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The Bott obstruction to the existence of nice polarizations

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Abstract

In this note, we prove and discuss the following theorem: if the symplectic or almost symplectic manifoldM admits a nice polarizationF with dim\((F \cap \bar F) = k \ne 0\), we must have Chernq M=0 and Pontq M=0, forq>2n−k, and, in particular, the Euler class ofM vanishes.

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References

  1. Arnold, V. I., Avez, A.: Problèmes Ergodiques de la Mécanique Classique. Paris: Gauthier-Villars. 1967.

    Google Scholar 

  2. Blattner, R. J.: Quantization and representation theory. In: Harmonic Analysis on Homogeneous Spaces, pp. 147–165. A. M. S. Proc. Symp. Pure Math. XXVI, Williamstown 1972. Providence, R. I.: Amer Math. Soc. 1974.

    Google Scholar 

  3. Block, S., Giesieker, D.: The positivity of the Chern classes of an ample vector bundle. Invent. Math.12, 112–117 (1971).

    Google Scholar 

  4. Bott, R.: Lectures on characteristic classes and foliations. In: Lectures on Algebraic and Differential Topology. ByR. Bott, S. Gitler, I. M. Jones. pp. 1–94. Lect. Notes Math. 279. New York-Berlin-Heidelberg: Springer. 1972.

    Google Scholar 

  5. Hirzebruch, F.: Topological Methods in Algebraic Geometry. Berlin-Heidelberg-New York: Springer. 1966.

    Google Scholar 

  6. Kamber, F. W., Tondeur, Ph.: Foliated Bundles and Characteristic Classes. Lect. Notes Math. 493. Berlin-Heidelberg-New York: Springer 1975.

    Google Scholar 

  7. Nirenberg, L.: A complex Frobenius theorem. Sem. Anal. Funct.1, 172–189 (1958).

    Google Scholar 

  8. Rawnsley, J. H.: On the pairing of polarizations. Comm. Math. Phys.58, 1–8 (1978).

    Google Scholar 

  9. Simms, J. D.: Geometric quantization of energy levels in the Kepler problem. Symp. Math.14, 125–137 (1974).

    Google Scholar 

  10. Sniatycki, J., Toporowski, S.: On representation spaces in geometric quantization. Int. J. Theor. Phys.16, 615–633 (1977).

    Google Scholar 

  11. Thurston, W. P.: Some simple examples of symplectic manifolds. Proc. Amer. Math. Soc.55, 467–468 (1976).

    Google Scholar 

  12. Vaisman, I.: On locally conformal almost Kähler manifolds. Isr. J. Math.24, 338–351 (1976).

    Google Scholar 

  13. Vaisman, I.: A coordinatewise formulation of geometric quantization. Ann. Inst. H. Poincaré31, 5–24 (1979).

    Google Scholar 

  14. Vaisman, I.: On locally and globally conformal Kähler manifolds. Trans. Amer. Math. Soc.262, 533–541 (1980).

    Google Scholar 

  15. Weinstein, A.: Symplectic manifolds and their Lagrangian submanifolds. Adv. Math.6, 329–346 (1971).

    Google Scholar 

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  1. Department of Mathematics, University of Haifa, Mount Carmel, 31999, Haifa, Israel

    Izu Vaisman

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  1. Izu Vaisman
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Vaisman, I. The Bott obstruction to the existence of nice polarizations. Monatshefte für Mathematik 92, 231–238 (1981). https://doi.org/10.1007/BF01442487

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