Holomorphic curves in Stein domains and the tau-invariant
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911270831128576 |
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| author | Alfieri, Antonio Cavallo, Alberto |
| author_facet | Alfieri, Antonio Cavallo, Alberto |
| contents | The scope of the paper is threefold. First, we build on recent work by Hayden to compute Hedden's tau-invariant $τ_ξ(L)$ in the case when $ξ$ is a Stein fillable contact structure on a rational homology sphere, and $L$ is a transverse link arising as the boundary of a pseudo-holomorphic curve. This leads to a new proof of the relative Thom conjecture for Stein domains. Secondly, we compare the invariant $τ_ξ$ to the Grigsby-Ruberman-Strle topological tau-invariant $τ_{\mathfrak s}$, associated to the $\text{Spin}^c$-structure $\mathfrak s=\mathfrak s_ξ$ of the contact structure $ξ$, to obtain topological obstructions for a link type to admit a holomorphically fillable transverse representative. Finally, we use our main result together with methods from lattice cohomology to compute the $τ_{\mathfrak s}$-invariants of certain links in lens spaces, and estimate their PL slice genus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08657 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Holomorphic curves in Stein domains and the tau-invariant Alfieri, Antonio Cavallo, Alberto Geometric Topology Symplectic Geometry 57K18, 57K33, 57K43 The scope of the paper is threefold. First, we build on recent work by Hayden to compute Hedden's tau-invariant $τ_ξ(L)$ in the case when $ξ$ is a Stein fillable contact structure on a rational homology sphere, and $L$ is a transverse link arising as the boundary of a pseudo-holomorphic curve. This leads to a new proof of the relative Thom conjecture for Stein domains. Secondly, we compare the invariant $τ_ξ$ to the Grigsby-Ruberman-Strle topological tau-invariant $τ_{\mathfrak s}$, associated to the $\text{Spin}^c$-structure $\mathfrak s=\mathfrak s_ξ$ of the contact structure $ξ$, to obtain topological obstructions for a link type to admit a holomorphically fillable transverse representative. Finally, we use our main result together with methods from lattice cohomology to compute the $τ_{\mathfrak s}$-invariants of certain links in lens spaces, and estimate their PL slice genus. |
| title | Holomorphic curves in Stein domains and the tau-invariant |
| topic | Geometric Topology Symplectic Geometry 57K18, 57K33, 57K43 |
| url | https://arxiv.org/abs/2310.08657 |