Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916818408439808 |
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| author | Hicks, Jeff |
| author_facet | Hicks, Jeff |
| contents | We say that a tropical subvariety $V\subset \mathbb R^n$ is $B$-realizable if it can be lifted to an analytic subset of $(Λ^*)^n$. When $V$ is a smooth curve or hypersurface, there always exists a Lagrangian submanifold lift $L_V\subset (\mathbb C^*)^n$. We prove that whenever $L_V$ has well-defined Floer cohomology, we can find for each point of $V$ a Lagrangian torus brane whose Lagrangian intersection Floer cohomology with $L_V$ is non-vanishing. Assuming an appropriate homological mirror symmetry result holds for toric varieties, it follows that whenever $L_V$ is a Lagrangian submanifold that can be made unobstructed by a bounding cochain, the tropical subvariety $V$ is $B$-realizable. As an application, we show that the Lagrangian lift of a genus zero tropical curve is unobstructed, thereby giving a purely symplectic argument for Nishinou and Siebert's proof that genus-zero tropical curves are $B$-realizable. We also prove that tropical curves inside tropical abelian surfaces are $B$-realizable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_06432 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds Hicks, Jeff Symplectic Geometry Algebraic Geometry Combinatorics We say that a tropical subvariety $V\subset \mathbb R^n$ is $B$-realizable if it can be lifted to an analytic subset of $(Λ^*)^n$. When $V$ is a smooth curve or hypersurface, there always exists a Lagrangian submanifold lift $L_V\subset (\mathbb C^*)^n$. We prove that whenever $L_V$ has well-defined Floer cohomology, we can find for each point of $V$ a Lagrangian torus brane whose Lagrangian intersection Floer cohomology with $L_V$ is non-vanishing. Assuming an appropriate homological mirror symmetry result holds for toric varieties, it follows that whenever $L_V$ is a Lagrangian submanifold that can be made unobstructed by a bounding cochain, the tropical subvariety $V$ is $B$-realizable. As an application, we show that the Lagrangian lift of a genus zero tropical curve is unobstructed, thereby giving a purely symplectic argument for Nishinou and Siebert's proof that genus-zero tropical curves are $B$-realizable. We also prove that tropical curves inside tropical abelian surfaces are $B$-realizable. |
| title | Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds |
| topic | Symplectic Geometry Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2204.06432 |