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arXiv · 1801.05257

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

Maciej Dunajski·Paul Tod
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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.

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Maciej Dunajski, Paul Tod. 2018-01-16. Conics, Twistors, and anti-self-dual tri-K\"ahler metrics. https://arxiv.org/abs/1801.05257

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