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Sharp bounds on the height of K-semistable Fano varieties II, the log case
[Bornes optimales pour la hauteur des variétés de Fano K-semi-stables II : le cas logarithmique]
Rolf Andreasson1 ; Robert J. Berman1
1 Chalmers University of Technology and the University of Gothenburg, Chalmers tvärgata 3, SE-412 96 Göteborg, Sweden
Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 983-1018.

In our previous work we conjectured—inspired by an algebro-geometric result of Fujita—that the height of an arithmetic Fano variety $\mathcal{X}$ of relative dimension $n$ is maximal when $\mathcal{X}$ 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.

Dans un travail antérieur, nous avons conjecturé — en nous inspirant d’un résultat algébro-géométrique de Fujita — que la hauteur d’une variété de Fano arithmétique $\mathcal{X}$ de dimension relative $n$ est maximale lorsque $\mathcal{X}$ est l’espace projectif $\mathbb{P}_{\mathbb{Z}}^{n}$ sur les entiers, muni de la métrique de Fubini-Study, à condition que la variété de Fano complexe correspondante soit K-semi-stable. Dans ce travail, nous démontrons cette conjecture pour les hypersurfaces diagonales dans $\mathbb{P}_{\mathbb{Z}}^{n+1}$. La démonstration repose sur une extension logarithmique de notre conjecture précédente — d’un intérêt indépendant — que nous établissons pour les variétés de Fano logarithmiques toriques de dimension relative au plus $3$, les arrangements d’hyperplans dans $\mathbb{P}_{\mathbb{Z}}^{n}$, ainsi que pour les surfaces de Fano arithmétiques orbifoldes générales.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.304
Classification : 14G40, 32Q20, 53C25, 11G50, 14J45
Keywords: Arakelov geometry, heights, Kähler–Einstein metrics, Fano varieties, K-stability
Mots-clés : Géométrie d’Arakelov, hauteurs, métriques de Kähler–Einstein, variétés de Fano, K-stabilité

Rolf Andreasson 1 ; Robert J. Berman 1

1 Chalmers University of Technology and the University of Gothenburg, Chalmers tvärgata 3, SE-412 96 Göteborg, Sweden
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Rolf Andreasson; Robert J. Berman. Sharp bounds on the height of K-semistable Fano varieties II, the log case. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 983-1018. doi : 10.5802/jep.304. https://jep.centre-mersenne.org/articles/10.5802/jep.304/

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