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

Belgun, Florin

doi:10.1090/S0002-9947-03-03157-X

On the Weyl tensor of a self-dual complex 4-manifold
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by Florin Alexandru Belgun

Trans. Amer. Math. Soc. 356 (2004), 853-880

DOI: https://doi.org/10.1090/S0002-9947-03-03157-X

Published electronically: October 21, 2003

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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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