Nearby Special Lagrangians
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866911212340510720 |
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| author | Abouzaid, Mohammed Imagi, Yohsuke |
| author_facet | Abouzaid, Mohammed Imagi, Yohsuke |
| contents | Let $X$ be a Calabi--Yau manifold and $Q\subset X$ a closed connected embedded special Lagrangian; closed Lagrangians mean compact Lagrangian submanifolds without boundary. We prove that if the fundamental group $π_1Q$ is abelian then there exists a Weinstein neighbourhood of $Q\subset X$ in which every closed irreducibly immersed special Lagrangian with unobstructed Floer cohomology is $C^1$ close to $Q.$ We prove also that if $π_1Q$ is virtually solvable then for every positive integer $R$ there exists a Weinstein neighbourhood of $Q\subset X$ in which every closed irreducibly immersed special Lagrangian of degree $\le R$ and with unobstructed Floer cohomology is unbranched; that is, the projection $L\to Q$ is a covering map. We prove a stronger statement when $π_1Q$ is finite and a weaker statement when $π_1Q$ has no non-abelian free subgroups. The $π_1Q$ conditions, the Floer cohomology condition and the special Lagrangian condition are all essential as we show by counterexamples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_10385 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Nearby Special Lagrangians Abouzaid, Mohammed Imagi, Yohsuke Symplectic Geometry Differential Geometry Let $X$ be a Calabi--Yau manifold and $Q\subset X$ a closed connected embedded special Lagrangian; closed Lagrangians mean compact Lagrangian submanifolds without boundary. We prove that if the fundamental group $π_1Q$ is abelian then there exists a Weinstein neighbourhood of $Q\subset X$ in which every closed irreducibly immersed special Lagrangian with unobstructed Floer cohomology is $C^1$ close to $Q.$ We prove also that if $π_1Q$ is virtually solvable then for every positive integer $R$ there exists a Weinstein neighbourhood of $Q\subset X$ in which every closed irreducibly immersed special Lagrangian of degree $\le R$ and with unobstructed Floer cohomology is unbranched; that is, the projection $L\to Q$ is a covering map. We prove a stronger statement when $π_1Q$ is finite and a weaker statement when $π_1Q$ has no non-abelian free subgroups. The $π_1Q$ conditions, the Floer cohomology condition and the special Lagrangian condition are all essential as we show by counterexamples. |
| title | Nearby Special Lagrangians |
| topic | Symplectic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2112.10385 |