Semi-Local Exotic Lagrangian Tori in Dimension Four
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arXiv
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| Format: | Preprint |
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2024
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| author | Brendel, Joé Hauber, Johannes Schmitz, Joel |
| author_facet | Brendel, Joé Hauber, Johannes Schmitz, Joel |
| contents | We study exotic Lagrangian tori in dimension four. In certain Stein domains $B_{dpq}$ (which naturally appear in almost toric fibrations) we find $d+1$ families of monotone Lagrangian tori which are mutually distinct, up to symplectomorphisms. We prove that these remain distinct under embeddings of $B_{dpq}$ into geometrically bounded symplectic four-manifolds. We show that there are infinitely many different such embeddings when $X$ is compact and (almost) toric and hence conclude that $X$ contains arbitrarily many Lagrangian tori which are distinct up to symplectomorphisms of $X$. In dimension four arbitrarily many different Lagrangian tori were previously known only in del Pezzo surfaces.
Neither the embedded tori, nor the ambient space $X$ needs to be monotone for our methods to work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00408 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semi-Local Exotic Lagrangian Tori in Dimension Four Brendel, Joé Hauber, Johannes Schmitz, Joel Symplectic Geometry Algebraic Geometry 53D12 (Primary) 53D20, 14J17 (Secondary) We study exotic Lagrangian tori in dimension four. In certain Stein domains $B_{dpq}$ (which naturally appear in almost toric fibrations) we find $d+1$ families of monotone Lagrangian tori which are mutually distinct, up to symplectomorphisms. We prove that these remain distinct under embeddings of $B_{dpq}$ into geometrically bounded symplectic four-manifolds. We show that there are infinitely many different such embeddings when $X$ is compact and (almost) toric and hence conclude that $X$ contains arbitrarily many Lagrangian tori which are distinct up to symplectomorphisms of $X$. In dimension four arbitrarily many different Lagrangian tori were previously known only in del Pezzo surfaces. Neither the embedded tori, nor the ambient space $X$ needs to be monotone for our methods to work. |
| title | Semi-Local Exotic Lagrangian Tori in Dimension Four |
| topic | Symplectic Geometry Algebraic Geometry 53D12 (Primary) 53D20, 14J17 (Secondary) |
| url | https://arxiv.org/abs/2403.00408 |