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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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000902/type/journal_article |
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| author | Jennifer Li |
| author_facet | Jennifer Li |
| author_sort | Jennifer Li |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-80697fff78934367ab002f639ca977ec |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-80697fff78934367ab002f639ca977ec2025-01-24T05:20:14ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.90A cone conjecture for log Calabi-Yau surfacesJennifer Li0https://orcid.org/0000-0003-4123-886XDepartment of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USAWe 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.https://www.cambridge.org/core/product/identifier/S2050509424000902/type/journal_article14J5014J32 |
| spellingShingle | Jennifer Li A cone conjecture for log Calabi-Yau surfaces Forum of Mathematics, Sigma 14J50 14J32 |
| title | A cone conjecture for log Calabi-Yau surfaces |
| title_full | A cone conjecture for log Calabi-Yau surfaces |
| title_fullStr | A cone conjecture for log Calabi-Yau surfaces |
| title_full_unstemmed | A cone conjecture for log Calabi-Yau surfaces |
| title_short | A cone conjecture for log Calabi-Yau surfaces |
| title_sort | cone conjecture for log calabi yau surfaces |
| topic | 14J50 14J32 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000902/type/journal_article |
| work_keys_str_mv | AT jenniferli aconeconjectureforlogcalabiyausurfaces AT jenniferli coneconjectureforlogcalabiyausurfaces |