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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2605.28624 |
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| _version_ | 1866914608623648768 |
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| author | Pirozhkov, Dmitrii |
| author_facet | Pirozhkov, Dmitrii |
| contents | Let $X$ be a smooth proper variety over an algebraically closed field of characteristic zero, and let $\mathcal{A} \subset D^{b}_{\mathrm{coh}}(X)$ be an admissible subcategory. Let $Z \subset X$ be the union of set-theoretical supports of all objects in $\mathcal{A}$ and assume that $Z \neq X$. We show that for any morphism from $Z$ to an abelian variety each fiber has no isolated points; this implies, for example, that $Z$ cannot be isomorphic to an abelian variety. The key input is the fact that while not all line bundles on $Z$ lift to infinitesimal thickenings of $Z$, sufficiently many do: specifically, we show that for any infinitesimal thickening $Z \subset \widetilde{Z}$ the restriction morphism $\mathrm{Pic}^0(\widetilde{Z}) \to \mathrm{Pic}^0(Z)$ on the connected components of Picard schemes induces an isogeny between Albanese group schemes of those connected components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28624 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the support of admissible subcategories Pirozhkov, Dmitrii Algebraic Geometry Let $X$ be a smooth proper variety over an algebraically closed field of characteristic zero, and let $\mathcal{A} \subset D^{b}_{\mathrm{coh}}(X)$ be an admissible subcategory. Let $Z \subset X$ be the union of set-theoretical supports of all objects in $\mathcal{A}$ and assume that $Z \neq X$. We show that for any morphism from $Z$ to an abelian variety each fiber has no isolated points; this implies, for example, that $Z$ cannot be isomorphic to an abelian variety. The key input is the fact that while not all line bundles on $Z$ lift to infinitesimal thickenings of $Z$, sufficiently many do: specifically, we show that for any infinitesimal thickening $Z \subset \widetilde{Z}$ the restriction morphism $\mathrm{Pic}^0(\widetilde{Z}) \to \mathrm{Pic}^0(Z)$ on the connected components of Picard schemes induces an isogeny between Albanese group schemes of those connected components. |
| title | On the support of admissible subcategories |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2605.28624 |