A cone conjecture for log Calabi-Yau surfaces
We consider log Calabi-Yau surfaces $(Y, D)$ with singular boundary. In each deformation type, there is a distinguished surface $(Y_e,D_e)$ such that the mixed Hodge structure on $H_2(Y \setminus D)$ is split. We prove that (1) the action of the automorphism group of $(Y_e,D...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000902/type/journal_article |
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| Summary: | We consider log Calabi-Yau surfaces
$(Y, D)$
with singular boundary. In each deformation type, there is a distinguished surface
$(Y_e,D_e)$
such that the mixed Hodge structure on
$H_2(Y \setminus D)$
is split. We prove that (1) the action of the automorphism group of
$(Y_e,D_e)$
on its nef effective cone admits a rational polyhedral fundamental domain; and (2) the action of the monodromy group on the nef effective cone of a very general surface in the deformation type admits a rational polyhedral fundamental domain. These statements can be viewed as versions of the Morrison cone conjecture for log Calabi–Yau surfaces. In addition, if the number of components of D is no greater than six, we show that the nef cone of
$Y_e$
is rational polyhedral and describe it explicitly. This provides infinite series of new examples of Mori Dream Spaces. |
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| ISSN: | 2050-5094 |