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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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