Online Talk

A toric vector bundle is a vector bundle over a toric variety which is equipped with a lift of the action action of the associated torus. As a source of examples, toric vector bundles and their projectivizations provide a rich class of spaces that still manage to admit a combinatorial characterization. Toric vector bundles were first classified by Kaneyama, and later by Klyachko using the data of decorated subspace arrangements. I will give an introduction to some recent joint work with Kiumars Kaveh reformulating Klyachko's classification in terms of piecewise-linear maps to the spherical building of the general linear group. I'll describe how we extend this characterization to give a classification of toric principal $G$-bundles for a reductive algebraic group $G$, and toric vector bundles over a toric scheme over a DVR. I'll show how this classification allows us to give descriptions of characteristic classes, notions of positivity for the respective bundles, and interesting connections with tropical geometry. I will then point to some applications and open problems. This is joint work with Kiumars Kaveh, Boris Tsvelikhovskiy, and Ana Botero.