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

We show that odd-dimensional projective varieties with tilting objects and only ADE-singularities are nodal, i.e. they only have A₁-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.