The higher Du Bois and higher rational properties for isolated singularities

TL;DR

This paper explores higher rational and Du Bois singularities for isolated cases, especially lci, providing characterizations, relations between invariants, and new proofs for existing theorems.

Contribution

It offers a complete characterization of higher Du Bois and rational singularities for isolated lci singularities and establishes new relations between their invariants.

Findings

$k$-rational isolated singularities are $k$-Du Bois.

$k$-Du Bois singularities are $(k-1)$-rational in the lci case.

Many invariants of isolated lci singularities vanish.

Abstract

Higher rational and higher Du Bois singularities have recently been introduced as natural generalizations of the standard definitions of rational and Du Bois singularities. In this note, we discuss these properties for isolated singularities, especially in the locally complete intersection (lci) case. First, we reprove the fact that a $k$-rational isolated singularity is $k$-Du Bois without any lci assumption. For isolated lci singularities, we give a complete characterization of the $k$-Du Bois and $k$-rational singularities in terms of standard invariants of singularities. In particular, we show that $k$-Du Bois singularities are $(k-1)$-rational for isolated lci singularities. In the course of the proof, we establish some new relations between invariants of isolated lci singularities and show that many of these vanish. The methods also lead to a quick proof of an inversion of adjunction theorem in the isolated lci case. Finally, we discuss some results specific to the hypersurface case.