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

Mallory, Devlin

doi:10.2140/ant.2021.15.2019

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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

DOI: 10.2140/ant.2021.15.2019

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

Milestones

Received: 2 March 2020

Revised: 23 November 2020

Accepted: 22 December 2020

Published: 10 November 2021

Authors

Devlin Mallory
Department of Mathematics University of Michigan Ann Arbor, MI United States

Vol. 15, No. 8, 2021