Advances in Mathematics· 2026Q1
Toric vector bundles over a discrete valuation ring and Bruhat-Tits buildings
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- Q1SCImago
- 2026year
Short summary
A classification of rank r torus equivariant vector bundles on a toric scheme over a discrete valuation ring (O) is established, using graded piecewise linear maps to the building of GL(r).
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Key points
- Classification of rank r torus equivariant vector bundles on toric schemes over a discrete valuation ring (O).
- Classification is based on graded piecewise linear maps to the building of GL(r).
- Extends prior classifications by Klyachko (toric varieties over a field) and Mumford (line bundles over O).
- Includes a criterion for splitting bundles into sums of toric line bundles.
AI-generated from the title and abstract; the full text is not read.
Abstract
We give a classification of rank r torus equivariant vector bundles E on a toric scheme X over a discrete valuation ring O , in terms of graded piecewise linear maps Φ from the fan of X to the (extended) building of GL ( r ) . This is an extension of Klyachko's classification of torus equivariant vector bundles on toric varieties over a field on one hand, and Mumford's classification of equivariant line bundles on toric schemes over O on the other hand. We also give a simple criterion for equivariant splitting of E into a sum of toric line bundles in terms of its piecewise linear map. Among other things, this work lays the foundations for study of arithmetic geometry of toric vector bundles.
The authors' abstract, as published at the source. Advances in Mathematics, 2026 · DOI ↗
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Field: Geometry and Topology
Geometry and TopologyMathematics