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Some Penrose transforms in complex differential geometry

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Abstract

In this article, we review a construction in the complex geometry often known as the Penrose transform. We then present two new applications of this transform. One concerns the construction of symmetries of the massless field equations from mathematical physics. The other concerns obstructions to the embedding of CR structures on the three-sphere.

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References

  1. Andreotti A, Norguet F. La convexité holomorphe dans l’espace analytique des cycles d’une variété algébrique. Ann Scuola Norm Sup Pisa, 1967, 21: 31–82

    MathSciNet  MATH  Google Scholar 

  2. Andreotti A, Norguet F. Cycles of algebraic manifolds and ∂\(\bar \partial \)-cohomology. Ann Scuola Norm Sup Pisa, 1971, 25: 59–114

    MathSciNet  MATH  Google Scholar 

  3. Penrose R. Solutions of the zero-rest-mass equations. Jour Math Phys, 1969, 10: 38–39

    Article  MathSciNet  Google Scholar 

  4. Penrose R. Twistor functions and sheaf cohomology. In: Hughston L P, Ward R S, eds Advances in Twistor Theory, Research Notes in Math vol. 37, Pitman, 1979: 25–36

  5. Gindikin S G, Henkin G. Integral geometry for \(\bar \partial \)-cohomology in q-linearly concave domains in ℂℙn. Funct Anal Appl, 1978, 12(4): 6–23

    MathSciNet  MATH  Google Scholar 

  6. Bateman H. The solution of partial differential equations by means of definite integrals. Proc Lond Math Soc, 1904, 1: 451–458

    Google Scholar 

  7. Bott R. Homogeneous vector bundles. Ann Math, 1957, 60: 203–248

    MathSciNet  Google Scholar 

  8. Schmid W. Homogeneous complex manifolds and representations of semisimple Lie groups. Proc Nat Acad Sci, USA, 1968, 59: 56–59

    Article  MATH  Google Scholar 

  9. Schmid W. L 2-cohomology and the discrete series. Ann Math, 1971, 93: 1–42

    Article  Google Scholar 

  10. Wolf J A. The action of a real semisimple group on a complex flag manifold I: orbit structure and holomorphic arc components. Bull Amer Math Soc, 1969, 75: 1121–1237

    Article  MathSciNet  MATH  Google Scholar 

  11. Baston R J, Eastwood M G. The Penrose Transform: its Interaction with Representation Theory. Oxford: Oxford University Press, 1989

    MATH  Google Scholar 

  12. Akhiezer D N, Gindikin S G. On Stein extensions of real symmetric spaces. Math Ann, 1990, 286: 1–12

    Article  MathSciNet  MATH  Google Scholar 

  13. Fels G, Huckleberry A, Wolf J A. Cycles Spaces of Flag Domains. Progress in Math vol. 245, Birkhäuser, 2006

  14. Huckleberry A, Wolf J A. Injectivity of the double fibration transform for cycle spaces of flag domains. Jour Lie Theory, 2004, 14: 509–522

    MathSciNet  MATH  Google Scholar 

  15. Eastwood M G, Penrose R, Wells Jr R O. Cohomology and massless fields. Commun Math Phys, 1981, 78: 305–351

    Article  MathSciNet  MATH  Google Scholar 

  16. Penrose R, Rindler W. Spinors and Space-time, vol. I. Cambridge: Cambridge University Press, 1984

    Google Scholar 

  17. Hitchin N J. Linear field equations on self-dual spaces. Proc Roy Soc Lond, 1980, A370: 173–191

    MathSciNet  Google Scholar 

  18. Wells Jr R O. The conformally invariant Laplacian and the instanton vanishing theorem. Seminar on Differential Geometry, Ann of Math Stud, vol. 102. Princeton: Princeton University Press, 1982: 483–498

