Tropical Intersection Theory from Toric Varieties

We apply ideas from intersection theory on toric varieties to tropical intersection theory. We introduce mixed Minkowski weights on toric varieties which interpolate between equivariant and ordinary Chow cohomology classes on complete toric varieties. These objects fit into the framework of tropical intersection theory developed by Allermann and Rau. Standard facts about intersection theory on toric varieties are applied to show that the definitions of tropical intersection product on tropical cycles in $\R^n$ given by Allermann-Rau and Mikhalkin are equivalent. We introduce an induced tropical intersection theory on subvarieties on a toric variety. This gives a conceptional proof that the intersection of tropical $ψ$-classes on $\cmbar_{0,n}$ used by Kerber and Markwig computes classical intersection numbers.