Compactifications of the Eisenstein ancestral Deligne-Mostow variety
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911617243938816 |
|---|---|
| author | Hulek, Klaus Kondo, Shigeyuki Maeda, Yota |
| author_facet | Hulek, Klaus Kondo, Shigeyuki Maeda, Yota |
| contents | All arithmetic non-compact ball quotients by Deligne-Mostow's unitary monodromy group arise as sub-ball quotients of either of two spaces called ancestral cases, corresponding to Gaussian or Eisenstein Hermitian forms respectively. In a previous paper, we investigated the compactifications of the Gaussian Deligne-Mostow variety. Here we work on the remaining case, namely the ring of Eisenstein integers. This variety is related to the moduli space of unordered 12 points on $\mathbb{P}^1$. In particular, we show that Kirwan's partial resolution of the moduli space is not a semi-toroidal compactification and Deligne-Mostow's period map does not lift to the unique toroidal compactification. We give two interpretations of these phenomena in terms of the log minimal model program and automorphic forms. As an application, we prove that the above two compactifications are not (stacky) derived equivalent, as the $DK$-conjecture predicts. Furthermore, we construct an automorphic form on the moduli space of non-hyperelliptic curves of genus 4, which is isogenous to the Eisenstein Deligne-Mostow variety, giving another intrinsic proof, independent of lattice embeddings, of a result by Casalaina-Martin, Jensen and Laza. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18345 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Compactifications of the Eisenstein ancestral Deligne-Mostow variety Hulek, Klaus Kondo, Shigeyuki Maeda, Yota Algebraic Geometry Number Theory 14D20, 14G35, 11F03, 14E30, 14F08 All arithmetic non-compact ball quotients by Deligne-Mostow's unitary monodromy group arise as sub-ball quotients of either of two spaces called ancestral cases, corresponding to Gaussian or Eisenstein Hermitian forms respectively. In a previous paper, we investigated the compactifications of the Gaussian Deligne-Mostow variety. Here we work on the remaining case, namely the ring of Eisenstein integers. This variety is related to the moduli space of unordered 12 points on $\mathbb{P}^1$. In particular, we show that Kirwan's partial resolution of the moduli space is not a semi-toroidal compactification and Deligne-Mostow's period map does not lift to the unique toroidal compactification. We give two interpretations of these phenomena in terms of the log minimal model program and automorphic forms. As an application, we prove that the above two compactifications are not (stacky) derived equivalent, as the $DK$-conjecture predicts. Furthermore, we construct an automorphic form on the moduli space of non-hyperelliptic curves of genus 4, which is isogenous to the Eisenstein Deligne-Mostow variety, giving another intrinsic proof, independent of lattice embeddings, of a result by Casalaina-Martin, Jensen and Laza. |
| title | Compactifications of the Eisenstein ancestral Deligne-Mostow variety |
| topic | Algebraic Geometry Number Theory 14D20, 14G35, 11F03, 14E30, 14F08 |
| url | https://arxiv.org/abs/2403.18345 |