Flat partial connections and holomorphic structures in vector bundles
J. H. Rawnsley
Proc. Amer. Math. Soc. 73 (1979), 391-397
Primary 58A30; Secondary 32L10, 58F06
Full-text PDF Free Access
Similar Articles |
Abstract: The notion of a flat partial connection D in a vector bundle E, defined on an integrable subbundle F of the complexified tangent bundle of a manifold X is defined. It is shown that E can be trivialized by local sections s satisfying . The sheaf of germs of sections s of E satisfying has a natural fine resolution, giving the de Rham and Dolbeault resolutions as special cases.
If X is a complex manifold and F the tangents of type (0, 1), the flat partial connections in a vector bundle E are put in correspondence with the holomorphic structures in E.
If X, E are homogeneous and F invariant, then invariant flat connections in E can be characterized as extensions of the representation of the isotropic subgroup to which E is associated, extending results of Tirao and Wolf in the holomorphic case.
F. Atiyah, N.
J. Hitchin, and I.
M. Singer, Self-duality in four-dimensional Riemannian
geometry, Proc. Roy. Soc. London Ser. A 362 (1978),
no. 1711, 425–461. MR 506229
Bott, Lectures on characteristic classes and foliations,
Lectures on algebraic and differential topology (Second Latin American
School in Math., Mexico City, 1971), Springer, Berlin, 1972,
pp. 1–94. Lecture Notes in Math., Vol. 279. Notes by Lawrence
Conlon, with two appendices by J. Stasheff. MR 0362335
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
Goldschmidt and D.
Spencer, Submanifolds and over-determined differential
operators, Complex analysis and algebraic geometry, Iwanami Shoten,
Tokyo, 1977, pp. 319–356. MR 0458512
Hitchin, Harmonic spinors, Advances in Math.
14 (1974), 1–55. MR 0358873
Nijenhuis and William
B. Woolf, Some integration problems in almost-complex and complex
manifolds., Ann. of Math. (2) 77 (1963),
424–489. MR 0149505
L. Nirenberg, A complex Frobenius theorem, Seminars on Analytic Functions. I, Princeton Univ. Press, Princeton, N. J., 1957, pp. 172-189.
H. Rawnsley, On the cohomology groups of a
polarisation and diagonal quantisation, Trans.
Amer. Math. Soc. 230 (1977), 235–255. MR 0648775
(58 #31147), http://dx.doi.org/10.1090/S0002-9947-1977-0648775-2
J. Sternberg and Turner
L. Smith, The Theory of Potential and Spherical Harmonics,
Mathematical Expositions, no. 3, University of Toronto Press, Toronto,
Ont., 1944. MR
A. Tirao and Joseph
A. Wolf, Homogeneous holomorphic vector bundles, Indiana Univ.
Math. J. 20 (1970/1971), 15–31. MR 0263110
- M. F. Atiyah, N. J. Hitchin and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Oxford preprint, 1977. MR 506229 (80d:53023)
- R. Bott, Lectures on characteristic classes and foliations, Lecture Notes in Math., vol. 279, Springer-Verlag, Berlin and New York, 1972. MR 0362335 (50:14777)
- S. S. Chern, Complex manifolds without potential theory, Math. Studies No. 15, Van Nostrand, Princeton, N. J., 1967. MR 0225346 (37:940)
- H. Goldschmidt and D. Spencer, Submanifolds and over-determined differential operators, Complex Analysis and Algebraic Geometry, edited by W. L. Bailey and T. Shioda, Cambridge Univ. Press, Cambridge, 1977. MR 0458512 (56:16712)
- N. J. Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1-55. MR 0358873 (50:11332)
- A. Nijenhuis and W. Woolf, Some integration problems in almost-complex and complex manifolds, Ann. of Math. (2) 77 (1963), 424-489. MR 0149505 (26:6992)
- L. Nirenberg, A complex Frobenius theorem, Seminars on Analytic Functions. I, Princeton Univ. Press, Princeton, N. J., 1957, pp. 172-189.
- J. H. Rawnsley, On the cohomology groups of a polarization and diagonal quantization, Trans. Amer. Math. Soc. 230 (1977), 235-255. MR 0648775 (58:31147)
- W. Sternberg and T. Smith, The theory of potential and spherical harmonics, Univ. of Toronto Press, Toronto, 1961. MR 0011513 (6:175d)
- J. A. Tirao and J. A. Wolf, Homogeneous holomorphic vector bundles, Indiana Univ. Math. J. 20 (1970), 15-31. MR 0263110 (41:7715)
Retrieve articles in Proceedings of the American Mathematical Society
Retrieve articles in all journals
Flat partial connections,
holomorphic vector bundles,
© Copyright 1979 American Mathematical Society