Half-spherical twists on derived categories of coherent sheaves
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916752909139968 |
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| author | Arai, Hayato |
| author_facet | Arai, Hayato |
| contents | For a flat morphism $π\colon X \to T$ between smooth quasi-projective varieties and its fiber $X_0$, we prove that spherical objects on $D^b(X)$ pushed-forward from $D^b(X_0)$ induce autoequivalences of $D^b(X_0)$ itself. Our construction provides new derived symmetries for some singular varieties, which include singular fibers of elliptic surfaces (commonly referred to as Kodaira fibers) and type $III$ degenerations of K3 surfaces. In the case of Kodaira fibers of type $I_n$, we also show the induced autoequivalences of $D^b(X_0)$ correspond to the half twists on the $n$-punctured $2$-torus via homological mirror symmetry. As an application, we describe the autoequivalence groups of elliptic surfaces in terms of mapping class groups of punctured tori. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_12501 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Half-spherical twists on derived categories of coherent sheaves Arai, Hayato Algebraic Geometry For a flat morphism $π\colon X \to T$ between smooth quasi-projective varieties and its fiber $X_0$, we prove that spherical objects on $D^b(X)$ pushed-forward from $D^b(X_0)$ induce autoequivalences of $D^b(X_0)$ itself. Our construction provides new derived symmetries for some singular varieties, which include singular fibers of elliptic surfaces (commonly referred to as Kodaira fibers) and type $III$ degenerations of K3 surfaces. In the case of Kodaira fibers of type $I_n$, we also show the induced autoequivalences of $D^b(X_0)$ correspond to the half twists on the $n$-punctured $2$-torus via homological mirror symmetry. As an application, we describe the autoequivalence groups of elliptic surfaces in terms of mapping class groups of punctured tori. |
| title | Half-spherical twists on derived categories of coherent sheaves |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2302.12501 |