Self-duality of Kähler surfaces.(English)Zbl 0546.53044

The notion of self-duality of an oriented Riemannian 4-manifold is the differential geometric condition that makes the Penrose twistor space construction work. The author here considers when a Kähler metric is self-dual (with respect to either orientation). With respect to the natural orientation the manifold is locally symmetric. This was proved by J. P. Bourguignon [Invent. Math. 63, 263-286 (1981; Zbl 0456.53033)], B.-Y. Chen [J. Differ. Geom. 13, 547-558 (1978; Zbl 0427.53033)] and A. Derdzinski [Compos. Math. 49, 405-433 (1983; Zbl 0527.53030)] and also by the author’s own approach. With the other orientation, this paper contains the theorem that a Kähler surface is anti-self-dual if and only if its scalar curvature is zero. Furthermore, a classification may be given, the only unknown metrics being those defined on rational surfaces with $$(c_ 1)^ 2\leq 0$$. It remains an intriguing possibility whether these metrics really exist.
Reviewer: N.Hitchin

MSC:

 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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References:

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