PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1998 | 45 | 1 | 137-154
Tytuł artykułu

Dolbeault homotopy theory and compact nilmanifolds

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper we study the degeneration of both the cohomology and the cohomotopy Frölicher spectral sequences in a special class of complex manifolds, namely the class of compact nilmanifolds endowed with a nilpotent complex structure. Whereas the cohomotopy spectral sequence is always degenerate for such a manifold, there exist many nilpotent complex structures on compact nilmanifolds for which the classical Frölicher spectral sequence does not collapse even at the second term.
Słowa kluczowe
Rocznik
Tom
45
Numer
1
Strony
137-154
Opis fizyczny
Daty
wydano
1998
Twórcy
  • Departamento de Geometría y Topología, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15705 Santiago de Compostela, Spain
  • Departamento de Matemáticas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain
autor
  • Department of Mathematics, University of Maryland, College Park, Maryland 20742, U.S.A.
autor
  • Departamento de Matemáticas (Geometría y Topología), Facultad de Ciencias, Universidad de Zaragoza, Campus Plaza San Francisco, 50009 Zaragoza, Spain
Bibliografia
  • [1] L. C. de Andrés, M. Fernández, A. Gray and 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 61, 25-50, Univ. Santiago de Compostela (Spain) 1989.
  • [2] L. C. de Andrés, M. Fernández, A. Gray and J. J. Mencía, Moduli spaces of complex structures on compact four dimensional nilmanifolds, Bolletino U.M.I. (7) 5-A (1991), 381-389.
  • [3] E. Abbena, S. Garbiero and S. Salamon, private communication.
  • [4] W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Ergebnisse der Mathematik (3) 4, Springer-Verlag, Berlin, Heidelberg, 1984.
  • [5] A. K. Bousfield and V. K. A. M. Gugenheim, On PL de Rham theory and rational homotopy type, Memoirs Amer. Math. Soc. 8 (179) (1976).
  • [6] C. Benson and C. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), 513-518.
  • [7] L. A. Cordero, M. Fernández and A. Gray, Symplectic manifolds with no Kähler structure, Topology 25 (1986), 375-380.
  • [8] L. A. Cordero, M. Fernández and A. Gray, The Frölicher spectral sequence for compact nilmanifolds, Illinois J. Math. 35 (1991), 56-67.
  • [9] L. A. Cordero, M. Fernández, A. Gray and L. Ugarte, A general description of the terms in the Frölicher spectral sequence, Diff. Geom. Appl. 7 (1997), 75-84.
  • [10] L. A. Cordero, M. Fernández, A. Gray and L. Ugarte, Compact nilmanifolds with nilpotent complex structure: Dolbeault cohomology, preprint 1997.
  • [11] L. A. Cordero, M. Fernández, A. Gray and L. Ugarte, Frölicher spectral sequence of compact nilmanifolds with nilpotent complex structure, Proc. Conference on Differential Geometry, Budapest, July 27-30, 1996 (to appear).
  • [12] P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-274.
  • [13] M. Fernández, M. J. Gotay and A. Gray, Four-dimensional compact parallelizable symplectic and complex manifolds, Proc. Amer. Math. Soc. 103 (1988), 1209-1212.
  • [14] A. Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 641-644.
  • [15] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978.
  • [16] K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), 65-71.
  • [17] K. Kodaira, On the structure of compact complex analytic surfaces, I, Amer. J. Math. 86 (1964), 751-798.
  • [18] G. Lupton and J. Oprea, Symplectic manifolds and formality, J. Pure Appl. Algebra 91 (1994), 193-207.
  • [19] G. Lupton and J. Oprea, Cohomologically symplectic spaces: toral actions and the Gottlieb group, Trans. Amer. Math. Soc. 347 (1995), 261-288.
  • [20] I. A. Mal'cev, A class of homogeneous spaces, Amer. Math. Soc. Transl. No. 39 (1951).
  • [21] I. Nakamura, Complex parallelisable manifolds and their small deformations, J. Differential Geom. 10 (1975), 85-112.
  • [22] J. Neisendorfer and L. Taylor, Dolbeault Homotopy Theory, Trans. Amer. Math. Soc. 245 (1978), 183-210.
  • [23] K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. 59 (1954), 531-538.
  • [24] J. Oprea and A. Tralle, Symplectic Manifolds with no Kähler Structure, Lecture Notes in Math. 1661, Springer, Berlin, 1997.
  • [25] H. Pittie, The nondegeneration of the Hodge-de Rham spectral sequence, Bull. Amer. Math. Soc. 20 (1989), 19-22.
  • [26] Y. Sakane, On compact parallelisable solvmanifolds, Osaka J. Math. 13 (1976), 187-212.
  • [27] D. Tanré, Modèle de Dolbeault et fibré holomorphe, J. Pure Appl. Algebra 91 (1994), 333-345.
  • [28] W. P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), 467-468.
  • [29] A. Tralle, Applications of rational homotopy to geometry (results, problems, conjectures), Expo. Math. 14 (1996), 425-472.
  • [30] H. C. Wang, Complex parallelisable manifolds, Proc. Amer. Math. Soc. 5 (1954), 771-776.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv45i1p137bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.