#### Vol. 15, No. 8, 2021

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Bigness of the tangent bundle of del Pezzo surfaces and $D$-simplicity

### Devlin Mallory

Vol. 15 (2021), No. 8, 2019–2036
##### Abstract

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.

##### Keywords
$D$-simplicity, tangent bundle, bigness of tangent bundle, positivity of vector bundles, differential operators, Fano varieties
##### Mathematical Subject Classification 2010
Primary: 13N10
Secondary: 14B05, 14J60