F = * F , A review

  • J. Madore
  • J. L. Richard
  • R. Stora
Section VI
Part of the Lecture Notes in Physics book series (LNP, volume 106)


Algebraic Geometry Dirac Equation Gauge Field Spinor Field Hermitean Structure 
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-References and Footnotes-

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    The derivation of H1(P 3, E(−2)) = 0 has been given by several authors, see ref. [1], Dolbeaut cohomology is usually preferred rather than Cečh cohomology (see e.g. J.H. Rawnsley: “Differential Geometry of Instantons”, “On the Atiyah Hitchin Drinfeld Manin vanishing theorem”, Dublin Institute for Advanced Study preprint). A spectral sequence argument has been given by J.L. Verdier in Séminaire ENS 1977–1978, A. Douady, J.L. Verdier Ed. to be published in Astérisque, and communication in “Journées sur les champs de Yang et Mills”, S.M.F. May 24–26, (1978). One can prove [1] by recursion that if H°(P3, E(−1)) = 0, H1(P3, E(−2)) = 0, H°(P3, E(−l)) = H1(P3, E (−l −1)) = 0, for 1 > 1 by using the fact that on a two plane P2 c P3 containing a real line, E is trivial on almost all straightlines in P2, hence H°(P3, Elp2(− e)) = 0 Actually, H°(P3, E(−1)) = 0 follows from H°(P3, E) = 0. Let P2 be the equation of the two plane (a section of O(1)), where O p2 is the sheaf of germs of functions holomorphic in P2. It follows : Hence from which the recursion is established.Google Scholar
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    This can also be seen directly from algebraic geometry and Serre duality, this [7] [10], [12]: From the sequence it follows see e.g. J.L. Verdier in Séminaire ENS 1977–1978, A. Douady, J.L. Verdier, Ed.Google Scholar
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Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • J. Madore
    • 1
  • J. L. Richard
    • 1
  • R. Stora
    • 1
  1. 1.Centre de Physique Théorique du CNRSMarseille

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