Conics, Twistors, and anti-self-dual tri-K\"ahler metrics

TL;DR

This paper characterizes the Radon transform on irreducible conics in complex projective space, linking it to anti-self-dual conformal structures and scalar-flat Kähler metrics, with applications to twistor theory and Poncelet pairs.

Contribution

It describes the Radon transform's range on conics in terms of differential operators and explores the geometric structures of the resulting four-manifolds and their twistor spaces.

Findings

Range of Radon transform characterized by differential operators.

Zero locus of functions in the range admits anti-self-dual conformal structures.

Special case relates to Poncelet pairs and twistor space structures.

Abstract

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