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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

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
Robert J. Blattner and John H. Rawnsley, Quantization of the action of ${\rm U}(k,\,l)$ on ${\bf R}^{2(k+l)}$, J. Funct. Anal. 50 (1983), no. 2, 188–214. MR 693228, DOI https://doi.org/10.1016/0022-1236%2883%2990067-8
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
J. H. Rawnsley, On the pairing of polarizations, Comm. Math. Phys. 58 (1978), no. 1, 1–8. MR 468930

---, Non-positive polarizations and half-forms, Lecture Notes in Math., vol. 836, Springer-Verlag, 1980, pp. 145-152.

---, The Bargmann-Segal approach to symplectic spinors and half-forms for ${\text {M}}{{\text {P}}^c}$ structures, Warwick preprint, 1983.

P. L. Robinson, ${\text {M}}{{\text {P}}^c}$ structures and applications, Warwick thesis, 1984.

P. L. Robinson and J. H. Rawnsley, The metaplectic representation, ${\text {M}}{{\text {P}}^c}$ structures and geometric quantization (in preparation (1985)).

André Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211 (French). MR 165033, DOI https://doi.org/10.1007/BF02391012
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, DOI https://doi.org/10.1090/S0002-9904-1969-12359-1