Hermitian metric being both pluriclosed and balanced is in fact Kähler, and by [FPS04, Theorem 1.2] the pluriclosed property is satisfied by either all left-invariant Hermitian metrics or by none.

  2. 2.

    Recall that the nilradical of a solvable Lie algebra is the maximal nilpotent ideal in \(\mathfrak{g}\).

  3. 3.

    Recall that, given a vector space V, any linear map \(A \in \mathfrak{g}\mathfrak{l}(V )\) admits a unique Jordan decomposition \(A = A_{\mathrm{s}} + A_{\mathrm{n}}\), where \(A_{\mathrm{s}} \in \mathfrak{g}\mathfrak{l}(V )\) is semi-simple (that is, each A s-invariant sub-space of V admits an A s-invariant complementary sub-space in V ), and \(A_{\mathrm{n}} \in \mathfrak{g}\mathfrak{l}(V )\) is nilpotent (that is, there exists \(N \in \mathbb{N}\) such that \(A_{\mathrm{n}}^{N} = 0\)), and A s and A n commute, see, e.g., [DtER03, II.1.10], [Kir08, Theorem 5.59].

  4. 4.

    Note that the case \(b \in \left (2\,\mathbb{Z}\right )\cdot \pi\) and \(d \in \left (2\,\mathbb{Z}\right )\cdot \pi\) can be identified with the case (i) of the completely-solvable Nakamura manifold, see [Yam05, Sect. 3].

References

  1. L. Alessandrini, G. Bassanelli, Small deformations of a class of compact non-Kähler manifolds. Proc. Am. Math. Soc. 109(4), 1059–1062 (1990)

    MathSciNet  MATH  Google Scholar 

  2. 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. D. Angella, M.G. Franzini, F.A. Rossi, Degree of non-kählerianity for 6-dimensional nilmanifolds, arXiv:1210.0406 [math.DG], 2012

    Google Scholar 

  4. 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. B. Alexandrov, S. Ivanov, Vanishing theorems on Hermitian manifolds. Differ. Geom. Appl. 14(3), 251–265 (2001)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. D. Angella, H. Kasuya, Bott-chern cohomology of solvmanifolds, arXiv: 1212.5708v3 [math.DG], 2012

    Google Scholar 

  7. D. Angella, H. Kasuya, Cohomologies of deformations of solvmanifolds and closedness of some properties, arXiv:1305.6709v1 [math.CV], 2013

    Google Scholar 

  8. D. Angella, H. Kasuya, Symplectic Bott-Chern cohomology of solvmanifolds. arXiv:1308.4258v1 [math.SG] (2013)

    Google Scholar 

  9. D. Angella, The cohomologies of the Iwasawa manifold and of its small deformations. J. Geom. Anal. 23(3), 1355–1378 (2013)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. D. Angella, A. Tomassini, On cohomological decomposition of almost-complex manifolds and deformations. J. Symplectic Geom. 9(3), 403–428 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. G. Bassanelli, Area-minimizing Riemann surfaces on the Iwasawa manifold. J. Geom. Anal. 9(2), 179–201 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. 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)

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. F.A. Belgun, On the metric structure of non-Kähler complex surfaces. Math. Ann. 317(1), 1–40 (2000)

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. J.-M. Bismut, A local index theorem for non-Kähler manifolds. Math. Ann. 284(4), 681–699 (1989)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. C. Bock, On low-dimensional solvmanifolds, arXiv:0903.2926v4 [math.DG], 2009

    Google Scholar 

  16. S. Console, A. Fino, Dolbeault cohomology of compact nilmanifolds. Transform. Groups 6(2), 111–124 (2001)

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. 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. 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)

    CrossRef  MATH  Google Scholar 

  19. S. Console, A.M. Fino, H. Kasuya, Modifications and cohomologies of solvmanifolds, arXiv:1301.6042v1 [math.DG], 2013

    Google Scholar 

  20. S. Console, M. Macrì, Lattices, cohomology and models of six dimensional almost abelian solvmanifolds, arXiv:1206.5977v1 [math.DG], 2012

    Google Scholar 

  21. S. Console, Dolbeault cohomology and deformations of nilmanifolds. Rev. Un. Mat. Argentina 47(1), 51–60 (2006)

    MathSciNet  MATH  Google Scholar 

  22. M. Ceballos, A. Otal, L. Ugarte, R. Villacampa, Classification of complex structures on 6-dimensional nilpotent Lie algebras, arXiv:1111.5873v3 [math.DG], 2011

