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

TL;DR

This paper investigates obstructions to semiorthogonal decompositions in singular projective varieties, showing that certain geometric and categorical conditions restrict the types of singularities and tilting objects possible.

Contribution

It establishes new links between singularity categories, tilting objects, and quiver properties, revealing restrictions on the existence of semiorthogonal decompositions in singular varieties.

Findings

Odd-dimensional varieties with tilting objects are nodal.

Categorical obstructions prevent tilting objects in certain hypersurface singularities.

Quivers of cluster-tilting objects lack loops and 2-cycles.

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.