Abstract
Moser proved in 1965 in his seminal paper [15] that two volume forms on a compact manifold can be conjugated by a diffeomorphism, that is to say they are equivalent, if and only if their associated cohomology classes in the top cohomology group of a manifold coincide. In particular, this yields a classification of compact symplectic surfaces in terms of De Rham cohomology. In this paper we generalize these results for volume forms admitting transversal zeroes. In this case there is also a cohomology capturing the classification: the relative cohomology with respect to the critical hypersurface. We compare this classification scheme with the classification of Poisson structures on surfaces which are symplectic away from a hypersurface where they fulfill a transversality assumption (b-Poisson structures). We do this using the desingularization technique introduced in [10] and extend it to bm-Nambu structures.
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References
Arnol’d, V. I., Remarks on Poisson Structures on a Plane and on Other Powers of Volume Elements, J. Soviet Math., 1989, vol. 47, no. 3, pp. 2509–2516; see also: Trudy Sem. Petrovsk., 1987, no. 12, pp. 37–46, 242.
Arnol’d, V. I., Critical Points of Smooth Functions, in Proc. of ICM’74 (Vancouver, BC, 1974): Vol. 1, pp. 19–39.
Braddell, R., Delshams, A., Miranda, E., Oms, C., and Planas, A., An Invitation to Singular Symplectic Geometry, Int. J. Geom. Methods Mod. Phys., 2019, vol. 16, suppl. 1, 1940008, 16 pp.
Cannas da Silva, A., Guillemin, V., and Woodward, Ch., On the Unfolding of Folded Symplectic Structures, Math. Res. Lett., 2000, vol. 7, no. 1, pp. 35–53.
Cannas da Silva, A., Guillemin, V., and Pires, A. R., Symplectic Origami, Int. Math. Res. Not. IMRN, 2011, no. 18, pp. 4252–4293.
Cannas da Silva, A., Fold-Forms for Four-Folds, J. Symplectic Geom., 2010, vol. 8, no. 2, pp. 189–203.
Delshams, A., Kiesenhofer, A., and Miranda, E., Examples of Integrable and Non-Integrable Systems on Singular Symplectic Manifolds, J. Geom. Phys., 2017, vol. 115, pp. 89–97.
Godbillon, C., Éléments de topologie algébrique, Hermann, 1971.
Guillemin, V., Miranda, E., and Pires, A.R., Symplectic and Poisson Geometry on b-Manifolds, Adv. Math., 2014, vol. 264, pp. 864–896.
Guillemin, V., Miranda, E., and Weitsman, J., Desingularizing bm-Symplectic Structures, Int. Math. Res. Notices, 2017, rnx126.
Guillemin, V. and Sternberg, Sh., Geometric Asymptotics, Math. Surveys, No. 14, Providence,R.I.: AMS, 1977.
Martinet, J., Sur les singularités des formes différentielles, Ann. Inst. Fourier (Grenoble), 1970, vol. 20, fasc. 1, pp. 95–178.
Martínez Torres, D., Global Classification of Generic Multi-Vector Fields of Top Degree, J. London Math. Soc. (2), 2004, vol. 69, no. 3, pp. 751–766.
Melrose, R. B., The Atiyah–Patodi–Singer Index Theorem, Res. Notes Math., vol. 4, Wellesley,Mass.: A. K.Peters, 1993.
Moser, J., On the Volume Elements on a Manifold, Trans. Amer. Math. Soc., 1965, vol. 120, pp. 286–294.
Miranda, E. and Oms, C., Contact Structures with Singularities, arXiv:1806.05638v1 (2018).
Miranda, E. and Planas, A., Classification of bm-Nambu Structures of Top Degree, C. R. Math. Acad. Sci. Paris, 2018, vol. 356, no. 1, pp. 92–96.
Miranda, E. and Planas, A., Equivariant Classification of bm-Symplectic Surfaces, Regul. Chaotic Dyn., 2018, vol. 23, no. 4, pp. 355–371.
Nambu, Y., Generalized Hamiltonian Dynamics, Phys. Rev. D (3), 1973, vol. 7, pp. 2405–2412.
Radko, O., A Classification of Topologically Stable Poisson Structures on a Compact Oriented Surface, J. Symplectic Geom., 2002, vol. 1, no. 3, pp. 523–542.
Scott, G., The Geometry of bk Manifolds, J. Symplectic Geom., 2016, vol. 14, no. 1, pp. 71–95.
Weinstein, A., Lectures on Symplectic Manifolds, CBMS Reg. Conf. Ser. Math., vol. 29, Providence,R.I.: AMS, 1979.
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Cardona, R., Miranda, E. On the Volume Elements of a Manifold with Transverse Zeroes. Regul. Chaot. Dyn. 24, 187–197 (2019). https://doi.org/10.1134/S1560354719020047
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DOI: https://doi.org/10.1134/S1560354719020047