Obstructions to semiorthogonal decompositions for singular projective varieties II: Representation theory
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Kalck, Martin Klapproth, Carlo Pavic, Nebojsa |
| author_facet | Kalck, Martin Klapproth, Carlo Pavic, Nebojsa |
| contents | We show that odd-dimensional projective varieties with tilting objects and only ADE-hypersurface singularities are nodal, i.e. they only have $A_1$-singularities. This is a very special case of more general obstructions to the existence of semiorthogonal decompositions for projective Gorenstein varieties. More precisely, for many isolated hypersurface singularities, we show that Kuznetsov-Shinder's categorical absorptions of singularities cannot contain tilting objects. The key idea is to compare singularity categories of projective varieties to singularity categories of finite-dimensional associative Gorenstein algebras. The former often contain special generators, called cluster-tilting objects, which typically have loops and $2$-cycles in their quivers. In contrast, quivers of cluster-tilting objects in the latter categories, can never have loops or $2$-cycles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07816 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Obstructions to semiorthogonal decompositions for singular projective varieties II: Representation theory Kalck, Martin Klapproth, Carlo Pavic, Nebojsa Algebraic Geometry Commutative Algebra Representation Theory 14F08, 14B05, 18G80, 13C14, 16G70 We show that odd-dimensional projective varieties with tilting objects and only ADE-hypersurface singularities are nodal, i.e. they only have $A_1$-singularities. This is a very special case of more general obstructions to the existence of semiorthogonal decompositions for projective Gorenstein varieties. More precisely, for many isolated hypersurface singularities, we show that Kuznetsov-Shinder's categorical absorptions of singularities cannot contain tilting objects. The key idea is to compare singularity categories of projective varieties to singularity categories of finite-dimensional associative Gorenstein algebras. The former often contain special generators, called cluster-tilting objects, which typically have loops and $2$-cycles in their quivers. In contrast, quivers of cluster-tilting objects in the latter categories, can never have loops or $2$-cycles. |
| title | Obstructions to semiorthogonal decompositions for singular projective varieties II: Representation theory |
| topic | Algebraic Geometry Commutative Algebra Representation Theory 14F08, 14B05, 18G80, 13C14, 16G70 |
| url | https://arxiv.org/abs/2404.07816 |