Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access October 18, 2021

Almost complex manifolds with small Nijenhuis tensor

  • Luis Fernandez , Tobias Shin and Scott O. Wilson EMAIL logo
From the journal Complex Manifolds


We give several explicit examples of compact manifolds with a 1-parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the C0-norm. The 4-dimensional examples possess no complex structure, whereas the 6-dimensional example does not possess a left invariant complex structure, and whether it possesses a complex structure appears to be unknown.

MSC 2010: 32Q60; 53C15


[1] L. Auslander, L. Green, and F. Hahn. Flows on some three-dimensional homogeneous spaces. Bull. Amer. Math. Soc., 67:494–497, 1961.10.1090/S0002-9904-1961-10653-8Search in Google Scholar

[2] C. Bock. On low-dimensional solvmanifolds. Asian J. Math., 20(2):199–262, 2016.10.4310/AJM.2016.v20.n2.a1Search in Google Scholar

[3] J.P. Demailly and H. Gaussier. Algebraic embeddings of smooth almost complex structures. J. Eur. Math. Soc. (JEMS), 19(11):3391–3419, 2017.10.4171/JEMS/742Search in Google Scholar

[4] M. Fernández and A. Gray. Compact symplectic solvmanifolds not admitting complex structures. Geom. Dedicata, 34(3):295–299, 1990.10.1007/BF00181691Search in Google Scholar

[5] T. Fei, D. Phong, S. Picard, and X. Zhang. Geometric flows for the type IIA string. Preprint arxiv:2011.03662v1, 2020.Search in Google Scholar

[6] M. Goze and E. Remm. Non existence of complex structures on filiform Lie algebras. Comm. Algebra, 30(8):3777–3788, 2002.10.1081/AGB-120005819Search in Google Scholar

[7] K. Hasegawa. Minimal models of nilmanifolds. Proc. Amer. Math. Soc., 106(1):65–71, 1989.10.1090/S0002-9939-1989-0946638-XSearch in Google Scholar

[8] K. Hasegawa. Complex and Kähler structures on compact solvmanifolds. J. Symplectic Geom., 3(4):749–767, 2005. Conference on Symplectic Topology.10.4310/JSG.2005.v3.n4.a9Search in Google Scholar

[9] A. Hattori. Spectral sequence in the de Rham cohomology of fibre bundles. J. Fac. Sci. Univ. Tokyo Sect. I, 8:289–331 (1960), 1960.Search in Google Scholar

[10] A. I. Mal’cev. On a class of homogeneous spaces. Izvestiya Akad. Nauk. SSSR. Ser. Mat. Amer. Math. Soc. Transl., 39 (1951), 13:9–32, 1949.Search in Google Scholar

[11] Aleksandar Milivojević. On the realization of symplectic algebras and rational homotopy types by closed symplectic manifolds. Proc. Amer. Math. Soc., 149(5):2257–2263, 2021.10.1090/proc/15397Search in Google Scholar

[12] G. D. Mostow. Factor spaces of solvable groups. Ann. of Math. (2), 60:1–27, 1954.10.2307/1969700Search in Google Scholar

[13] K. Nomizu. On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. of Math. (2), 59:531–538, 1954.10.2307/1969716Search in Google Scholar

Received: 2021-09-10
Accepted: 2021-10-02
Published Online: 2021-10-18

© 2021 Luis Fernandez et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 10.12.2023 from
Scroll to top button