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

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.

Keywords

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

2010 Mathematics Subject Classification

32L25, 53C28, 53C65

Full Text (PDF format)

Received 17 May 2019

Accepted 9 December 2019

Published 18 February 2021