CONICS, TWISTORS, AND ANTI-SELF-DUAL TRI-KAHLER METRICS

We describe the range of the Radon transform on the space M of irreducible conics in CP2 in terms of natural differential operators associated to the SO(3)-structure on M = SL(3, R)/SO(3) and its complexification. Following [27] we show that for any function F in this range, the zero locus of F is a four-manifold admitting an anti-self-dual conformal structure which contains three different scalar-flat Kahler metrics. The corresponding twistor space Z admits a holomorphic fibration over CP2. In the special case where Z = CP3 \ CP1 the twistor lines project down to a four-parameter family of conics which form triangular Poncelet pairs with a fixed base conic.