A Hyperresolution-Free Characterization of the Deligne-Du Bois Complex

Abstract

Let $Y$ be a reduced, finite type scheme over $\mathbb{C}$, $X$ a closed subscheme of $Y$ and $\pi:\widetilde{Y} \to Y$ a projective morphism which is an isomorphism outside of $X$ with $E=(\pi^{-1}(X))_\text{red}$. In this paper, we provide a construction of the Deligne-Du Bois complex of $X$ in terms of the Deligne-Du Bois complexes of $Y,\widetilde{Y}$ and $E$. In the case that $Y$ is smooth and $\pi$ is a log resolution of $X$ in $Y$, this will provide a hyperresolution-free construction of $\underline{\Omega}^\bullet_X$ and its graded pieces.