$E_1$-Degeneration for Irregular Hodge Filtrations on Deligne--Mumford Stacks

Abstract

Let $(\mathscr U,w)$ be a smooth separated Deligne--Mumford stack over $\mathbb{C}$. Assume that its coarse space is quasi-projective. We prove $E_1$-degeneration at every rational index for its irregular Hodge filtration. The result holds on every good stack compactification and every NC rational stack compactification. In particular, the filtration and its graded dimensions can be computed on any such compactification. The proof uses finite flat descent for Kontsevich complexes. It reduces the projective stack case to the theorem of Esnault--Sabbah--Yu for smooth projective varieties.