Musson, Ian M. Differential operators on toric varieties. (English) Zbl 0824.14044 J. Pure Appl. Algebra 95, No. 3, 303-315 (1994). The author considers the surjectivity of the canonical map \(\theta : \mathbb{D} (Y)^ G \to \mathbb{D} (Y // G)\) where \(\mathbb{D} (Y // G)\) is the ring of differential operators on the quotient variety \(Y // G = \text{Spec} {\mathcal O} (Y)^ G\), \(G\) a reductive algebraic group and \(\mathbb{D} (Y)^ G\) the fixed ring. – The author shows that if \(X\) is a toric variety then it is isomorphic to a quotient \(Y // G\) where \(G\) is a torus acting on an affine space \(K^ s\) and \(Y\) is a \(G\) invariant open subset of \(K^ s\). Then he shows that any ring of differential operators on \(X\) twisted by an invertible sheaf is a factor ring of the fixed ring \(D(Y)^ G\) by an ideal generated by central elements. Reviewer: R.Salvi (Milano) Cited in 1 ReviewCited in 18 Documents MSC: 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14M17 Homogeneous spaces and generalizations 13N10 Commutative rings of differential operators and their modules Keywords:fixed ring; toric variety; ring of differential operators PDFBibTeX XMLCite \textit{I. M. Musson}, J. Pure Appl. Algebra 95, No. 3, 303--315 (1994; Zbl 0824.14044) Full Text: DOI References: [1] Audin, M., The Topology of Torus Actions on Symplectic Manifolds, (Progress in Mathematics, Vol. 93 (1991), Birkhäuser: Birkhäuser Basel) · Zbl 0726.57029 [2] D.A. Cox, The homogeneous coordinate ring of a toric variety, Preprint, Amherst College.; D.A. Cox, The homogeneous coordinate ring of a toric variety, Preprint, Amherst College. · Zbl 1285.14055 [3] Danilov, V. I., The geometry of toric varieties, Russian Math. Surveys, 33, 2, 97-154 (1978) · Zbl 0425.14013 [4] J. Fine, in preparation.; J. Fine, in preparation. [5] Hartshorne, R., Algebraic Geometry, (Graduate Texts in Mathematics, Vol. 52 (1977), Springer: Springer Berlin) · Zbl 0532.14001 [6] Jones, A. G., Rings of differential operators on toric varieties, Proc. Edinburgh Math. Soc., 37, 143-160 (1993) · Zbl 0805.16024 [7] Kantor, J.-M., Formes et opérateurs différentiels sur les espaces analytiques complexes, Bull. Soc. Math. France, 53, 5-80 (1977) · Zbl 0376.32001 [8] Levasseur, T., Anneaux d’opérateurs différentiels, (Dubreil, P.; Malliavn, M.-P., Séminaire d’Algébre. Séminaire d’Algébre, Lecture Notes in Mathematics, Vol. 867 (1981), Springer: Springer Berlin), 157-173 · Zbl 0507.14012 [9] Levasseur, T., Relèvements d’opérateurs différentiels sur les anneaux d’invariants, (Operator Algebras, Unitary Representations, Enveloping Algebras and Representation Theory. Operator Algebras, Unitary Representations, Enveloping Algebras and Representation Theory, Progress in Mathematics, Vol. 92 (1990), Birkhäuser: Birkhäuser Basel), 449-470 · Zbl 0733.16009 [10] Levasseur, T.; Stafford, J. T., Rings of Differential Operators on Classical Rings of Invariants, Mem. Amer. Math. Soc., 412 (1989) · Zbl 0691.16019 [11] Musson, I. M., Rings of differential operators on invariant rings of tori, Trans. Amer. Math. Soc., 303, 805-827 (1987) · Zbl 0628.13019 [12] Oda, T., Convex Bodies and Algebraic Geometry, (An Introduction to the Theory of Toric Varieties (1985), Springer: Springer Berlin) [13] G. Schwarz, Lifting differential operators from orbit spaces, Preprint Brandeis University.; G. Schwarz, Lifting differential operators from orbit spaces, Preprint Brandeis University. · Zbl 0836.14032 [14] Van den Bergh, M., Differential operators on semi-invariants for tori and weighted projective spaces, (Malliavin, M.-P., Topics in Invariant Theory. Topics in Invariant Theory, Lecture Notes in Mathematics, Vol. 1478 (1991), Springer: Springer Berlin), 255-272 · Zbl 0802.13005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.