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Compact self-dual manifolds with torus actions. (English) Zbl 1032.57036
Let $$M$$ be a compact connected oriented 4-manifold which admits a smooth and effective action of a 2-torus $$K$$. In the case $$M=m{\mathbb{C} P}^2$$, the connected sum of complex projective planes, D. Joyce [Duke Math. J. 77, 519-552 (1995; Zbl 0855.57028)] constructed examples of conformal self-dual metrics which are invariant under the $$K$$-action, and conjectured that these are the only examples for simply connected $$M$$.
The main theorem of this paper confirms Joyce’s conjecture by proving that any simply connected $$M$$ with non-zero Euler characteristic admitting a $$K$$-invariant self-dual metric is necessarily a connected sum of complex projective planes, with the metric being one of Joyce’s.
The lengthy and impressive proof considers the lift of the $$K$$-action on $$M$$ to a holomorphic $$G={\mathbb C}^*\times {\mathbb C}^*$$ action on the twistor space $$Z$$ of $$M$$. The author shows that every $$G$$-orbit has an analytic closure and that there exists a canonical meromorphic quotient $$\overline{f}:Z\to {\mathbb{C} P}^1$$ of the $$G$$-action on $$Z$$. The general fiber of $$\overline{f}$$ is a smooth toric surface determined by an invariant of the given torus action on $$M$$, and the proof demonstrates that this invariant completely determines $$Z$$.

##### MSC:
 57S25 Groups acting on specific manifolds 32J18 Compact complex $$n$$-folds 53C28 Twistor methods in differential geometry
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