On twistor spaces of anti-self-dual Hermitian surfaces

Pontecorvo, Massimiliano

doi:10.1090/S0002-9947-1992-1050087-0

On twistor spaces of anti-self-dual Hermitian surfaces
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by Massimiliano Pontecorvo

Trans. Amer. Math. Soc. 331 (1992), 653-661

DOI: https://doi.org/10.1090/S0002-9947-1992-1050087-0

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Abstract:

We consider a complex surface $M$ with anti-self-dual hermitian metric $h$ and study the holomorphic properties of its twistor space $Z$. We show that the naturally defined divisor line bundle $[X]$ is isomorphic to the $- \frac {1} {2}$ power of the canonical bundle of $Z$, if and only if there is a Kähler metric of zero scalar curvature in the conformal class of $h$. This has strong consequences on the geometry of $M$, which were also found by C. Boyer $[3]$ using completely different methods. We also prove the existence of a very close relation between holomorphic vector fields on $M$ and $Z$ in the case that $M$ is compact and Kähler.

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