Tropical and non-Archimedean Monge-Ampère equations for a class of Calabi-Yau hypersurfaces
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866909060383637504 |
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| author | Hultgren, Jakob Jonsson, Mattias Mazzon, Enrica McCleerey, Nicholas |
| author_facet | Hultgren, Jakob Jonsson, Mattias Mazzon, Enrica McCleerey, Nicholas |
| contents | For a class of maximally degenerate families of Calabi-Yau hypersurfaces of complex projective space, we study associated non-Archimedean and tropical Monge-Ampère equations, taking place on the associated Berkovich space, and the essential skeleton therein, respectively. For a symmetric measure on the skeleton, we prove that the tropical equation admits a unique solution, up to an additive constant. Moreover, the solution to the non-Archimedean equation can be derived from the tropical solution, and is the restriction of a continuous semipositive toric metric on projective space. Together with the work of Yang Li, this implies the weak metric SYZ conjecture on the existence of special Lagrangian fibrations in our setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_13697 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Tropical and non-Archimedean Monge-Ampère equations for a class of Calabi-Yau hypersurfaces Hultgren, Jakob Jonsson, Mattias Mazzon, Enrica McCleerey, Nicholas Differential Geometry Algebraic Geometry 14J32, 14J33, 32P05, 32Q25, 35J96, 53A15 (primary) 05E14, 14T90 (secondary) For a class of maximally degenerate families of Calabi-Yau hypersurfaces of complex projective space, we study associated non-Archimedean and tropical Monge-Ampère equations, taking place on the associated Berkovich space, and the essential skeleton therein, respectively. For a symmetric measure on the skeleton, we prove that the tropical equation admits a unique solution, up to an additive constant. Moreover, the solution to the non-Archimedean equation can be derived from the tropical solution, and is the restriction of a continuous semipositive toric metric on projective space. Together with the work of Yang Li, this implies the weak metric SYZ conjecture on the existence of special Lagrangian fibrations in our setting. |
| title | Tropical and non-Archimedean Monge-Ampère equations for a class of Calabi-Yau hypersurfaces |
| topic | Differential Geometry Algebraic Geometry 14J32, 14J33, 32P05, 32Q25, 35J96, 53A15 (primary) 05E14, 14T90 (secondary) |
| url | https://arxiv.org/abs/2208.13697 |