德利涅-芒福德栈上非正则霍奇滤子的$E_1$退化
$E_1$-Degeneration for Irregular Hodge Filtrations on Deligne--Mumford Stacks
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中文总结 AI 辅助
针对粗空间拟射影的光滑分离德利涅-芒福德栈，利用Kontsevich复形的有限平展下降，将射影栈情形归约为Esnault–Sabbah–Yu的定理，证明其非正则霍奇滤子在所有有理指标处$E_1$退化，结果适用于各类栈紧化。
中文摘要 AI 辅助
设$(\ucCcal,w)$是$\bbC$上的光滑分离德利涅-芒福德栈，假设其粗空间拟射影。我们证明该栈的非正则霍奇滤子在所有有理指标处均为$E_1$退化，该结果对所有好的栈紧化及所有NC有理栈紧化均成立。特别地，可在任意此类紧化上计算该滤子及其分次维数。证明使用Kontsevich复形的有限平展下降，将射影栈情形归约为Esnault–Sabbah–Yu关于光滑射影簇的定理。
英文摘要
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.
