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The generalized Penrose-Ward transform. (English) Zbl 0581.32035
The first part of this paper brings in the following unifying conceptual set-up: For complex manifolds Z, Y, X and surjective holomorphic mappings \(\mu\), \(\nu\) of maximal rank that embed Y as a submanifold of \(Z\times X\), \(Z\leftarrow^{\mu}Y\to^{\nu}X\) is called a correspondence. The transforms studied by the author transform objects on Z into objects on X. Thus, an analytic cohomology class, or a holomorphic vector bundle, on Z is pulled back to Y (first stage) and then (second stage) a spectral sequence is formed through the direct image of the data on Y obtained from the first stage. The differentials of the spectral sequence are induced from certain connections. In this way, if E is a vector bundle on Z, the cohomology \(H^ r(Z,{\mathcal O}(E))\) obtained through the spectral sequence can be interpreted as the kernel of a differential operator on X. This formulation allows a far-reaching generalisation of the Penrose and Ward transforms arising in mathematical physics.
The second part of the paper contains an explicit realisation of this program for several particular cases, mostly from twistor theory. In these examples, Y is a flag manifold. These interesting calculations are most welcome. As the author states, there should be many other uses of these ideas.
Reviewer: E.J.Akutowicz

32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14M15 Grassmannians, Schubert varieties, flag manifolds
32C35 Analytic sheaves and cohomology groups
32L25 Twistor theory, double fibrations (complex-analytic aspects)
51M35 Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations)
53C65 Integral geometry
53C80 Applications of global differential geometry to the sciences
Full Text: DOI
[1] Penrose, Quantum Gravity: an Oxford Symposium pp 268– (1975)
[2] DOI: 10.1007/BF00762011 · Zbl 0354.53025 · doi:10.1007/BF00762011
[3] Littlewood, The Theory of Group Characters and Matrix Representations of Groups (1950) · Zbl 0038.16504
[4] DOI: 10.1215/S0012-7094-81-04812-2 · Zbl 0483.55004 · doi:10.1215/S0012-7094-81-04812-2
[5] DOI: 10.1007/BF01942327 · Zbl 0465.58031 · doi:10.1007/BF01942327
[6] Buchdahl, Trans. A.M.S.
[7] DOI: 10.2307/2043718 · Zbl 0511.58001 · doi:10.2307/2043718
[8] DOI: 10.2307/1969996 · Zbl 0094.35701 · doi:10.2307/1969996
[9] Atiyah, Geometry of Yang-Mills Fields (1979)
[10] Wybourne, Symmetry Principles and Atomic Spectroscopy (1970)
[11] Atiyah, Proc. Roy. Soc. Lond. 362 pp 425– (1978)
[12] DOI: 10.1016/0375-9601(83)90715-6 · doi:10.1016/0375-9601(83)90715-6
[13] DOI: 10.1007/BF01626514 · Zbl 0362.14004 · doi:10.1007/BF01626514
[14] Witten, Phys. Lett. 77 pp 394– (1978) · doi:10.1016/0370-2693(78)90585-3
[15] Wells, Complex Geometry in Mathematical Physics (1982)
[16] DOI: 10.1007/BF00419314 · Zbl 0518.58023 · doi:10.1007/BF00419314
[17] Isenberg, Phys. Lett. 78 pp 462– (1978) · doi:10.1016/0370-2693(78)90486-0
[18] DOI: 10.1007/BF01208717 · Zbl 0502.58017 · doi:10.1007/BF01208717
[19] DOI: 10.1007/BFb0066025 · doi:10.1007/BFb0066025
[20] Hitchin, Proc. Boy. Soc. 370 pp 173– (1980)
[21] Henkin, Phys. Lett. 95 pp 405– (1980) · doi:10.1016/0370-2693(80)90178-1
[22] DOI: 10.1007/BF01208497 · doi:10.1007/BF01208497
[23] DOI: 10.1016/0375-9601(77)90842-8 · Zbl 0964.81519 · doi:10.1016/0375-9601(77)90842-8
[24] Salamon, Symposia Mathematica (1982)
[25] Penbose, General Relativity and Gravitation (1980)
[26] Manin, Gauge Fields and Holomorphic Geometry 17 pp 3– (1981) · Zbl 0486.53051
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