Higher Du Bois and Higher Rational Pairs

TL;DR

This paper extends the concepts of higher Du Bois and higher rational singularities to pairs within the minimal model program, establishing key properties and technical results for these generalized notions.

Contribution

It introduces the extension of higher Du Bois and higher rational pairs, proving their properties and stability, and develops a generalized injectivity theorem for pairs.

Findings

Higher Du Bois and higher rational pairs are extended to pairs.

Numerous properties like Bertini theorems and stability are established.

A generalized Kovács-Schwede-type injectivity theorem for pairs is proved.

Abstract

We extend the notions of higher Du Bois and higher rational singularities to pairs in the sense of the minimal model program. We extend numerous results to these higher pairs, including Bertini type theorems, stability under finite maps and that m-rational pairs are m-Du Bois. We prove these using a generalized Kov\'acs-Schwede-type injectivity theorem for pairs, the main technical result of this paper.