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

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Title: SHARP BOUNDS ON THE HEIGHT OF K-SEMISTABLE FANO VARIETIES II, THE LOG CASE
Authors: Andreasson, Rolf, 1997, Berman, Robert, 1976
Source: Journal de l'Ecole Polytechnique - Mathematiques. 12:983-1018
Subject Terms: heights, Fano varieties, Arakelov geometry, hler-Einstein metrics, K-stability, K & auml
Description: 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.
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Access URL: https://research.chalmers.se/publication/547509
https://research.chalmers.se/publication/547509/file/547509_Fulltext.pdf
Database: SwePub
Description
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 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.
ISSN:24297100
2270518X
DOI:10.5802/jep.304