Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology

Authors: Luis A. Cordero, Marisa Fernández, Alfred Gray and Luis Ugarte
Journal: Trans. Amer. Math. Soc. 352 (2000), 5405-5433
MSC (2000): Primary 32L05, 55P62, 57T15; Secondary 53C56, 32Q99
Published electronically: June 28, 2000
MathSciNet review: 1665327
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if $\Gamma \backslash G$ is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology $H^{\ast ,\ast }_{{\bar{\partial}}}(\Gamma \backslash G)$ is canonically isomorphic to the ${\bar{\partial}}$-cohomology $H^{\ast ,\ast }_{{\bar{\partial}}}({\mathfrak g}^{{\mathbb C}})$of the bigraded complex $(\Lambda ^{\ast ,\ast } ({\mathfrak g}^{{\mathbb C}})^{\ast }, {\bar{\partial}})$of complex valued left invariant differential forms on the nilpotent Lie group $G$.

References [Enhancements On Off] (What's this?)

  • [A] M.F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181-207. MR 19:172c
  • [AFGM] L.C. de Andrés, M. Fernández, A. Gray, J.J. Mencía, Compact manifolds with indefinite Kähler metrics, Proc. VIth Int. Coll. Differential Geometry (Ed. L.A. Cordero), Santiago (Spain) 1988, Cursos y Congresos Univ. Santiago de Compostela (Spain) 61 (1989), 25-50. MR 91e:53066
  • [AGS] E. Abbena, S. Garbiero, S. Salamon, Hermitian geometry on the Iwasawa manifold, Boll. U.M.I. B (7) 11, no. 2, suppl. (1997), 231-249. MR 98e:53121
  • [BG] C. Benson, C. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), 513-518. MR 90b:53042
  • [Co] L.A. Cordero, Holomorphic principal torus bundles, curvature and compact complex nilmanifolds, Proc. Workshop on Curvature and Geometry (Ed. C.T.J. Dodson), Lancaster Univ. (U.K.) January 1989 (1989), 107-149. MR 92b:53111
  • [CFG1] L.A. Cordero, M. Fernández, A. Gray, Symplectic manifolds with no Kähler structure, Topology 25 (1986), 375-380. MR 87j:53051
  • [CFG2] L.A. Cordero, M. Fernández, A. Gray, Compact symplectic manifolds not admitting positive definite Kähler metrics, to appear, Proc. Int. Conf. on Topology and its Appl., Baku 3-9 October 1987, U.R.S.S..
  • [CFG3] L.A. Cordero, M. Fernández, A. Gray, The Frölicher spectral sequence and complex compact nilmanifolds, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 753-756. MR 89c:32081
  • [CFG4] L.A. Cordero, M. Fernández, A. Gray, The Frölicher spectral sequence for compact nilmanifolds, Illinois J. Math. 35 (1991), 56-67. MR 92b:53112
  • [CFG5] L.A. Cordero, M. Fernández, A. Gray, The de Rham cohomology ring of a compact nilmanifold, preprint.
  • [CFGU] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte, A general description of the terms in the Frölicher spectral sequence, Diff. Geom. and its Applic. 7 (1997), 75-84. MR 98m:53092
  • [DGMS] P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-274. MR 52:3584
  • [FG] M. Fernández, A. Gray, The Iwasawa manifold, (Differential Geometry, Peñíscola, 1985), Lect. Notes in Math. 1209 (1986), 157-159. MR 87k:53162
  • [F] A. Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 641-644. MR 17:409a
  • [GM] P. Griffiths, J. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Math. 16, Birkhäuser, 1981. MR 82m:35014
  • [Ha] K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), 65-71. MR 89i:32015
  • [Hi] F. Hirzebruch, Topological Methods in Algebraic Geometry, third enlarged ed., Springer-Verlag, New York, 1966. MR 34:2573
  • [KN] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, I, II, Interscience Publ., New York, 1963, 1969. MR 27:2945; MR 38:6501
  • [Ko] K. Kodaira, On the structure of compact complex analytic surfaces, I, Amer. J. Math. 86 (1964), 751-798. MR 32:4708
  • [Ks] J.L. Koszul, Lectures on Fibre Bundles and Differential Geometry, Tata Inst., Bombay 1960. (Reissued Springer 1986.). MR 42:3698; MR 88e:53040
  • [Ma] I.A. Mal'cev, A class of homogeneous spaces, Amer. Math. Soc. Transl. 39 (1951). MR 10:50701; MR 12:589a
  • [McC] J. McCleary, User's Guide to Spectral Sequences, Publish or Perish, 1985. MR 87f:55014
  • [Mu] S. Murakami, Sur certain espaces fibrés principaux holomorphes dont le groupe est abélien connexe, Osaka Math. J. 13 (1961), 143-167. MR 26:788
  • [Na] I. Nakamura, Complex parallelisable manifolds and their small deformations, J. Differential Geometry 10 (1975), 85-112. MR 52:14389
  • [NT] J. Neisendorfer, L. Taylor, Dolbeault Homotopy Theory, Trans. Amer. Math. Soc. 245 (1978), 183-210. MR 80f:32004
  • [No] K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. 59 (1954), 531-538. MR 16:218c
  • [R] M.S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, Berlin, 1972. MR 58:22394a
  • [Sa] Y. Sakane, On compact complex parallelisable solvmanifolds, Osaka J. Math. 13 (1976), 187-212. MR 54:10692
  • [Th] W.P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), 467-468. MR 53:6578
  • [Tr] A. Tralle, Applications of rational homotopy to geometry (results, problems, conjectures), Expo. Math. 14 (1996), 425-472. MR 97k:55015
  • [Va] V.S. Varadarajan, Lie Groups, Lie Algebras, and their Representations, Graduate Texts in Math. 102, Springer-Verlag, Berlin, 1984. MR 85e:22001
  • [Wa] H.C. Wang, Complex parallelisable manifolds, Proc. Amer. Math. Soc. 5 (1954), 771-776. MR 17:531a

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 32L05, 55P62, 57T15, 53C56, 32Q99

Retrieve articles in all journals with MSC (2000): 32L05, 55P62, 57T15, 53C56, 32Q99

Additional Information

Luis A. Cordero
Affiliation: Departamento de Geometría y Topología, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15705 Santiago de Compostela, Spain

Marisa Fernández
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain

Alfred Gray
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Luis Ugarte
Affiliation: Departamento de Matemáticas (Geometría y Topología), Facultad de Ciencias, Universidad de Zaragoza, Campus Plaza San Francisco, 50009 Zaragoza, Spain

Received by editor(s): March 29, 1998
Received by editor(s) in revised form: December 20, 1998
Published electronically: June 28, 2000
Additional Notes: Partially supported by Xunta de Galicia (Spain), Research Project XUGA20701B93, U.P.V. Project 127.310–EA 191/94, DGICYT Projects PB94-0633-C02-01 and PB94-0633-C02-02, and by DGICYT Sabbatical Program 1995-0735.
The revised version was submitted just after the unexpected death of A. Gray in Bilbao (Spain) on October 27, 1998.
Article copyright: © Copyright 2000 American Mathematical Society