Lagrangian multi-sections and their toric equivariant mirror
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866909124176904192 |
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| author | Oh, Yong-Geun Suen, Yat-Hin |
| author_facet | Oh, Yong-Geun Suen, Yat-Hin |
| contents | The SYZ conjecture suggests a folklore that "Lagrangian multi-sections are mirror to holomorphic vector bundles". In this paper, we prove this folklore for Lagrangian multi-sections inside the cotangent bundle of a vector space, which are equivariantly mirror to complete toric varieties by the work of Fang-Liu-Treumann-Zaslow. We also introduce the Lagrangian realization problem, which asks whether one can construct an unobstructed Lagrangian multi-section with asymptotic conditions prescribed by a tropical Lagrangian multi-section. We solve the realization problem for tropical Lagrangian multi-sections over a complete 2-dimensional fan that satisfy the so-called $N$-generic condition with $N\geq 3$. As an application, we show that every rank 2 toric vector bundle on the projective plane is mirror to a Lagrangian multi-section. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_12191 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Lagrangian multi-sections and their toric equivariant mirror Oh, Yong-Geun Suen, Yat-Hin Symplectic Geometry The SYZ conjecture suggests a folklore that "Lagrangian multi-sections are mirror to holomorphic vector bundles". In this paper, we prove this folklore for Lagrangian multi-sections inside the cotangent bundle of a vector space, which are equivariantly mirror to complete toric varieties by the work of Fang-Liu-Treumann-Zaslow. We also introduce the Lagrangian realization problem, which asks whether one can construct an unobstructed Lagrangian multi-section with asymptotic conditions prescribed by a tropical Lagrangian multi-section. We solve the realization problem for tropical Lagrangian multi-sections over a complete 2-dimensional fan that satisfy the so-called $N$-generic condition with $N\geq 3$. As an application, we show that every rank 2 toric vector bundle on the projective plane is mirror to a Lagrangian multi-section. |
| title | Lagrangian multi-sections and their toric equivariant mirror |
| topic | Symplectic Geometry |
| url | https://arxiv.org/abs/2211.12191 |