Stability condition on a singular surface and its resolution
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911183039102976 |
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| author | Chou, Tzu-Yang |
| author_facet | Chou, Tzu-Yang |
| contents | Let $X$ be a surface with an ADE-singularity and let $\widetilde{X}$ be its crepant resolution. In this paper, we show that there exists a Bridgeland stability condition $σ_X$ on ${\rm D}^b(X)$ and a weak stability condition $σ_{\widetilde{X}}$ on the derived category of the desingularisation ${\rm D}^b(\widetilde{X})$, such that pushforward of $σ_{\widetilde{X}}$-semistable objects are $σ_X$-semistable
We first construct Bridgeland stability conditions on ${\rm D}^b(\widetilde{X})$ associated to the contraction $\widetilde{X} \longrightarrow X$, generalizing the results of Tramel and Xia in \cite{TX22}, Then we deform it to a weak stability condition $σ_{\widetilde{X}}$ and show that it descends to ${\rm D}^b(X)$, producing the stability condition $σ_X$.
Finally, we study the moduli spaces of $σ_{π^\ast H,β,z}$, of $σ_{\widetilde{X}}$, and of $σ_X$-semistable objects, and we show that the moduli spaces satisfy boundedness and openness, and hence are all Artin stacks of finite type over $\mathbb{C}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19768 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability condition on a singular surface and its resolution Chou, Tzu-Yang Algebraic Geometry 14F08 (Primary) 14J17, 18E40, 18G80 (Secondary) Let $X$ be a surface with an ADE-singularity and let $\widetilde{X}$ be its crepant resolution. In this paper, we show that there exists a Bridgeland stability condition $σ_X$ on ${\rm D}^b(X)$ and a weak stability condition $σ_{\widetilde{X}}$ on the derived category of the desingularisation ${\rm D}^b(\widetilde{X})$, such that pushforward of $σ_{\widetilde{X}}$-semistable objects are $σ_X$-semistable We first construct Bridgeland stability conditions on ${\rm D}^b(\widetilde{X})$ associated to the contraction $\widetilde{X} \longrightarrow X$, generalizing the results of Tramel and Xia in \cite{TX22}, Then we deform it to a weak stability condition $σ_{\widetilde{X}}$ and show that it descends to ${\rm D}^b(X)$, producing the stability condition $σ_X$. Finally, we study the moduli spaces of $σ_{π^\ast H,β,z}$, of $σ_{\widetilde{X}}$, and of $σ_X$-semistable objects, and we show that the moduli spaces satisfy boundedness and openness, and hence are all Artin stacks of finite type over $\mathbb{C}$. |
| title | Stability condition on a singular surface and its resolution |
| topic | Algebraic Geometry 14F08 (Primary) 14J17, 18E40, 18G80 (Secondary) |
| url | https://arxiv.org/abs/2411.19768 |