    Google Scholar 

  19. Eastwood M G, Singer M A. A conformally invariant Maxwell gauge. Lett Math Phys, 1985, 107A: 73–74

    MathSciNet  Google Scholar 

  20. Paneitz S M. A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. Preprint 1983

  21. Riegert R. A nonlocal action for the trace anomaly. Phys Lett, 1984, B134: 56–60

    MathSciNet  Google Scholar 

  22. Eastwood M G. Higher symmetries of the Laplacian. Ann Math, 2005, 161: 1645–1665

    Article  MathSciNet  MATH  Google Scholar 

  23. Dynkin E B. The maximal subgroups of the classical groups. Amer Math Soc Transl Series 2, 1957, 6: 245–378

    MATH  Google Scholar 

  24. Joseph A. The minimal orbit in a simple Lie algebra and its associated maximal ideal. Ann Sci Ecole Norm Sup, 1976, 9: 1–30

    MathSciNet  MATH  Google Scholar 

  25. Huggett S A, Tod K P. An Introduction to Twistor Theory. Lond Math Soc Student Texts vol. 4. Cambridge: Cambridge University Press, 1985

    Google Scholar 

  26. Penrose R. Twistor algebra. Jour Math Phys, 1967, 8: 345–366

    Article  MathSciNet  MATH  Google Scholar 

  27. Eastwood M G. On rasing and lowering helicity. In: Hughston L P, Ward R S, eds Advances in Twistor Theory. Research Notes in Math vol. 37, Pitman, 1979: 97–100

  28. Ginsberg M L. A Cohomological Approach to Scattering Theory. D Phil Thesis, University of Oxford, 1980

  29. Eastwood M G, Somberg P, Souček V. The uniqueness of the Joseph ideal for the classical groups. arViv: math.RT/0512296

  30. Fox D J F. Projectively invariant star products. Int Math Res Pap, 2005, 9: 461–510

    Article  Google Scholar 

  31. Garfinkle D. A New Construction of the Joseph Ideal. Ph D thesis, Massachusetts Institute of Technology, 1982

  32. Anco S C, Pohjanpelto J. Conserved currents of massless fields of spin s ≥ 1/2. Proc Roy Soc Lond, 2003, A459: 1215–1239

    MathSciNet  Google Scholar 

  33. Anco S C, Pohjanpelto J. Symmetries and currents of massless neutrino fields, electromagnetic and graviton fields. Symmetry in Physics, CRM Proc. Lecture Notes vol. 34, Amer Math Soc, 2004, 1–12

  34. Eastwood M G, Somberg P, Souček V. Higher symmetries of the Dirac operator. in preparation.

  35. Hitchin N J. Complex manifolds and Einstein’s equations. Twistor Geometry and Nonlinear Systems, Lecture Notes in Math vol. 970, Springer 1982: 73–99

  36. Kodaira K. On stability of compact submanifolds of complex manifolds. Amer Jour Math, 1963, 85: 79–94

    Article  MathSciNet  MATH  Google Scholar 

  37. Kiremidjian G K. A direct extension method for CR structures. Math Ann, 1979, 242: 1–19

    Article  MathSciNet  MATH  Google Scholar 

  38. Epstein C L, Henkin G. Extension of CR-structures for 3-dimensional pseudoconcave manifolds. Multidimensional Complex Analysis and Partial Differential Equations, Contemp Math, vol. 205, Amer Math Soc, 1997: 51–67

    MathSciNet  Google Scholar 

  39. Bland J S. Contact geometry and CR structures on S 3. Acta Math, 1994, 172: 1–49

    Article  MathSciNet  MATH  Google Scholar 

  40. Eastwood M G, Ginsberg M L. Duality in twistor theory. Duke Math Jour, 1981, 48: 177–196

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Brock University, 500 Glenridge Avenue, St. Catharines, ON, L2S 3A1, Canada

    Anco Stephen

  2. Department of Mathematics, University of Toronto, 100 St. George Street, Toronto, ON, M5S 3G3, Canada

    Bland John

  3. Department of Mathematics, University of Adelaide, SA, 5005, Australia

    Eastwood Michael

Authors
  1. Anco Stephen
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  2. Bland John
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  3. Eastwood Michael
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Correspondence to Eastwood Michael.

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Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

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Anco, S., Bland, J. & Eastwood, M. Some Penrose transforms in complex differential geometry. SCI CHINA SER A 49, 1599–1610 (2006). https://doi.org/10.1007/s11425-006-2066-5

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  • DOI: https://doi.org/10.1007/s11425-006-2066-5

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