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
From the journal Complex Manifolds

Abstract

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

References

[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.

Scroll Up Arrow