We consider the question of simplicity of a $ℂ$-algebra $R$ under the action of its ring of differential operators $D_{R∕ℂ}$. We give examples to show that even when $R$ is Gorenstein and has rational singularities, $R$ need not be a simple $D_{R∕ℂ}$-module; for example, this is the case when $R$ is the homogeneous coordinate ring of a smooth cubic surface. Our examples are homogeneous coordinate rings of smooth Fano varieties, and our proof proceeds by showing that the tangent bundle of such a variety need not be big. We also give a partial converse showing that when $R$ is the homogeneous coordinate ring of a smooth projective variety $X$, embedded by some multiple of its canonical divisor, then simplicity of $R$ as a $D_{R∕ℂ}$-module implies that $X$ is Fano and thus $R$ has rational singularities.

我们考虑一个简单的问题 $ℂ$-代数 $\mathrm{电阻}$ 在其微分算子环的作用下 $D_{\mathrm{电阻}∕ℂ}$. 我们举例说明，即使当$\mathrm{电阻}$ 是 Gorenstein 并且具有有理奇点， $\mathrm{电阻}$ 不必是一个简单的 $D_{\mathrm{电阻}∕ℂ}$-模块; 例如，当$\mathrm{电阻}$是光滑三次曲面的齐次坐标环。我们的例子是光滑 Fano 簇的齐次坐标环，我们的证明是通过证明这种簇的切丛不需要很大来进行的。我们还给出了部分逆向证明，当$\mathrm{电阻}$ 是光滑射影簇的齐次坐标环 $X$，嵌入其规范除数的一些倍数，然后简单 $\mathrm{电阻}$ 作为一个 $D_{\mathrm{电阻}∕ℂ}$-module 意味着 $X$ 是法诺，因此 $\mathrm{电阻}$ 有理性的奇点。