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

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 $\mathrm{\Phi }$ 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.