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.

中文翻译：

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圆锥曲线，扭曲和反自双重Tri-Kähler度量

我们根据与$ M = SL（$$ {$}（3）$$-结构相关的自然微分算子，描述$ \ mathbb {CP} ^ 2 $中不可约圆锥空间$ M $上Radon变换的范围。 3，\ mathbb {R}）/ SO（3）$及其复杂性。根据[27]，我们表明，对于此范围内的任何函数$ F $，$ F $的零轨迹都是一个四流形，它承认一个包含三个不同标量平坦Kähler度量的反自对偶共形结构。相应的扭曲空间$ \ mathcal {Z} $承认$ \ mathbb {CP} ^ 2 $上的全纯纤维。在$ \ mathcal {Z} = \ mathbb {CP} ^ 3 \ setminus \ mathbb {CP} ^ 1 $的特殊情况下，扭曲线向下投射到一个四参数圆锥曲线族，这些圆锥曲线形成具有固定点的三角Poncelet对基本圆锥。