Tropical cycles and Chow polytopes

Abstract

The Chow polytope of an algebraic cycle in a torus depends only on its tropicalisation. Generalising this, we associate a Chow polytope to any abstract tropical variety X in a tropicalised toric variety: the normal tropical hypersurface to this Chow polytope is the Minkowski sum of X and a reflected skeleton of the fan of the toric variety. Several significant polyhedra associated to tropical varieties are special cases of our Chow polytope. We show also that Chow polytope subdivisions do not fully distinguish the combinatorial types of tropical variety, and record a proof of the equivalence of two standard definitions of tropical linear spaces.