Constructions of symplectic forms on 4-manifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911791760539648 |
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| author | Lambert-Cole, Peter |
| author_facet | Lambert-Cole, Peter |
| contents | Given a symplectic 4-manifold $(X,ω)$ with rational symplectic form, Auroux constructed branched coverings to $(CP^2,ω_{FS})$. By modifying a previous construction of Lambert-Cole--Meier--Starkston, we prove that the branch locus in $CP^2$ can be assumed holomorphic in a neighborhood of the spine of the standard trisection of $CP^2$. Consequently, the symplectic 4-manifold $(X,ω)$ admits a cohomologous symplectic form that is Kähler in a neighborhood of the 2-skeleton of $X$. We define the Picard group of holomorphic line bundles over the holomorphic 2-skeleton. We then investigate Hodge theory and apply harmonic spinors to construct holomorphic sections over the Kähler subset. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05512 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructions of symplectic forms on 4-manifolds Lambert-Cole, Peter Geometric Topology Symplectic Geometry Given a symplectic 4-manifold $(X,ω)$ with rational symplectic form, Auroux constructed branched coverings to $(CP^2,ω_{FS})$. By modifying a previous construction of Lambert-Cole--Meier--Starkston, we prove that the branch locus in $CP^2$ can be assumed holomorphic in a neighborhood of the spine of the standard trisection of $CP^2$. Consequently, the symplectic 4-manifold $(X,ω)$ admits a cohomologous symplectic form that is Kähler in a neighborhood of the 2-skeleton of $X$. We define the Picard group of holomorphic line bundles over the holomorphic 2-skeleton. We then investigate Hodge theory and apply harmonic spinors to construct holomorphic sections over the Kähler subset. |
| title | Constructions of symplectic forms on 4-manifolds |
| topic | Geometric Topology Symplectic Geometry |
| url | https://arxiv.org/abs/2403.05512 |