D-modules on Smooth Toric Varieties

Details D-modules on Smooth Toric Varieties D-modules on Smooth Toric Varieties

2000-07-16

arxiv.org/abs/math/0007099v1

Journal of Algebra 240 (2001) 744-770

Mathematics

Algebraic Geometry

AMS-LaTeX, 28 pages

Scientific paper

10.1006/jabr.2001.8731

Let X be a smooth toric variety. David Cox introduced the homogeneous coordinate ring S of X and its irrelevant ideal B. Extending well-known results on projective space, Cox established the following: (1) the category of quasi-coherent sheaves on X is equivalent to the category of graded S-modules modulo B-torsion, (2) the variety X is a geometric quotient of Spec(S) V(B) by a suitable torus action. We provide the D-module version of these results. More specifically, let A denote the ring of differential operators on Spec(S). We show that the category of D-modules on X is equivalent to a subcategory of graded A-modules modulo B-torsion. Additionally, we prove that the characteristic variety of a D-module is a geometric quotient of an open subset of the characteristic variety of the associated A-module and that holonomic D-modules correspond to holonomic A-modules.

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