Asian Journal of Mathematics

Volume 24 (2020)

Number 4

Conics, twistors, and anti-self-dual tri-Kähler metrics

Pages: 621 – 652



Maciej Dunajski (Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom)

Paul Tod (Mathematical Institute, University of Oxford, United Kingdom)


We describe the range of the Radon transform on the space $M$ of irreducible conics in $\mathbb{CP}^2$ in terms of natural differential operators associated to the $SO(3)$-structure on $M = SL(3,\mathbb{R})/SO(3)$ 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 Kähler metrics. The corresponding twistor space $\mathcal{Z}$ admits a holomorphic fibration over $\mathbb{CP}^2$. In the special case where $\mathcal{Z} = \mathbb{CP}^3 \setminus \mathbb{CP}^1$ the twistor lines project down to a four-parameter family of conics which form triangular Poncelet pairs with a fixed base conic.


twistor theory, anti-self-duality, tri-Kähler metrics, Radon transform

2010 Mathematics Subject Classification

32L25, 53C28, 53C65

Received 17 May 2019

Accepted 9 December 2019

Published 18 February 2021