Logarithmic Stable Toric Varieties and their Moduli

The Chow quotient of a toric variety by a subtorus, as defined by Kapranov–Sturmfels– Zelevinsky, coarsely represents the main component of the moduli space of stable toric varieties with a map to a fixed projective toric variety, as constructed by Alexeev and Brion. We show that, after we endow both spaces with the structure of a logarithmic stack, the spaces are isomorphic. Along the way, we construct the Chow quotient stack and demonstrate several properties it satisfies.