SHARP BOUNDS ON THE HEIGHT OF K-SEMISTABLE FANO VARIETIES II, THE LOG CASE

Journal article, 2025

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 lln 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 lln+1 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 lln Z, as well as for general arithmetic orbifold Fano surfaces.