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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Notes
- 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.
Recall that the nilradical of a solvable Lie algebra is the maximal nilpotent ideal in \(\mathfrak{g}\).
- 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.
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
L. Alessandrini, G. Bassanelli, Small deformations of a class of compact non-Kähler manifolds. Proc. Am. Math. Soc. 109(4), 1059–1062 (1990)
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)
D. Angella, M.G. Franzini, F.A. Rossi, Degree of non-kählerianity for 6-dimensional nilmanifolds, arXiv:1210.0406 [math.DG], 2012
E. Abbena, S. Garbiero, S. Salamon, Hermitian geometry on the Iwasawa manifold. Boll. Un. Mat. Ital. B (7) 11(2, Suppl.), 231–249 (1997)
B. Alexandrov, S. Ivanov, Vanishing theorems on Hermitian manifolds. Differ. Geom. Appl. 14(3), 251–265 (2001)
D. Angella, H. Kasuya, Bott-chern cohomology of solvmanifolds, arXiv: 1212.5708v3 [math.DG], 2012
D. Angella, H. Kasuya, Cohomologies of deformations of solvmanifolds and closedness of some properties, arXiv:1305.6709v1 [math.CV], 2013
D. Angella, H. Kasuya, Symplectic Bott-Chern cohomology of solvmanifolds. arXiv:1308.4258v1 [math.SG] (2013)
D. Angella, The cohomologies of the Iwasawa manifold and of its small deformations. J. Geom. Anal. 23(3), 1355–1378 (2013)
D. Angella, A. Tomassini, On cohomological decomposition of almost-complex manifolds and deformations. J. Symplectic Geom. 9(3), 403–428 (2011)
G. Bassanelli, Area-minimizing Riemann surfaces on the Iwasawa manifold. J. Geom. Anal. 9(2), 179–201 (1999)
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)
F.A. Belgun, On the metric structure of non-Kähler complex surfaces. Math. Ann. 317(1), 1–40 (2000)
J.-M. Bismut, A local index theorem for non-Kähler manifolds. Math. Ann. 284(4), 681–699 (1989)
C. Bock, On low-dimensional solvmanifolds, arXiv:0903.2926v4 [math.DG], 2009
S. Console, A. Fino, Dolbeault cohomology of compact nilmanifolds. Transform. Groups 6(2), 111–124 (2001)
S. Console, A. Fino, On the de Rham cohomology of solvmanifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) X(4), 801–818 (2011)
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)
S. Console, A.M. Fino, H. Kasuya, Modifications and cohomologies of solvmanifolds, arXiv:1301.6042v1 [math.DG], 2013
S. Console, M. Macrì, Lattices, cohomology and models of six dimensional almost abelian solvmanifolds, arXiv:1206.5977v1 [math.DG], 2012
S. Console, Dolbeault cohomology and deformations of nilmanifolds. Rev. Un. Mat. Argentina 47(1), 51–60 (2006)
M. Ceballos, A. Otal, L. Ugarte, R. Villacampa, Classification of complex structures on 6-dimensional nilpotent Lie algebras, arXiv:1111.5873v3 [math.DG], 2011
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)
P. de Bartolomeis, A. Tomassini, On solvable generalized Calabi-Yau manifolds. Ann. Inst. Fourier (Grenoble) 56 (5), 1281–1296 (2006)
K. Dekimpe, Semi-simple splittings for solvable Lie groups and polynomial structures. Forum Math. 12(1), 77–96 (2000)
P. Deligne, Ph.A. Griffiths, J. Morgan, D.P. Sullivan, Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)
S.G. Dani, M.G. Mainkar, Anosov automorphisms on compact nilmanifolds associated with graphs. Trans. Am. Math. Soc. 357(6), 2235–2251 (2005)
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)
M.G. Eastwood, M.A. Singer, The Fröhlicher spectral sequence on a twistor space. J. Differ. Geom. 38(3), 653–669 (1993)
M. Fernández, A. Gray, The Iwasawa manifold, in Differential Geometry, Peñíscola 1985. Lecture Notes in Mathematics, vol. 1209 (Springer, Berlin, 1986), pp. 157–159
A. Fino, G. Grantcharov, Properties of manifolds with skew-symmetric torsion and special holonomy. Adv. Math. 189(2), 439–450 (2004)
M. Fernández, V. Muñoz, J.A. Santisteban, Cohomologically Kähler manifolds with no Kähler metrics. Int. J. Math. Math. Sci. 2003(52), 3315–3325 (2003)
A. Fino, M. Parton, S. Salamon, Families of strong KT structures in six dimensions. Comment. Math. Helv. 79(2), 317–340 (2004)
M.G. Franzini, Deformazioni di varietà bilanciate e loro proprietà coomologiche. Tesi di Laurea Magistrale in Matematica, Università degli Studi di Parma, 2011
D. Guan, Modification and the cohomology groups of compact solvmanifolds. Electron. Res. Announc. Am. Math. Soc. 13, 74–81 (2007)
K. Hasegawa, A note on compact solvmanifolds with Kähler structures. Osaka J. Math. 43(1), 131–135 (2006)
K. Hasegawa, Small deformations and non-left-invariant complex structures on six-dimensional compact solvmanifolds. Differ. Geom. Appl. 28(2), 220–227 (2010)
