Compact self-dual manifolds with torus actions.

*(English)*Zbl 1032.57036Let \(M\) 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 \(Z\).

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 \(Z\).

Reviewer: Terry Fuller (Northridge)