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On twistor spaces of anti-self-dual Hermitian surfaces


Author: Massimiliano Pontecorvo
Journal: Trans. Amer. Math. Soc. 331 (1992), 653-661
MSC: Primary 32L25; Secondary 32J15, 32J17, 53C25, 53C55
DOI: https://doi.org/10.1090/S0002-9947-1992-1050087-0
MathSciNet review: 1050087
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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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DOI: https://doi.org/10.1090/S0002-9947-1992-1050087-0
Article copyright: © Copyright 1992 American Mathematical Society

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