Toric Anti-self-dual 4-manifolds via Complex Geometry

Using the twistor correspondence, this article gives a oneto-one correspondence between germs of toric anti-self-dual conformal classes and certain holomorphic data determined by the induced action on twistor space. Recovering the metric from the holomorphic data leads to the classical problem of prescribing the Čech coboundary of 0-cochains on an elliptic curve covered by two annuli. The classes admitting Kähler representatives are described; each such class contains a circle of Kähler metrics. This gives new local examples of scalar flat Kähler surfaces and generalises work of Joyce [Joy95] who considered the case where the distribution orthogonal to the torus action is integrable.