Toric Mirror Symmetry for Homotopy Theorists
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917889937768448 |
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| author | Bai, Qingyuan Hu, Yuxuan |
| author_facet | Bai, Qingyuan Hu, Yuxuan |
| contents | We construct functors sending torus-equivariant quasi-coherent sheaves on toric schemes over the sphere spectrum to constructible sheaves of spectra on real vector spaces. This provides a spectral lift of the toric homolgoical mirror symmetry theorem of Fang-Liu-Treumann-Zaslow (arXiv:1007.0053). Along the way, we obtain symmetric monoidal structures and functoriality results concerning those functors, which are new even over a field $k$. We also explain how the `non-equivariant' version of the theorem would follow from this functoriality via the de-equivariantization technique. As a concrete application, we obtain an alternative proof of Beilinson's linear algebraic description of quasi-coherent sheaves on projective spaces with spectral coefficients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_06649 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Toric Mirror Symmetry for Homotopy Theorists Bai, Qingyuan Hu, Yuxuan Algebraic Geometry Algebraic Topology Category Theory We construct functors sending torus-equivariant quasi-coherent sheaves on toric schemes over the sphere spectrum to constructible sheaves of spectra on real vector spaces. This provides a spectral lift of the toric homolgoical mirror symmetry theorem of Fang-Liu-Treumann-Zaslow (arXiv:1007.0053). Along the way, we obtain symmetric monoidal structures and functoriality results concerning those functors, which are new even over a field $k$. We also explain how the `non-equivariant' version of the theorem would follow from this functoriality via the de-equivariantization technique. As a concrete application, we obtain an alternative proof of Beilinson's linear algebraic description of quasi-coherent sheaves on projective spaces with spectral coefficients. |
| title | Toric Mirror Symmetry for Homotopy Theorists |
| topic | Algebraic Geometry Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2501.06649 |