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

Abstract

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.

28 pages. Main changes in version 2: Included Lemma 4.3. In Thm 1.5 we now assume that the divisor has at most three irreducible components (which is automatic in the orbifold case). This assumption is needed in Lemma 6.2