Embeddings of symplectic balls into the complex projective plane
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909098294902784 |
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| author | Anjos, Sílvia Kędra, Jarek Pinsonnault, Martin |
| author_facet | Anjos, Sílvia Kędra, Jarek Pinsonnault, Martin |
| contents | We investigate spaces of symplectic embeddings of $n\leq 4$ balls into the complex projective plane. We prove that they are homotopy equivalent to explicitly described algebraic subspaces of the configuration spaces of $n$ points. We compute the rational homotopy type of these embedding spaces and their cohomology with rational coefficients. Our approach relies on the comparison of the action of $\mathrm{PGL}(3,\mathbb{C})$ on the configuration space of $n$ ordered points in $\mathbf{CP}^2$ with the action of the symplectomorphism group $\mathrm{Symp}(\mathbf{CP}^2)$ on the space of $n$ embedded symplectic balls. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_00556 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Embeddings of symplectic balls into the complex projective plane Anjos, Sílvia Kędra, Jarek Pinsonnault, Martin Symplectic Geometry Algebraic Geometry Primary 57K43, Secondary 57R17, 57S05, 57R40 We investigate spaces of symplectic embeddings of $n\leq 4$ balls into the complex projective plane. We prove that they are homotopy equivalent to explicitly described algebraic subspaces of the configuration spaces of $n$ points. We compute the rational homotopy type of these embedding spaces and their cohomology with rational coefficients. Our approach relies on the comparison of the action of $\mathrm{PGL}(3,\mathbb{C})$ on the configuration space of $n$ ordered points in $\mathbf{CP}^2$ with the action of the symplectomorphism group $\mathrm{Symp}(\mathbf{CP}^2)$ on the space of $n$ embedded symplectic balls. |
| title | Embeddings of symplectic balls into the complex projective plane |
| topic | Symplectic Geometry Algebraic Geometry Primary 57K43, Secondary 57R17, 57S05, 57R40 |
| url | https://arxiv.org/abs/2307.00556 |