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 \(\mathbb{P}^n_{\mathbb{Z}}\) 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 \(\mathbb{P}^{n+1}_{\mathbb{Z}}\). 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 \(\mathbb{P}^n_{\mathbb{Z}}\), as well as for general arithmetic orbifold Fano surfaces.