We show that the solutions to the equations, defining the so-called Calabi–Yau condition for fourth-order operators of degree two, define a variety that consists of ten irreducible components. These can be described completely in parametric form, but only two of the components seem to admit arithmetically interesting operators. We include a description of the 69 essentially distinct fourth-order Calabi–Yau operators of degree two that are presently known to us.

中文翻译：

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二级卡拉比-丘算子

我们证明了方程的解，定义了二阶四阶算子的所谓卡拉比-丘条件，定义了由十个不可约分量组成的簇。这些可以完全以参数形式描述，但似乎只有两个组件允许算术上有趣的运算符。我们描述了目前已知的 69 个本质上不同的二阶四阶 Calabi-Yau 算子。