The effective cone conjecture for Calabi--Yau pairs
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866910023656931328 |
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| author | Gachet, Cécile Lin, Hsueh-Yung Stenger, Isabel Wang, Long |
| author_facet | Gachet, Cécile Lin, Hsueh-Yung Stenger, Isabel Wang, Long |
| contents | We formulate an effective cone conjecture for klt Calabi--Yau pairs $(X,Δ)$, pertaining to the structure of the cone of effective divisors $\mathrm{Eff}(X)$ modulo the action of the subgroup of pseudo-automorphisms $\mathrm{PsAut}(X,Δ)$. Assuming the existence of good minimal models in dimension $\dim(X)$, known to hold in dimension up to $3$, we prove that the effective cone conjecture for $(X,Δ)$ is equivalent to the Kawamata--Morrison--Totaro movable cone conjecture for $(X,Δ)$, among other statements. As an application, we show that the movable cone conjecture unconditionally holds for the smooth Calabi--Yau threefolds introduced by Schoen and studied by Namikawa, Grassi and Morrison. We also show that for such a Calabi--Yau threefold $X$, all of its minimal models, apart from $X$ itself, have rational polyhedral nef cones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07307 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The effective cone conjecture for Calabi--Yau pairs Gachet, Cécile Lin, Hsueh-Yung Stenger, Isabel Wang, Long Algebraic Geometry We formulate an effective cone conjecture for klt Calabi--Yau pairs $(X,Δ)$, pertaining to the structure of the cone of effective divisors $\mathrm{Eff}(X)$ modulo the action of the subgroup of pseudo-automorphisms $\mathrm{PsAut}(X,Δ)$. Assuming the existence of good minimal models in dimension $\dim(X)$, known to hold in dimension up to $3$, we prove that the effective cone conjecture for $(X,Δ)$ is equivalent to the Kawamata--Morrison--Totaro movable cone conjecture for $(X,Δ)$, among other statements. As an application, we show that the movable cone conjecture unconditionally holds for the smooth Calabi--Yau threefolds introduced by Schoen and studied by Namikawa, Grassi and Morrison. We also show that for such a Calabi--Yau threefold $X$, all of its minimal models, apart from $X$ itself, have rational polyhedral nef cones. |
| title | The effective cone conjecture for Calabi--Yau pairs |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2406.07307 |