Symplectic submanifolds in dimension $6$ from hyperelliptic Lefschetz fibrations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915344069689344 |
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| author | Oba, Takahiro |
| author_facet | Oba, Takahiro |
| contents | We provide a closed, simply connected, symplectic $6$-manifold having infinitely many codimension $2$ symplectic submanifolds. These are mutually homologous but homotopy inequivalent, and furthermore, they cannot admit complex structures. The key ingredient for the construction is hyperelliptic Lefschetz fibrations on $4$-manifolds. As a corollary, we present a similar result on symplectic submanifolds of codimension $2$ in higher dimensions. In the appendix, we give a proof of the well-known fact that all symplectic submanifolds of codimension $2$ in $(\mathbb{CP}^3, ω_{\mathrm{FS}})$ of a fixed degree $\leq 3$ are mutually diffeomorphic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_12146 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Symplectic submanifolds in dimension $6$ from hyperelliptic Lefschetz fibrations Oba, Takahiro Symplectic Geometry Geometric Topology We provide a closed, simply connected, symplectic $6$-manifold having infinitely many codimension $2$ symplectic submanifolds. These are mutually homologous but homotopy inequivalent, and furthermore, they cannot admit complex structures. The key ingredient for the construction is hyperelliptic Lefschetz fibrations on $4$-manifolds. As a corollary, we present a similar result on symplectic submanifolds of codimension $2$ in higher dimensions. In the appendix, we give a proof of the well-known fact that all symplectic submanifolds of codimension $2$ in $(\mathbb{CP}^3, ω_{\mathrm{FS}})$ of a fixed degree $\leq 3$ are mutually diffeomorphic. |
| title | Symplectic submanifolds in dimension $6$ from hyperelliptic Lefschetz fibrations |
| topic | Symplectic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2302.12146 |