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On the Weyl tensor of a self-dual complex 4-manifold


Author: Florin Alexandru Belgun
Journal: Trans. Amer. Math. Soc. 356 (2004), 853-880
MSC (2000): Primary 53C21, 53A30, 32Qxx
DOI: https://doi.org/10.1090/S0002-9947-03-03157-X
Published electronically: October 21, 2003
MathSciNet review: 1984459
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Abstract: We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-geodesics. We also show that the projective structure of the $\beta$-surfaces of a self-dual manifold is flat. All these results are illustrated in detail in the case of the complexification of $\mathbb{CP} ^2$.


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Additional Information

Florin Alexandru Belgun
Affiliation: Centre de Mathématiques, UMR 7640 CNRS, Ecole Polytechnique, 91128 Palaiseau cedex, France
Address at time of publication: Mathematisches Institut, Augustusplatz 10/11, 04109 Leipzig, Germany
Email: belgun@math.polytechnique.fr, Florin.Belgun@math.uni-leipzig.de

DOI: https://doi.org/10.1090/S0002-9947-03-03157-X
Received by editor(s): February 27, 2000
Published electronically: October 21, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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