Filtrations on the log de Rham complex

Abstract

Given a log smooth log scheme $X$ over $\operatorname{Spec} \mathbb{C}$, in this article we analyze and compare different filtrations defined on the log de Rham complex $\omega^{\bullet}_X$ associated to $X$. We mainly refer to the articles of Ogus ([23]), Danilov ([1]), Ishida ([16]). In this context, we analyze two filtrations on $\omega^{\bullet}_X$: the decreasing Ogus filtration $\tilde{L}^{\bullet}$, which is a sort of extension of the Deligne weight filtration $W_{\bullet}$ to log smooth log schemes over $\Spec \mathbb{C}$, and an increasing filtration, which we call the Ishida filtration and denote by $I_{\bullet}$, defined by using the Ishida complex $\tilde{\Omega}^{\bullet}_X$ of $X$. Moreover, we have the Danilov de Rham complex $\Omega^{\bullet}_X(\log D)$ with logarithmic poles along $D= X - X_{\mathrm{triv}}$ ($X_{\mathrm{triv}}$ being the trivial locus for the log structure on $X$), endowed with an increasing weight filtration (the Danilov weight filtration $\mathcal{W}_{\bullet}$). Then we prove that the Danilov de Rham complex $\Omega^{\bullet}_X(\log D)$ coincides with the log de Rham complex $\omega^{\bullet}_X$ and the Ishida filtration $I_{\bullet}$ (which is a globalization of the Danilov weight filtration $\mathcal W_{\bullet}$) coincides with the opposite Ogus filtration $\tilde{L}^{-\bullet}$.