Graded rings and equivariant sheaves on toric varieties

By Markus Perling. 2002

66 citations recorded by the source corpus.

In this note we derive a formalism for describing equivariant sheaves over toric varieties. This formalism is a generalization of a correspondence due to Klyachko, which states that equivariant vector bundles on toric varieties are equivalent to certain sets of filtrations of vector spaces. We systematically construct the theory from the point of view of graded ring theory and this way we clarify earlier constructions of Kaneyama and Klyachko. We also connect the formalism to the theory of fine‐graded modules over Cox' homogeneous coordinate ring of a toric variety. As an application we construct minimal resolutions of equivariant vector bundles of rank two on toric surfaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Subfields: Algebra and Number Theory, Geometry and Topology, Pharmacology

View Paper: https://www.semanticscholar.org/paper/2cc4c6f2326aa2202b14732d660888dd7f405837

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Hendrik Suss
2. Hendrik Süß
3. Sam Payne
4. Jürgen Hausen
5. Eric Katz
6. Chi Li
7. Lars Petersen
8. David Jensen

In Geometry and Topology

1. Hendrik Suss
2. Hendrik Süß
3. Sam Payne
4. Jürgen Hausen
5. Chi Li
6. Eric Katz
7. David Jensen
8. Dhruv Ranganathan

In Pharmacology

1. Hendrik Suss
2. Sam Payne
3. Jürgen Hausen

Citations in the corpus

• EQUIVARIANT CHOW COHOMOLOGY OF TORIC VARIETIES (2005, background): “inally motivated by a desire to have a simple, combinatorial formula for the equivariant Chern classes of equivariant vector bundles on toric varieties. We refer the reader to [Kan], [Kly], [KS], and [Per] for the basic facts about equivariant vector bundles on toric varieties, and simply recall tha”
• The Cox ring of an algebraic variety with torus action (2009, methodology): “We will use Klyachko’s description [13] of equivariant reflexive sheaves over X ; we will follow Perling’s notation [19] in terms of families of complete increasing filtrations.”
• Fixed point loci of moduli spaces of sheaves on toric varieties (2008, unknown): “Perling has given a general description of equivariant quasi-coherent sheaves on toric varieties in [32,30].”
• Moduli of toric vector bundles (2007, background): “ecently, Perling has systematically developed the approach to toric sheaves through T-eigenspace decompositions of modules of sections, giving a classification of arbitrary toric quasicoherent sheaves [Per04a]. Perling’s classification data, which he calls ∆-families of vector spaces, explicitly encod”
• Toric vector bundles, branched covers of fans, and the resolution property (2006, methodology): “ut since there seems to be no single accessible source for this material, we have included proofs for the convenience of the reader. Our presentation draws on ideas from [Kan75], [Kly91], [KS98], and [Per04], as well as from Klyachko’sseminal paper [Kly90a], and we have attempted to simplify the ori”
• Gluing Affine Torus Actions Via Divisorial Fans (2006, background): “Recall that Klyachko [Kly90], and later Perling [ Per04 ], gave a combinatorial description of the T-equivariant, reflexive sheaves E on X. In Perling’s notation, E is given by a K-vector space E together with Z-labeled increasing filtations E̺(i) for every ̺ ∈ �(1).”