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Equivariant Chern Classes of Toric Vector Bundles over a DVR and Bruhat--Tits Buildings

Ana María Botero, Kiumars Kaveh, Christopher Manon

Abstract

We define equivariant Chern classes of a toric vector bundle over a proper toric scheme over a DVR. We provide a combinatorial description of them in terms of piecewise polynomial functions on the polyhedral complex associated to the toric scheme, which factorize through to an extended Bruhat--Tits building. We further motivate this definition from an arithmetic perspective, connecting to the non-Archimedean Arakelov theory of toric varieties.

Equivariant Chern Classes of Toric Vector Bundles over a DVR and Bruhat--Tits Buildings

Abstract

We define equivariant Chern classes of a toric vector bundle over a proper toric scheme over a DVR. We provide a combinatorial description of them in terms of piecewise polynomial functions on the polyhedral complex associated to the toric scheme, which factorize through to an extended Bruhat--Tits building. We further motivate this definition from an arithmetic perspective, connecting to the non-Archimedean Arakelov theory of toric varieties.
Paper Structure (11 sections, 16 theorems, 32 equations)

This paper contains 11 sections, 16 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Toric schemes over a DVR
  4. Equivariant Chow groups of special fibers of toric schemes
  5. Tits and Bruhat--Tits buildings of $\operatorname{GL}(r)$
  6. Link of a vertex
  7. Piecewise linear and affine maps
  8. Classification of toric vector bundles over a field and over a DVR
  9. Toric vector bundles over irreducible components of the special fiber
  10. Equivariant Chern classes of toric vector bundles over a DVR
  11. Connections to the Arakelov theory of toric varieties

Key Result

Theorem 1.1

Let ${\mathcal{E}}$ be a rank $r$ toric vector bundle over a toric variety $X_{\Sigma}$ with $\Phi_{{\mathcal{E}}} \colon |\Sigma| \to \widetilde{{\mathfrak B}}_{\textup{sph}}\left(\operatorname{GL}(r)\right)$ its corresponding piecewise linear map. Then for any $1 \leq i \leq r$, the $i$-th equivar

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2: Piecewise linear map corresponding to restriction of ${\mathcal{E}}$ to ${\mathfrak X}_{s,\nu}$
  • Theorem 1.3
  • Conjecture 1.4
  • Remark 3.1
  • Definition 3.2
  • Theorem 4.1
  • Definition 4.2
  • Remark 4.3
  • Theorem 4.4
  • ...and 33 more