Trace for the Du Bois complex

TL;DR

The paper addresses constructing a trace morphism for the Du Bois complex under finite morphisms between normal varieties. It proves that for a finite group quotient $\pi: X \to X'$, the composition $\underline{\Omega}_{X'}^{\bullet} \to \mathbf{R}\pi_{*} \underline{\Omega}_{X}^{\bullet} \to \mathbf{R}\underline{\Gamma}^{G} \mathbf{R}\pi_{*} \underline{\Omega}_{X}^{\bullet}$ is an isomorphism in the filtered derived category, with corresponding graded-piece isomorphisms $\underline{\Omega}_{X'}^{p} \to \mathbf{R}\pi_{*} \underline{\Omega}_{X}^{p} \to \mathbf{R}\underline{\Gamma}^{G} \mathbf{R}\pi_{*} \underline{\Omega}_{X}^{p}$ for all $p$. As corollaries, lcdef does not increase under finite quotients, and various higher Du Bois and higher rationality properties are preserved under finite morphisms, while Hodge–Du Bois numbers satisfy $\underline{h}^{p,q}(Y) \ge \underline{h}^{p,q}(X)$ for proper $X,Y$. The methods combine equivariant Du Bois theory, resolutions, and Grothendieck duality, with an inductive dimension argument that reduces to the smooth case via Du Bois’ theorem. These results clarify how singularities and Hodge-theoretic invariants behave under finite maps and provide a trace-theoretic framework for further study.

Abstract

We construct some version of the trace morphism between the Du Bois complexes, with applications towards the behavior of the local cohomological dimension and some Hodge theoretic aspects of singularities under finite morphisms.

Theorems & Definitions (13)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Corollary 1.4
• Remark 1.5
• Remark 1.6
• Proposition 2.1: Mustata-Popa22:Hodge-filtration-local-cohomology*Corollary 5.3
• Theorem 2.2
• Definition 2.3
• proof : Proof of Corollary \ref{['coro:trace']}