Sharp bounds on the height of K-semistable Fano varieties II, the log case

In our previous work we conjectured - inspired by an algebro-geometric result of Fujita - that the height of an arithmetic Fano variety X of relative dimension $n$ is maximal when X is the projective space ${\mathbb{P}}_{\mathbb{Z}}^n$ over the integers, endowed with the Fubini-Study metric, if the corresponding complex Fano variety is K-semistable. In this work the conjecture is settled for diagonal hypersurfaces in ${\mathbb{P}}_{\mathbb{Z}}^{n+1}$. The proof is based on a logarithmic extension of our previous conjecture, of independent interest, which is established for toric log Fano varieties of relative dimension at most three, hyperplane arrangements on ${\mathbb{P}}_{\mathbb{Z}}^n$, as well as for general arithmetic orbifold Fano surfaces.