Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the cohomology groups of a polarisation and diagonal quantisation


Author: J. H. Rawnsley
Journal: Trans. Amer. Math. Soc. 230 (1977), 235-255
MSC: Primary 58A10; Secondary 58A30, 58F05, 81.58
MathSciNet review: 0648775
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The sheaf $ {\mathcal{S}_F}(L)$ of germs of sections of a line bundle L on a manifold X covariant constant with respect to a flat connection defined for vectors in a complex subbundle F of the tangent bundle has a resolution by differential forms defined on F with values in L provided F satisfies the integrability conditions of the complex Frobenius theorem. This includes as special cases the de Rham and Dolbeault resolutions.


References [Enhancements On Off] (What's this?)

  • [1] Robert J. Blattner, Pairing of half-form spaces, Géométrie symplectique et physique mathématique (Colloq. Internat. C.N.R.S., Aix-en-Provence, 1974) Éditions Centre Nat. Recherche Sci., Paris, 1975, pp. 175–186 (English, with French summary). With questions by A. Voros and J. Śniatycki and replies by the author. MR 0451295 (56 #9582)
  • [2] S. S. Chern, Complex manifolds without potential theory, Van Nostrand Mathematical Studies, No. 15, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225346 (37 #940)
  • [3] F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. Die Grundlehren der Mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. MR 0202713 (34 #2573)
  • [4] Lars Hörmander, The Frobenius-Nirenberg theorem, Ark. Mat. 5 (1965), 425–432 (1965). MR 0178222 (31 #2480)
  • [5] Joseph B. Keller, Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems., Ann. Physics 4 (1958), 180–188. MR 0099207 (20 #5650)
  • [6] Bertram Kostant, Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, Springer, Berlin, 1970, pp. 87–208. Lecture Notes in Math., Vol. 170. MR 0294568 (45 #3638)
  • [7] -, Symplectic spinors, Convegno di Geomettrica Simplettica e Fisica Matematica, INDAM, Rome, 1973.
  • [8] -, On the definition of quantization, Colloque Symplectique, Aix-en-Provence, 1974.
  • [9] Hans Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. (2) 66 (1957), 155–158. MR 0088629 (19,551d)
  • [10] L. Nirenberg, A complex Frobenius theorem, Seminars on Analytic Functions. I, Princeton Univ. Press, Princeton, N.J., 1957, pp. 172-189.
  • [11] E. Onofri and M. Pauri, Analyticity and quantization, Lett. Nuovo Cimento (2) 3 (1972), 35–42. MR 0384016 (52 #4893)
  • [12] E. Onofri and M. Pauri, Dynamical quantization, J. Mathematical Phys. 13 (1972), 533–543. MR 0314389 (47 #2941)
  • [13] E. Onofri, Quantization theory for homogeneous Kähler manifolds, Parma, 1974 (preprint).
  • [14] P. Renouard, Variétés symplectiques et quantification, Thèse, Orsay, 1969.
  • [15] D. J. Simms, Geometric quantization of symplectic manifolds, Internat. Sympos. Mathematical Physics, Warsaw, 1974.
  • [16] -, Geometric quantization of the harmonic oscillator with diagonalised Hamiltonian, Proc. 2nd Internat. Colloq. Group Theoretical Methods in Physics (Catholic Univ., Nijmegen, 1973).
  • [17] -, Metalinear structures and a geometric quantization of the harmonic oscillator, Colloque Symplectique, Aix-en-Provence, 1974.
  • [18] Jędrzej Śniatycki, Bohr-Sommerfeld conditions in geometric quantization, Rep. Mathematical Phys. 7 (1975), no. 2, 303–311. MR 0391824 (52 #12643)
  • [19] -, Bohr-Sommerfeld quantum systems, 3rd Internat. Colloq. Group Theoretical Methods in Physics (Marseille, 1974).
  • [20] J. Śniatycki, Wave functions relative to a real polarization, Internat. J. Theoret. Phys. 14 (1975), no. 4, 277–288. MR 0408632 (53 #12396)
  • [21] -, On cohomology groups appearing in geometric quantization, Calgary, 1975 (preprint).
  • [22] J.-M. Souriau, Structure des systèmes dynamiques, Ma\cflexıtrises de mathématiques, Dunod, Paris, 1970 (French). MR 0260238 (41 #4866)
  • [23] Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman and Co., Glenview, Ill.-London, 1971. MR 0295244 (45 #4312)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58A10, 58A30, 58F05, 81.58

Retrieve articles in all journals with MSC: 58A10, 58A30, 58F05, 81.58


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0648775-2
PII: S 0002-9947(1977)0648775-2
Keywords: Line bundle, integrable tangent subbundle, sheaf of covariant constant sections, Poincaré lemma, resolution, periodic Hamiltonian flow, quantisation, Bohr-Sommerfeld condition
Article copyright: © Copyright 1977 American Mathematical Society