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We give a combinatorial classification of torus equivariant vector bundles on a (normal) projective   $T$  -variety of complexity-one. This extends the classification of equivariant line bundles on complexity-one   $T$  -varieties by Petersen–Süß on one hand, and Klyachko’s classification of equivariant vector bundles on toric varieties on the other hand. A main ingredient in our classification is the classification of torus equivariant vector bundles on toric schemes over a DVR in terms of piecewise affine maps to the (extended) Bruhat–Tits building of the general linear group.

Equivariant Vector Bundles on Complexity-One T -Varieties and Bruhat–Tits Buildings