Toric vector bundles with trivial Chern class and flag decorations

RESUMO

We study toric vector bundles on complete toric varieties whose total equivariant Chern classes are trivial. Our approach is tropical, using the notion of tropical toric vector bundles as piecewise linear maps introduced by Kaveh and Manon. We prove that any toric vector bundle of rank $r$ with trivial Chern class and affinely independent equivariant Chern roots is equivariantly isomorphic to a toric vector bundle pulled back from one of a finite set of varieties with dimension at most $r-1$ after twisting by a character. This extends a theorem of Payne about toric vector bundles of rank $r\leq 3$ with trivial Chern class. As an application, we construct examples of complete toric varieties of dimension $n$ that admit no nontrivial toric vector bundles of rank $r\leq n+1$ with the aforementioned properties. We also introduce combinatorial gadgets we call flag decorations of permutohedra, whose convexity properties are key for our results.