Abstract
The linear holonomy of a Poisson structure, introduced in the present paper, generalizes the linearized holonomy of a regular symplectic foliation. For singular Poisson structures the linear holonomy is defined for the lifts of tangential paths to the cotangent bundle. The linear holonomy is closely related to the modular class. Namely, the logarithm of the determinant of the linear holonomy is equal to the integral of the modular vector field along such a lift. This assertion relies on the notion of the integral of a vector field along a cotangent path on a Poisson manifold, which is also introduced in the paper.
We then prove that for locally unimodular Poisson manifolds the modular class is an invariant of Morita equivalence.
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References
[Br] J.-L. Brylinski,A differential complex for Poisson manifolds, Journal of Differential Geometry28 (1988), 93–114.
[BZ] J.-L. Brylinski and G. Zuckerman,The outer derivation of a complex Poisson manifold, Journal für die reine und angewandte Mathematik506 (1999), 181–189.
[CW] A. Cannas da Silva and A. Weinstein,Lectures on geometric models for non-commutative algebras, Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Providence, RI, 1999.
[Co] A. Connes,Noncommutative Geometry, Academic Press, San Diego, 1994.
[Da1] P. Dazord,Holonomie des feuilletages singuliers, Comptes Rendus de l'Académie des Sciences, Paris, Série I, Mathématiques298 (1984), 27–30.
[Da2] P. Dazord,Feuilletages à singularités, Indagationes Mathematicae (N.S.)47 (1985), 21–39.
[ELW] S. Evens, J.-H. Lu and A. Weinstein,Transverse measures, the modular class, and a cohomology pairing for Lie algebroids, The Quarterly Journal of Mathematics. Oxford. Ser. (2)50 (1999), 417–436.
[GL] V. L. Ginzburg and J.-H. Lu,Poisson cohomology of Morita equivalent Poisson manifolds, International Mathematics Research Notices10 (1992), 199–205.
[Godb] C. Godbillon,Feuilletages. Études géométriques, Progress in Mathematics 98, Birkhauser, Boston, 1991.
[Gode] R. Godement,Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1958.
[Ko] J. L. Koszul,Crochet de Schouten-Nijenhuis et cohomologie, Astérisque, hors série, Sociéte Mathématiques de France, Paris (1985), 257–271.
[SM] C. L. Siegel and J. K. Moser,Lectures on Celestial Mechanics, Springer-Verlag, New York, 1971.
[Va] I. Vaisman,Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics 118, Birkhäuser, Boston, 1994.
[We1] A. Weinstein,The local structure of Poisson manifolds, Journal of Differential Geometry18 (1983), 523–557.
[We2] A. Weinstein,Lagrangian mechanics and groupoids, inMechanics Day (Waterloo, ON, 1992), Fields Institute Communications7, American Mathematical Society, Providence, RI, 1996, pp. 207–231.
[We3] A. Weinstein,The modular automorphism group of a Poisson manifold, Journal of Geometry and Physics23 (1997), 379–394.
[Xu] P. Xu,Morita equivalence of Poisson manifolds, Communications in Mathematical Physics142 (1991), 493–509.
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The work was partially supported by the NSF and by the faculty research funds of the UC Santa Cruz.
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Ginzburg, V.L., Golubev, A. Holonomy on poisson manifolds and the modular class. Isr. J. Math. 122, 221–242 (2001). https://doi.org/10.1007/BF02809901
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DOI: https://doi.org/10.1007/BF02809901