The effective cone conjecture for Calabi--Yau pairs

Research output: Working paper/Preprint › Preprint

Authors

  • Cécile Gachet
  • Hsueh-Yung Lin
  • Isabel Stenger
  • Long Wang

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Original languageEnglish
Publication statusE-pub ahead of print - 11 Jun 2024

Abstract

We formulate an {\it effective cone conjecture} for klt Calabi--Yau pairs $(X,\Delta)$, pertaining to the structure of the cone of effective divisors $\mathrm{Eff}(X)$ modulo the action of the subgroup of pseudo-automorphisms $\mathrm{PsAut}(X,\Delta)$. 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 $(X,\Delta)$ is equivalent to the Kawamata--Morrison--Totaro movable cone conjecture for $(X,\Delta)$. 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.

Keywords

    math.AG

Cite this

The effective cone conjecture for Calabi--Yau pairs. / Gachet, Cécile; Lin, Hsueh-Yung; Stenger, Isabel et al.
2024.

Research output: Working paper/Preprint › Preprint

Gachet, C, Lin, H-Y, Stenger, I & Wang, L 2024 'The effective cone conjecture for Calabi--Yau pairs'.
Gachet, C., Lin, H.-Y., Stenger, I., & Wang, L. (2024). The effective cone conjecture for Calabi--Yau pairs. Advance online publication.
Gachet C, Lin HY, Stenger I, Wang L. The effective cone conjecture for Calabi--Yau pairs. 2024 Jun 11. Epub 2024 Jun 11.
Gachet, Cécile ; Lin, Hsueh-Yung ; Stenger, Isabel et al. / The effective cone conjecture for Calabi--Yau pairs. 2024.
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abstract = " We formulate an {\it effective cone conjecture} for klt Calabi--Yau pairs $(X,\Delta)$, pertaining to the structure of the cone of effective divisors $\mathrm{Eff}(X)$ modulo the action of the subgroup of pseudo-automorphisms $\mathrm{PsAut}(X,\Delta)$. 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 $(X,\Delta)$ is equivalent to the Kawamata--Morrison--Totaro movable cone conjecture for $(X,\Delta)$. 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. ",
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AU - Lin, Hsueh-Yung

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AU - Wang, Long

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