Sharp bounds on the height of K-semistable Fano varieties II, the log case
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914700498829312 |
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| author | Andreasson, Rolf Berman, Robert J. |
| author_facet | Andreasson, Rolf Berman, Robert J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_05064 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sharp bounds on the height of K-semistable Fano varieties II, the log case Andreasson, Rolf Berman, Robert J. Algebraic Geometry Differential Geometry Number Theory 14G40 (Primary), 32Q20, 53C25 (Secondary) 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. |
| title | Sharp bounds on the height of K-semistable Fano varieties II, the log case |
| topic | Algebraic Geometry Differential Geometry Number Theory 14G40 (Primary), 32Q20, 53C25 (Secondary) |
| url | https://arxiv.org/abs/2312.05064 |