A cohomological pairing of half-forms

Robinson, P. L.

doi:10.1090/S0002-9947-1987-0879572-7

A cohomological pairing of half-forms

Author: P. L. Robinson
Journal: Trans. Amer. Math. Soc. 301 (1987), 251-261
MSC: Primary 58F06; Secondary 17B10, 53C57
DOI: https://doi.org/10.1090/S0002-9947-1987-0879572-7
MathSciNet review: 879572
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Abstract: Blattner and Rawnsley have constructed half-forms for regular polarizations of arbitrary index. We show how to pair these half-forms into a line bundle fashioned purely from the symplectic data, with no assumption on the intersection of the polarizations. Our pairing agrees with the regular BKS pairing when the polarizations are positive.

[1] Robert J. Blattner, The metalinear geometry of non-real polarizations, Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975) Springer, Berlin, 1977, pp. 11–45. Lecture Notes in Math., Vol. 570. MR 0451296
[2] Robert J. Blattner and John H. Rawnsley, Quantization of the action of 𝑈(𝑘,𝑙) on 𝑅^{2(𝑘+𝑙)}, J. Funct. Anal. 50 (1983), no. 2, 188–214. MR 693228, https://doi.org/10.1016/0022-1236(83)90067-8
[3] Bertram Kostant, Symplectic spinors, Symposia Mathematica, Vol. XIV (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973) Academic Press, London, 1974, pp. 139–152. MR 0400304
[4] J. H. Rawnsley, On the pairing of polarizations, Comm. Math. Phys. 58 (1978), no. 1, 1–8. MR 468930
[5] -, Non-positive polarizations and half-forms, Lecture Notes in Math., vol. 836, Springer-Verlag, 1980, pp. 145-152.
[6] -, The Bargmann-Segal approach to symplectic spinors and half-forms for ${\text{M}}{{\text{P}}^c}$ structures, Warwick preprint, 1983.
[7] P. L. Robinson, ${\text{M}}{{\text{P}}^c}$ structures and applications, Warwick thesis, 1984.
[8] P. L. Robinson and J. H. Rawnsley, The metaplectic representation, ${\text{M}}{{\text{P}}^c}$ structures and geometric quantization (in preparation (1985)).
[9] André Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211 (French). MR 165033, https://doi.org/10.1007/BF02391012
[10] Joseph A. Wolf, The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121–1237. MR 251246, https://doi.org/10.1090/S0002-9904-1969-12359-1