Combinatorial construction of symplectic 6-manifolds via bifibration structures
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912179833274368 |
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| author | Hayano, Kenta |
| author_facet | Hayano, Kenta |
| contents | A bifibration structure on a $6$-manifold is a map to either the complex projective plane $\mathbb{P}^2$ or a $\mathbb{P}^1$-bundle over $\mathbb{P}^1$, such that its composition with the projection to $\mathbb{P}^1$ is a ($6$-dimensional) Lefschetz fibration/pencil, and its restriction to the preimage of a generic $\mathbb{P}^1$-fiber is also a ($4$-dimensional) Lefschetz fibration/pencil. This object has been studied by Auroux, Katzarkov, Seidel, among others. From a pair consisting of a monodromy representation of a Lefschetz fibration/pencil on a $4$-manifold and a relation in a braid group, which are mutually compatible in an appropriate sense, we construct a bifibration structure on a closed symplectic $6$-manifold, producing the given compatible pair as its monodromies. We further establish methods for computing topological invariants of symplectic $6$-manifolds, including Chern numbers, from compatible pairs. Additionally, we provide an explicit example of a compatible pair, conjectured to correspond to a bifibration structure derived from the degree-$2$ Veronese embedding of the $3$-dimensional complex projective space. This example can be viewed as a higher-dimensional analogue of the lantern relation in the mapping class group of the four-punctured sphere. Our results not only extend the applicability of combinatorial techniques to higher-dimensional symplectic geometry but also offer a unified framework for systematically exploring symplectic $6$-manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04282 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Combinatorial construction of symplectic 6-manifolds via bifibration structures Hayano, Kenta Geometric Topology Symplectic Geometry A bifibration structure on a $6$-manifold is a map to either the complex projective plane $\mathbb{P}^2$ or a $\mathbb{P}^1$-bundle over $\mathbb{P}^1$, such that its composition with the projection to $\mathbb{P}^1$ is a ($6$-dimensional) Lefschetz fibration/pencil, and its restriction to the preimage of a generic $\mathbb{P}^1$-fiber is also a ($4$-dimensional) Lefschetz fibration/pencil. This object has been studied by Auroux, Katzarkov, Seidel, among others. From a pair consisting of a monodromy representation of a Lefschetz fibration/pencil on a $4$-manifold and a relation in a braid group, which are mutually compatible in an appropriate sense, we construct a bifibration structure on a closed symplectic $6$-manifold, producing the given compatible pair as its monodromies. We further establish methods for computing topological invariants of symplectic $6$-manifolds, including Chern numbers, from compatible pairs. Additionally, we provide an explicit example of a compatible pair, conjectured to correspond to a bifibration structure derived from the degree-$2$ Veronese embedding of the $3$-dimensional complex projective space. This example can be viewed as a higher-dimensional analogue of the lantern relation in the mapping class group of the four-punctured sphere. Our results not only extend the applicability of combinatorial techniques to higher-dimensional symplectic geometry but also offer a unified framework for systematically exploring symplectic $6$-manifolds. |
| title | Combinatorial construction of symplectic 6-manifolds via bifibration structures |
| topic | Geometric Topology Symplectic Geometry |
| url | https://arxiv.org/abs/2501.04282 |