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arXiv · 2404.07816

Obstructions to semiorthogonal decompositions for singular projective varieties II: Representation theory

Martin Kalck·Carlo Klapproth·Nebojsa Pavic
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Abstract

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.

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Martin Kalck, Carlo Klapproth, Nebojsa Pavic. 2024-04-11. Obstructions to semiorthogonal decompositions for singular projective varieties II: Representation theory. https://arxiv.org/abs/2404.07816

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