On $k$-Du Bois and $k$-rational singularities

TL;DR

This paper introduces generalized notions of $k$-Du Bois and $k$-rational singularities, extending existing definitions beyond local complete intersections, and explores their stability and relationships.

Contribution

It extends the definitions of $k$-Du Bois and $k$-rational singularities to broader classes of varieties and studies their stability under hyperplane sections.

Findings

Varieties with $k$-rational singularities are $k$-Du Bois.

The notions depend only on higher cohomologies of Du Bois complexes.

Extensions include natural examples outside local complete intersections.

Abstract

We introduce new notions of $k$-Du Bois and $k$-rational singularities, extending the previous definitions in the case of local complete intersections (lci), to include natural examples outside of this setting. We study the stability of these notions under general hyperplane sections and show that varieties with $k$-rational singularities are $k$-Du Bois, extending previous results in [MP22b] and [FL22b] in the lci and the isolated singularities cases. In the process, we identify the aspects of the theory that depend only on the vanishing of higher cohomologies of Du Bois complexes (or related objects), and not on the behaviour of the K\"ahler differentials.