Symplectomorphisms and spherical objects in the conifold smoothing
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911589465063424 |
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| author | Keating, Ailsa Smith, Ivan |
| author_facet | Keating, Ailsa Smith, Ivan |
| contents | Let $X$ denote the `conifold smoothing', the symplectic Weinstein manifold which is the complement of a smooth conic in $T^*S^3$, or equivalently the plumbing of two copies of $T^*S^3$ along a Hopf link. Let $Y$ denote the `conifold resolution', by which we mean the complement of a smooth divisor in $\mathcal{O}(-1) \oplus \mathcal{O}(-1) \to \mathbb{P}^1$. We prove that the compactly supported symplectic mapping class group of $X$ splits off a copy of an infinite rank free group, in particular is infinitely generated; and we classify spherical objects in the bounded derived category $D(Y)$ (the three-dimensional `affine $A_1$-case'). Our results build on work of Chan-Pomerleano-Ueda and Toda, and both theorems make essential use of working on the `other side' of the mirror. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10525 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Symplectomorphisms and spherical objects in the conifold smoothing Keating, Ailsa Smith, Ivan Symplectic Geometry Algebraic Geometry 53D37, 14F08 Let $X$ denote the `conifold smoothing', the symplectic Weinstein manifold which is the complement of a smooth conic in $T^*S^3$, or equivalently the plumbing of two copies of $T^*S^3$ along a Hopf link. Let $Y$ denote the `conifold resolution', by which we mean the complement of a smooth divisor in $\mathcal{O}(-1) \oplus \mathcal{O}(-1) \to \mathbb{P}^1$. We prove that the compactly supported symplectic mapping class group of $X$ splits off a copy of an infinite rank free group, in particular is infinitely generated; and we classify spherical objects in the bounded derived category $D(Y)$ (the three-dimensional `affine $A_1$-case'). Our results build on work of Chan-Pomerleano-Ueda and Toda, and both theorems make essential use of working on the `other side' of the mirror. |
| title | Symplectomorphisms and spherical objects in the conifold smoothing |
| topic | Symplectic Geometry Algebraic Geometry 53D37, 14F08 |
| url | https://arxiv.org/abs/2301.10525 |