Left-invariant almost complex structures on the higher dimensional Kodaira-Thurston manifolds. (English) Zbl 07887431
Summary: We develop computational techniques which allow us to calculate the Kodaira dimension as well as the dimension of spaces of Dolbeault harmonic forms for left-invariant almost complex structures on the generalised Kodaira-Thurston manifolds.
MSC:
| 32Q60 | Almost complex manifolds |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 58A14 | Hodge theory in global analysis |
Keywords:
almost complex manifolds; Kodaira dimension; Dolbeault harmonic forms; Kodaira-Thurston manifoldsReferences:
| [1] | Auslander L.: Lectures on Nil-theta Functions, CBMS Lectures no. 34, American Mathematical Society, Providence (1977) · Zbl 0421.22001 |
| [2] | Baldoni, M.W.: General representation theory of real reductive Lie groups. In: Proceedings of Symposia in Pure Mathematics, vol. 61, pp. 61-72 (1997) · Zbl 0895.22009 |
| [3] | Berndt, J., Tricerri, F.: Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces. Lecture Notes in Math, vol. 1598. Springer-Verlag, Berlin Heidelberg (1995) · Zbl 0818.53067 |
| [4] | Cattaneo, A.; Nannicini, A.; Tomassini, A., Kodaira dimension of almost Kähler manifolds and curvature of the canonical connection, Ann. Mat. Pura Appl., 199, 5, 1815-1842, 2020 · Zbl 1447.53061 · doi:10.1007/s10231-020-00944-z |
| [5] | Cattaneo, A.; Nannicini, A.; Tomassini, A., On Kodaira dimension of almost complex 4-dimensional solvmanifolds without complex structures, Int. J. Math., 32, 10, 2150075, 2021 · Zbl 1486.53081 · doi:10.1142/S0129167X21500750 |
| [6] | Cattaneo, A., Nannicini, A., Tomassini, A.: Almost complex parallelizable manifolds: Kodaira dimension and special structures. Manuscr. Math. (2023) |
| [7] | Chen, H.; Zhang, W., Kodaira dimensions of almost complex manifolds I, Am. J. Math., 145, 2, 477-514, 2023 · Zbl 1517.32083 · doi:10.1353/ajm.2023.0011 |
| [8] | Chen, H., Zhang, W.: Kodaira Dimensions of Almost Complex Manifolds II (2020). preprint arXiv:2004.12825 |
| [9] | Denhinger, C.; Singhof, W., The \(e\)-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by the Heisenberg group, Invent. Math., 78, 101-112, 1984 · Zbl 0558.55010 · doi:10.1007/BF01388716 |
| [10] | Frölicher, A., Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Nat. Acad. Sci. U.S.A., 41, 641-644, 1955 · Zbl 0065.16502 · doi:10.1073/pnas.41.9.641 |
| [11] | Hirzebruch, F., Some problems on differentiable and complex manifolds, Ann. Math., 2, 60, 213-236, 1954 · Zbl 0056.16803 · doi:10.2307/1969629 |
| [12] | Holt, T., Bott-Chern and \(\overline{\partial }\) harmonic forms on almost Hermitian 4-manifolds, Math. Z., 302, 1, 47-72, 2022 · Zbl 1496.32035 · doi:10.1007/s00209-022-03048-x |
| [13] | Holt, T.; Piovani, R., Primitive decomposition of Bott-Chern and Dolbeault harmonic \((k, k)\)-forms on compact almost Kähler manifolds, Eur. J. Math., 9, 73, 2023 · Zbl 1522.32060 · doi:10.1007/s40879-023-00666-5 |
| [14] | Holt, T.; Zhang, W., Harmonic forms on the Kodaira-Thurston manifold, Adv. Math., 400, 108277, 2022 · Zbl 1523.32046 · doi:10.1016/j.aim.2022.108277 |
| [15] | Holt, T.; Zhang, W., Almost Kähler Kodaira-Spencer problem, Math. Res. Lett., 29, 6, 1685-1700, 2022 · Zbl 1517.32086 · doi:10.4310/MRL.2022.v29.n6.a3 |
| [16] | Müller, D., Peloso, M.M., Ricci, F.: Eigenvalues of the Hodge Laplacian on the Heisenberg group. Collect. Math. 327-342 (2006) · Zbl 1213.43015 |
| [17] | Müller,D., Peloso, M.M., Ricci, F.: Analysis of the Hodge Laplacian on the Heisenberg Group, vol. 233, no. 1095, pp. vi+91. Memoirs of the American Mathematical Society (2015). ISBN: 978-1-4704-0939-5 · Zbl 1309.43005 |
| [18] | Nachbin, L., The Haar Integral, 1965, Princeton: Van Nostrand, Princeton · Zbl 0127.07602 |
| [19] | Palais, RS, Seminar on the Atiyah-Singer Index Theorem, 1965, Princeton: Princeton University Press, Princeton · Zbl 0137.17002 |
| [20] | Piovani, R., Dolbeault harmonic \((1,1)\)-forms on \(4\)-dimensional compact quotients of Lie groups with a left invariant almost Hermitian structure, J. Geom. Phys., 180, 2022 · Zbl 1514.53072 · doi:10.1016/j.geomphys.2022.104639 |
| [21] | Piovani, R.; Tomassini, A., On the dimension of Dolbeault harmonic \((1,1)\)-forms on almost Hermitian \(4\)-manifolds, Pure Appl. Math. Q., 18, 3, 1187-1201, 2022 · Zbl 1496.32041 · doi:10.4310/PAMQ.2022.v18.n3.a11 |
| [22] | Rollenske, S.: Dolbeault cohomology of nilmanifolds with left-invariant complex structure. In: Complex and Differential Geometry, vol. 8, pp. 369-392. Springer Proceedings of the Mathematics, Springer, Heidelberg (2011) · Zbl 1266.53066 |
| [23] | Tardini, N.; Tomassini, A., Almost-complex invariants of families of six-dimensional solvmanifolds, Complex Manifolds, 9, 238-260, 2022 · Zbl 1501.53037 · doi:10.1515/coma-2021-0139 |
| [24] | Tardini, N.; Tomassini, A., \( \overline{\partial } \)-harmonic forms on 4-dimensional almost-Hermitian manifolds, Math. Res. Lett., 30, 5, 1617-1637, 2023 · Zbl 07849872 · doi:10.4310/MRL.2023.v30.n5.a14 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
