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arXiv · 2312.05064

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

Rolf Andreasson·Robert J. Berman
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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 $\mathbb{P}^n_{\mathbb{Z}}$ 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.

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Rolf Andreasson, Robert J. Berman. 2024-03-04. Sharp bounds on the height of K-semistable Fano varieties II, the log case. https://arxiv.org/abs/2312.05064

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