Toric bundles, valuations, and tropical geometry over semifield of piecewise linear functions

Kiumars Kaveh, Christopher Manon

Published 2019-07-01Version 1

We initiate the algebro-geometric study of tropical geometry over the idemopotent semifield of piecewise linear functions. One of our main results shows that points on the tropical variety of a linear ideal over this semifield correspond to toric vector bundles. We introduce the notion of a valuation with values in the semifield of piecewise linear functions and we describe Khovanskii bases in this context. Far extending the Klyachko classification of toric vector bundles, we show that torus equivariant families over toric varieties are classified by such valuations. Finally, we see that the Gross-Hacking-Keel-Kontsevich toric degenerations of cluster varieties fit into our picture as a family over the toric scheme of the Fock-Goncharov fan.