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

Abstract

We consider the question of simplicity of a ring $R$ under the action of its ring of differential operators $D_R$. We give examples to show that even when $R$ is Gorenstein and has rational singularities $R$ need not be a simple $D_R$-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 $D_R$-module implies that $X$ is Fano and thus $R$ has rational singularities.

v2: an error concerning degree 4 del Pezzos is corrected (an alternative proof is given in the new Theorem 6.2); in Section 6 the main results are deduced from previous work of Bogomolov--De Oliveira and De Oliveira--Langdon; references added to the more recent work of Höring--Liu--Shao on the tangent bundle of del Pezzos (Remark 9.5)

References in corpus (1)

• Cohomological characterizations of projective spaces and hyperquadrics

Cited by in corpus (2)

• On moment map and bigness of tangent bundles of $G$-varieties
• On pseudoeffective thresholds and cohomology of twisted symmetric tensor fields on irreducible Hermitian symmetric spaces