Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Complex Manifolds

Ed. by Fino, Anna Maria

Covered by Web of Science - Emerging Sources Citation Index and Zentralblatt Math (zbMATH)

CiteScore 2018: 0.64

SCImago Journal Rank (SJR) 2018: 0.643
Source Normalized Impact per Paper (SNIP) 2018: 0.812

Mathematical Citation Quotient (MCQ) 2018: 0.61

Open Access
See all formats and pricing
More options …

A Family of Complex Nilmanifolds with in finitely Many Real Homotopy Types

Adela Latorre
  • Corresponding author
  • Centro Universitario de la Defensa - I.U.M.A., Academia General Militar, Crta. de Huesca s/n. 50090 Zaragoza, Spain
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Luis Ugarte
  • Departamento de Matemáticas - I.U.M.A., Universidad de Zaragoza, Campus Plaza San Francisco, 50009 Zaragoza, Spain
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Raquel Villacampa
  • Centro Universitario de la Defensa - I.U.M.A., Academia General Militar, Crta. de Huesca s/n. 50090 Zaragoza, Spain
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-02-16 | DOI: https://doi.org/10.1515/coma-2018-0004


We find a one-parameter family of non-isomorphic nilpotent Lie algebras ga, with a > [0,∞), of real dimension eight with (strongly non-nilpotent) complex structures. By restricting a to take rational values, we arrive at the existence of infinitely many real homotopy types of 8-dimensional nilmanifolds admitting a complex structure. Moreover, balanced Hermitian metrics and generalized Gauduchon metrics on such nilmanifolds are constructed.

Keywords: Nilmanifold; Nilpotent Lie algebra; Complex structure; Hermitian metric; Homotopy theory; Minimal model

MSC 2010: 55P62; 17B30; 53C55


  • [1] G. Bazzoni, V. Muñoz, Classification of minimal algebras over any field up to dimension 6, Trans. Amer. Math. Soc. 364 (2012), no. 2, 1007-1028.Google Scholar

  • [2] J.M. Bismut, A local index theorem for non-Kähler manifolds, Math. Ann. 284 (1989), 681-699.Google Scholar

  • [3] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte, Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5405-5433.Google Scholar

  • [4] P. Deligne, P. Grifiths, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-274.Google Scholar

  • [5] I. Dotti, A. Fino, Abelian hypercomplex 8-dimensional nilmanifolds, Ann. Global Anal. Geom. 18 (2000), no. 1, 47-59.Google Scholar

  • [6] I. Dotti, A. Fino, Hypercomplex eight-dimensional nilpotent Lie groups, J. Pure Appl. Algebra 184 (2003), no. 1, 41-57.Google Scholar

  • [7] N. Enrietti, A. Fino, L. Vezzoni, Tamed symplectic forms and SKT metrics, J. Symplectic Geom. 10 (2012), 203-224.Google Scholar

  • [8] Y. Félix, S. Halperin, J.C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics 205, Springer, 2001.Google Scholar

  • [9] A. Fino, L. Ugarte, On generalized Gauduchon metrics, Proc. Edinburgh Math. Soc. 56 (2013), 733-753.CrossrefGoogle Scholar

  • [10] J. Fu, Z. Wang, D. Wu, Semilinear equations, the k function, and generalized Gauduchon metrics, J. Eur. Math. Soc. 15 (2013), 659-680.Google Scholar

  • [11] P. Gauduchon, La 1-forme de torsion d’une variété hermitienne compacte, Math. Ann. 267 (1984), 495-518.Google Scholar

  • [12] P. Grifiths, J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics, Birkhäuser, 1981.Web of ScienceGoogle Scholar

  • [13] K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65-71.Google Scholar

  • [14] S. Ivanov, G. Papadopoulos, Vanishing theorems on (lSk)-strong Kähler manifolds with torsion, Adv. Math. 237 (2013), 147-164.Google Scholar

  • [15] J. Jost, S.-T. Yau, A non-linear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry, Acta Math. 170 (1993), 221-254; Corrigendum Acta Math. 173 (1994), 307.Google Scholar

  • [16] A. Latorre, L. Ugarte, On non-Kähler compact complex manifolds with balanced and astheno-Kähler metrics, C. R. Math. Acad. Sci. Paris 355 (2017), no. 1, 90-93.Google Scholar

  • [17] A. Latorre, L. Ugarte, R. Villacampa, On generalized Gauduchon nilmanifolds, Diff. Geom. Appl. 54 (2017), part A, 150-164.CrossrefGoogle Scholar

  • [18] A. Latorre, L. Ugarte, R. Villacampa, The ascending central series of nilpotent Lie algebras with complex structure, to appear in Trans. Amer. Math. Soc.Google Scholar

  • [19] I.A. Mal’cev, A class of homogeneous spaces, Amer. Math. Soc. Transl. 39 (1951).Google Scholar

  • [20] M.L. Michelsohn, On the existence of special metrics in complex geometry, Acta Math. 149 (1982), 261-295.Google Scholar

  • [21] K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. Math. 59 (1954), 531-538.Google Scholar

  • [22] J. Oprea, A. Tralle, Symplectic Manifolds with no Kähler Structure, Lecture Notes in Mathematics 1661, Springer, 1997.Google Scholar

  • [23] D. Popovici, Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics, Invent. Math. 194 (2013), 515-534.Google Scholar

  • [24] D. Popovici, Stability of strongly Gauduchon manifolds under modifications, J. Geom. Anal. 23 (2013), no. 2, 653-659.Web of ScienceCrossrefGoogle Scholar

  • [25] S. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311-333.Google Scholar

  • [26] D. Sullivan, Infinitesimal Computations in Topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269-331.Google Scholar

  • [27] L. Ugarte, Hermitian structures on six-dimensional nilmanifolds, Transform. Groups 12 (2007), no. 1, 175-202.Google Scholar

  • [28] L. Ugarte, R. Villacampa, Non-nilpotent complex geometry of nilmanifolds and heterotic supersymmetry, Asian J. Math. 18 (2014), no. 2, 229-246.Google Scholar

About the article

Received: 2017-12-14

Accepted: 2018-01-30

Published Online: 2018-02-16

Citation Information: Complex Manifolds, Volume 5, Issue 1, Pages 89–102, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2018-0004.

Export Citation

© 2018 Adela Latorre, et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in