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The canonical spectral sequences for poisson manifolds

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Abstract

For a compact symplectic manifoldM of dimension 2n, Brylinski proved that the canonical homology groupH can k (M) is isomorphic to the de Rham cohomology groupH 2n-k(M), and the first spectral sequence {E r(M)} degenerates atE 1(M). In this paper, we show that these isomorphisms do not exist for an arbitrary Poisson manifold. More precisely, we exhibit an example of a five-dimensional compact Poisson manifoldM 5 for whichH can1 (M 5) is not isomorphic toH 4(M 5), andE 1(M 5) is not isomorphic toE 2(M 5).

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Authors and Affiliations

  1. Departmento de Matemáticas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, 48080, Bilbao, Spain

    Marisa Fernández & Raúl Ibáñez

  2. Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006, Madrid, Spain

    Manuel de León

Authors
  1. Marisa Fernández
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  2. Raúl Ibáñez
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  3. Manuel de León
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Corresponding author

Correspondence to Marisa Fernández.

Additional information

This work has been partially supported through grants DGICYT (Spain), Projects PB91-0142 and PB89-0571, PB94-0633-C02-02; and through grants UPV, Project 127.310-EA 191/94, 127.310-EC248/96.

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Fernández, M., Ibáñez, R. & de León, M. The canonical spectral sequences for poisson manifolds. Isr. J. Math. 106, 133–155 (1998). https://doi.org/10.1007/BF02773464

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  • DOI: https://doi.org/10.1007/BF02773464

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