    Google Scholar 

  23. 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. P. de Bartolomeis, A. Tomassini, On solvable generalized Calabi-Yau manifolds. Ann. Inst. Fourier (Grenoble) 56 (5), 1281–1296 (2006)

    Google Scholar 

  25. K. Dekimpe, Semi-simple splittings for solvable Lie groups and polynomial structures. Forum Math. 12(1), 77–96 (2000)

    MathSciNet  MATH  Google Scholar 

  26. P. Deligne, Ph.A. Griffiths, J. Morgan, D.P. Sullivan, Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. S.G. Dani, M.G. Mainkar, Anosov automorphisms on compact nilmanifolds associated with graphs. Trans. Am. Math. Soc. 357(6), 2235–2251 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. 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. M.G. Eastwood, M.A. Singer, The Fröhlicher spectral sequence on a twistor space. J. Differ. Geom. 38(3), 653–669 (1993)

    MathSciNet  MATH  Google Scholar 

  30. M. Fernández, A. Gray, The Iwasawa manifold, in Differential Geometry, Peñíscola 1985. Lecture Notes in Mathematics, vol. 1209 (Springer, Berlin, 1986), pp. 157–159

    Google Scholar 

  31. A. Fino, G. Grantcharov, Properties of manifolds with skew-symmetric torsion and special holonomy. Adv. Math. 189(2), 439–450 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  32. M. Fernández, V. Muñoz, J.A. Santisteban, Cohomologically Kähler manifolds with no Kähler metrics. Int. J. Math. Math. Sci. 2003(52), 3315–3325 (2003)

    CrossRef  MATH  Google Scholar 

  33. A. Fino, M. Parton, S. Salamon, Families of strong KT structures in six dimensions. Comment. Math. Helv. 79(2), 317–340 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. M.G. Franzini, Deformazioni di varietà bilanciate e loro proprietà coomologiche. Tesi di Laurea Magistrale in Matematica, Università degli Studi di Parma, 2011

    Google Scholar 

  35. D. Guan, Modification and the cohomology groups of compact solvmanifolds. Electron. Res. Announc. Am. Math. Soc. 13, 74–81 (2007)

    CrossRef  MATH  Google Scholar 

  36. K. Hasegawa, A note on compact solvmanifolds with Kähler structures. Osaka J. Math. 43(1), 131–135 (2006)

    MathSciNet  MATH  Google Scholar 

  37. K. Hasegawa, Small deformations and non-left-invariant complex structures on six-dimensional compact solvmanifolds. Differ. Geom. Appl. 28(2), 220–227 (2010)

    CrossRef  MATH  Google Scholar 

  38. A. Hattori, Spectral sequence in the de Rham cohomology of fibre bundles. J. Fac. Sci. Univ. Tokyo Sect. I 8(2), 289–331 (1960)

    MathSciNet  MATH  Google Scholar 

  39. 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 edition

    Google Scholar 

  40. S. Iitaka, Genus and classification of algebraic varieties. I. Sûgaku 24(1), 14–27 (1972)

    MathSciNet  Google Scholar 

  41. H. Kasuya, Hodge symmetry and decomposition on non-Kähler solvmanifolds, arXiv:1109.5929v4 [math.DG], 2011

    Google Scholar 

  42. H. Kasuya, De Rham and Dolbeault cohomology of solvmanifolds with local systems, arXiv:1207.3988v3 [math.DG], 2012

    Google Scholar 

  43. H. Kasuya, Degenerations of the Frölicher spectral sequences of solvmanifolds, arXiv:1210.2661v2 [math.DG], 2012

    Google Scholar 

  44. H. Kasuya, Differential Gerstenhaber algebras and generalized deformations of solvmanifolds, arXiv:1211.4188v2 [math.DG], 2012

    Google Scholar 

  45. 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. 21–34

    Google Scholar 

  46. H. Kasuya, Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems. J. Differ. Geom. 93 (2), 269–297 (2013)

    MathSciNet  MATH  Google Scholar 

  47. H. Kasuya, Techniques of computations of Dolbeault cohomology of solvmanifolds. Math. Z. 273(1–2), 437–447 (2013)

    CrossRef  MathSciNet  MATH  Google Scholar 

  48. 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. 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. G. Ketsetzis, S. Salamon, Complex structures on the Iwasawa manifold. Adv. Geom. 4(2), 165–179 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  51. 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. 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. A. Latorre, L. Ugarte, R. Villacampa, On the Bott-Chern cohomology and balanced Hermitian nilmanifolds, arXiv:1210.0395v1 [math.DG], 2012

