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Toric vector bundles over a discrete valuation ring and Bruhat-Tits buildings

Kiumars Kaveh, Christopher Manon, Boris Tsvelikhovskiy

TL;DR

The paper extends Klyachko's classification of toric vector bundles from toric varieties over a field to toric schemes over a discrete valuation ring by realizing bundles through graded piecewise linear maps into the total extended Bruhat-Tits building. It shows that isomorphism classes of toric vector bundles on a proper toric DVR scheme correspond to integral graded piecewise linear maps $\Phi:|\Sigma|\to\widetilde{\mathfrak{B}}(E)$, with the generic fiber captured by the linear part $\Phi_0$ and, in the complete case, the piecewise affine data $\Phi_1$ describing the vector bundle on the generic fiber. A local equivariant triviality result, a precise construction of $\Phi$, and a splitting criterion extend Klyachko's field-based theory and Mumford's line-bundle classification to the DVR setting. The framework lays groundwork for arithmetic and tropical aspects of toric vector bundles, including future avenues like equivariant Chern classes and tropical analogues, and it suggests further extensions to higher-dimensional local fields via Bruhat-Tits theory.

Abstract

We give a classification of rank $r$ torus equivariant vector bundles $\mathcal{E}$ on a toric scheme $\mathfrak{X}$ over a discrete valuation ring $\mathcal{O}$, in terms of graded piecewise linear maps $Φ$ from the fan of $\mathfrak{X}$ 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 $\mathcal{O}$ on the other hand. We also give a simple criterion for equivariant splitting of $\mathcal{E}$ 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.

Toric vector bundles over a discrete valuation ring and Bruhat-Tits buildings

TL;DR

The paper extends Klyachko's classification of toric vector bundles from toric varieties over a field to toric schemes over a discrete valuation ring by realizing bundles through graded piecewise linear maps into the total extended Bruhat-Tits building. It shows that isomorphism classes of toric vector bundles on a proper toric DVR scheme correspond to integral graded piecewise linear maps , with the generic fiber captured by the linear part and, in the complete case, the piecewise affine data describing the vector bundle on the generic fiber. A local equivariant triviality result, a precise construction of , and a splitting criterion extend Klyachko's field-based theory and Mumford's line-bundle classification to the DVR setting. The framework lays groundwork for arithmetic and tropical aspects of toric vector bundles, including future avenues like equivariant Chern classes and tropical analogues, and it suggests further extensions to higher-dimensional local fields via Bruhat-Tits theory.

Abstract

We give a classification of rank torus equivariant vector bundles on a toric scheme over a discrete valuation ring , in terms of graded piecewise linear maps from the fan of to the (extended) building of . 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 on the other hand. We also give a simple criterion for equivariant splitting of 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.
Paper Structure (16 sections, 19 theorems, 76 equations, 6 figures)

This paper contains 16 sections, 19 theorems, 76 equations, 6 figures.

Table of Contents

  1. Preliminaries on toric varieties and toric schemes
  2. Klyachko's classification of toric vector bundles over a field
  3. Toric schemes over a discrete valuation ring
  4. Preliminaries on additive norms and valuations
  5. Additive norms, lattices and affine Grassmannian
  6. Valuations and labeled flags
  7. Floor, ceiling and pull-back of an additive norm
  8. Tits and Bruhat-Tits buildings of $\operatorname{GL}(E)$
  9. The Bruhat-Tits building of $\operatorname{GL}(r, K)$
  10. Bruhat-Tits building of $\operatorname{GL}(2, K)$
  11. The Tits building of $\operatorname{GL}(r, K)$
  12. Toric vector bundles over a discrete valuation ring
  13. Local equivariant triviality
  14. Graded piecewise linear maps
  15. Main theorems
  16. ...and 1 more sections

Key Result

Theorem 2

With notation as above, let $\mathfrak{X}_\Sigma$ be a proper toric scheme over $\operatorname{Spec}(\mathcal{O})$. Then there is a bijection between the isomorphism classes of toric vector bundles on $\mathfrak{X}_\Sigma$ and the piecewise affine maps from $|\Sigma_1|$ to $\widetilde{\mathfrak{B}}_

Figures (6)

  • Figure 1: A fan $\Sigma$ defining a toric scheme with $\mathbb{P}^1$ as general fiber
  • Figure 4: An apartment in the Bruhat-Tits/affine building of $\operatorname{GL}(2)$, a subdivision of the affine line into identical segments. An extended apartment is the Cartesian product of this picture with a copy of $\mathbb{R}$
  • Figure 6: The Bruhat-Tits buildings of $\operatorname{GL}(2, \mathbb{Q}_2)$ and $\operatorname{GL}(2, \mathbb{Q}_3)$
  • Figure 7: Two apartments (infinite paths) corresponding to the bases $B=\{b_1, b_2\}$ and $B' = \{b_1+b_2, b_2\}$ overlapping on a half-infinite path.
  • Figure 8: An apartment in the Tits building of $\operatorname{SL}(3)$, a subdivision of circle into $6$ arcs
  • ...and 1 more figures

Theorems & Definitions (55)

  • Definition 1
  • Theorem 2: Main theorem, first part
  • Theorem 3: Main theorem, second part
  • Theorem 4: Main theorem, general case
  • Proposition 1.1
  • Theorem 1.2: Klyachko
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Example 1.6
  • ...and 45 more