A. Hattori, Spectral sequence in the de Rham cohomology of fibre bundles. J. Fac. Sci. Univ. Tokyo Sect. I 8(2), 289–331 (1960)
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
S. Iitaka, Genus and classification of algebraic varieties. I. Sûgaku 24(1), 14–27 (1972)
H. Kasuya, Hodge symmetry and decomposition on non-Kähler solvmanifolds, arXiv:1109.5929v4 [math.DG], 2011
H. Kasuya, De Rham and Dolbeault cohomology of solvmanifolds with local systems, arXiv:1207.3988v3 [math.DG], 2012
H. Kasuya, Degenerations of the Frölicher spectral sequences of solvmanifolds, arXiv:1210.2661v2 [math.DG], 2012
H. Kasuya, Differential Gerstenhaber algebras and generalized deformations of solvmanifolds, arXiv:1211.4188v2 [math.DG], 2012
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
H. Kasuya, Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems. J. Differ. Geom. 93 (2), 269–297 (2013)
H. Kasuya, Techniques of computations of Dolbeault cohomology of solvmanifolds. Math. Z. 273(1–2), 437–447 (2013)
A. Kirillov Jr., An Introduction to Lie Groups and Lie Algebras. Cambridge Studies in Advanced Mathematics, vol. 113 (Cambridge University Press, Cambridge, 2008)
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)
G. Ketsetzis, S. Salamon, Complex structures on the Iwasawa manifold. Adv. Geom. 4(2), 165–179 (2004)
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)
J. Leray, L’homologie d’un espace fibré dont la fibre est connexe. J. Math. Pures Appl. (9) 29, 169–213 (1950)
A. Latorre, L. Ugarte, R. Villacampa, On the Bott-Chern cohomology and balanced Hermitian nilmanifolds, arXiv:1210.0395v1 [math.DG], 2012
M. Macrì, Cohomological properties of unimodular six dimensional solvable Lie algebras. Differ. Geom. Appl. 31(1), 112–129 (2013)
Y. Matsushima, On the discrete subgroups and homogeneous spaces of nilpotent Lie groups. Nagoya Math. J. 2, 95–110 (1951)
J. McCleary, A User’s Guide to Spectral Sequences, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 58 (Cambridge University Press, Cambridge, 2001)
M.L. Michelsohn, On the existence of special metrics in complex geometry. Acta Math. 149(3–4), 261–295 (1982)
J. Milnor, Curvatures of left invariant metrics on Lie groups. Adv. Math. 21(3), 293–329 (1976)
J. Morrow, K. Kodaira, Complex Manifolds (AMS Chelsea Publishing, Providence, 2006). Reprint of the 1971 edition with errata
G.D. Mostow, Factor spaces of solvable groups. Ann. Math. (2) 60(1), 1–27 (1954)
I. Nakamura, Complex parallelisable manifolds and their small deformations. J. Differ. Geom. 10, 85–112 (1975)
A. Newlander, L. Nirenberg, Complex analytic coordinates in almost complex manifolds. Ann. Math. (2) 65(3), 391–404 (1957)
K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. Math. (2) 59, 531–538 (1954)
H. Pouseele, P. Tirao, Compact symplectic nilmanifolds associated with graphs. J. Pure Appl. Algebra 213(9), 1788–1794 (2009)
M.S. Raghunathan, Discrete Subgroups of Lie Groups (Springer, New York, 1972). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68
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,
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)
S. Rollenske, Lie-algebra Dolbeault-cohomology and small deformations of nilmanifolds. J. Lond. Math. Soc. (2) 79(2), 346–362 (2009)
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
S. Rollenske, The Kuranishi space of complex parallelisable nilmanifolds. J. Eur. Math. Soc. (JEMS) 13(3), 513–531 (2011)
Y. Sakane, On compact complex parallelisable solvmanifolds. Osaka J. Math. 13(1), 187–212 (1976)
M. Schweitzer, Autour de la cohomologie de Bott-Chern, arXiv:0709.3528 [math.AG], 2007
J.-P. Serre, Homologie singulière des espaces fibrés. Applications. Ann. Math. (2) 54, 425–505 (1951)
A. Tralle, J. Oprea, Symplectic Manifolds with No Kähler Structure. Lecture Notes in Mathematics, vol. 1661 (Springer, Berlin, 1997)
L.-S. Tseng, S.-T. Yau, Generalized cohomologies and supersymmetry, arXiv:1111.6968v1 [hep-th], 2011
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.
L. Ugarte, Hermitian structures on six-dimensional nilmanifolds. Transform. Groups 12(1), 175–202 (2007)
L. Ugarte, R. Villacampa, Non-nilpotent complex geometry of nilmanifolds and heterotic supersymmetry, arXiv:0912.5110v2 [math.DG], 2009
H.-C. Wang, Complex parallisable manifolds. Proc. Am. Math. Soc. 5(5), 771–776 (1954)
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
T. Yamada, A pseudo-Kähler structure on a nontoral compact complex parallelizable solvmanifold. Geom. Dedicata 112, 115–122 (2005)
X. Ye, The jumping phenomenon of Hodge numbers. Pac. J. Math. 235(2), 379–398 (2008)
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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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