    Google Scholar 

  54. M. Macrì, Cohomological properties of unimodular six dimensional solvable Lie algebras. Differ. Geom. Appl. 31(1), 112–129 (2013)

    CrossRef  MATH  Google Scholar 

  55. Y. Matsushima, On the discrete subgroups and homogeneous spaces of nilpotent Lie groups. Nagoya Math. J. 2, 95–110 (1951)

    MathSciNet  MATH  Google Scholar 

  56. 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. M.L. Michelsohn, On the existence of special metrics in complex geometry. Acta Math. 149(3–4), 261–295 (1982)

    CrossRef  MathSciNet  MATH  Google Scholar 

  58. J. Milnor, Curvatures of left invariant metrics on Lie groups. Adv. Math. 21(3), 293–329 (1976)

    CrossRef  MathSciNet  MATH  Google Scholar 

  59. J. Morrow, K. Kodaira, Complex Manifolds (AMS Chelsea Publishing, Providence, 2006). Reprint of the 1971 edition with errata

    Google Scholar 

  60. G.D. Mostow, Factor spaces of solvable groups. Ann. Math. (2) 60(1), 1–27 (1954)

    Google Scholar 

  61. I. Nakamura, Complex parallelisable manifolds and their small deformations. J. Differ. Geom. 10, 85–112 (1975)

    MATH  Google Scholar 

  62. A. Newlander, L. Nirenberg, Complex analytic coordinates in almost complex manifolds. Ann. Math. (2) 65(3), 391–404 (1957)

    Google Scholar 

  63. K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. Math. (2) 59, 531–538 (1954)

    Google Scholar 

  64. H. Pouseele, P. Tirao, Compact symplectic nilmanifolds associated with graphs. J. Pure Appl. Algebra 213(9), 1788–1794 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  65. M.S. Raghunathan, Discrete Subgroups of Lie Groups (Springer, New York, 1972). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68

    Google Scholar 

  66. 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. 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. S. Rollenske, Lie-algebra Dolbeault-cohomology and small deformations of nilmanifolds. J. Lond. Math. Soc. (2) 79(2), 346–362 (2009)

    Google Scholar 

  69. 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. 369–392

    Google Scholar 

  70. S. Rollenske, The Kuranishi space of complex parallelisable nilmanifolds. J. Eur. Math. Soc. (JEMS) 13(3), 513–531 (2011)

    Google Scholar 

  71. Y. Sakane, On compact complex parallelisable solvmanifolds. Osaka J. Math. 13(1), 187–212 (1976)

    MathSciNet  MATH  Google Scholar 

  72. M. Schweitzer, Autour de la cohomologie de Bott-Chern, arXiv:0709.3528 [math.AG], 2007

    Google Scholar 

  73. J.-P. Serre, Homologie singulière des espaces fibrés. Applications. Ann. Math. (2) 54, 425–505 (1951)

    Google Scholar 

  74. A. Tralle, J. Oprea, Symplectic Manifolds with No Kähler Structure. Lecture Notes in Mathematics, vol. 1661 (Springer, Berlin, 1997)

    Google Scholar 

  75. L.-S. Tseng, S.-T. Yau, Generalized cohomologies and supersymmetry, arXiv:1111.6968v1 [hep-th], 2011

    Google Scholar 

  76. 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. L. Ugarte, Hermitian structures on six-dimensional nilmanifolds. Transform. Groups 12(1), 175–202 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  78. L. Ugarte, R. Villacampa, Non-nilpotent complex geometry of nilmanifolds and heterotic supersymmetry, arXiv:0912.5110v2 [math.DG], 2009

    Google Scholar 

  79. H.-C. Wang, Complex parallisable manifolds. Proc. Am. Math. Soc. 5(5), 771–776 (1954)

    CrossRef  MATH  Google Scholar 

  80. 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 edition

    Google Scholar 

  81. T. Yamada, A pseudo-Kähler structure on a nontoral compact complex parallelizable solvmanifold. Geom. Dedicata 112, 115–122 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  82. X. Ye, The jumping phenomenon of Hodge numbers. Pac. J. Math. 235(2), 379–398 (2008)

    CrossRef  MATH  Google Scholar 

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Angella, D. (2014). Cohomology of Nilmanifolds. In: Cohomological Aspects in Complex Non-Kähler Geometry. Lecture Notes in Mathematics, vol 2095. Springer, Cham. https://doi.org/10.1007/978-3-319-02441-7_3

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