On higher Du Bois singularities and $K$-regularity

TL;DR

The paper develops a deep link between higher Du Bois singularities and $K$-regularity, using Hodge-theoretic tools (Du Bois complexes, derived de Rham, cotangent complex) and Bass–Weibel $K$-theory to relate A1-invariance violations to singularity structure. A central advance is a numerical criterion for $K_m$-regularity via the minimal exponent for hypersurfaces and a strengthened Vorst-type regularity result for affine local complete intersections in characteristic zero. It provides projective- and affine-setting characterizations of $K_m$-regularity through hypercohomology comparisons between the derived cotangent powers $\mathbb L_{X/F}^p$ and the Du Bois complexes $\underline{\Omega}_{X/F}^p$, and uses vanishing theorems to derive regularity criteria and Bass-invariant consequences. The work also develops constructive methods to produce new examples illustrating $K$-regularity phenomena and explores the Bass question through the Du Bois invariants $b^{p,q}$, including explicit computations for surfaces and homogeneous hypersurfaces, with several open questions left for future research.

Abstract

We study the relationship between higher Du Bois singularities and $K$-regularity, a notion that measures the $\mathbb{A}^1$-invariance of the algebraic $K$-groups. Building on this relationship, we establish a strengthened form of Vorst's conjecture for local complete intersections in characteristic zero. Our work also provides tools to construct new examples that illustrate various phenomena in the study of $K$-regularity. The main inputs for our results are vanishing theorems for the Du Bois complexes.

Theorems & Definitions (79)

• Theorem 1
• Example
• Theorem 2
• Proposition 3
• Theorem 4
• Corollary 5
• Theorem 6
• Proposition 7
• Proposition 8
• Theorem 9