Even if the toric local height in the above definition differs from the local height of Definition 2.39, we will be able to use them to compute global heights because, for toric subvarieties and closures of orbits, the sum over all places of the local canonical heights is zero (see Proposition 6.35). This is the case, in particular, for the height of the total space $X_{\Sigma}$.

By Theorem 2.46(4), the right-hand side of equation (6.2) does not depend on the choice of refinement nor on the choice of sections, but the toric local height depends on the toric structure of the line bundles (see Definition 4.19), because the canonical metric depends on the toric structure.