Toric Vector Bundles, Branched Covers of Fans, and the Resolution Property

We associate to each toric vector bundle on a toric variety X(∆) a “branched cover” of the fan ∆ together with a piecewise-linear function on the branched cover. This construction generalizes the usual correspondence between toric Cartier divisors and piecewise-linear functions. We apply this combinatorial geometric technique to study the moduli of toric vector bundles with fixed equivariant Chern class and to investigate the existence of resolutions of coherent sheaves by vector bundles, using singular nonquasiprojective toric threefolds as a testing ground. Our main new result is the construction of complete toric threefolds that have no nontrivial toric vector bundles of rank less than or equal to three. The preliminary sections of the paper give a self-contained introduction to Klyachko’s classification of toric vector bundles. The combinatorial geometric sections, which develop a theory of cone complexes and their branched covers, can be read independently.