Filtrations on the log de Rham complex

Given a log smooth log scheme χ over Spec ℂ, in this article we analyze and compare different filtrations defined on the log de Rham complex ωX• associated to X. We mainly refer to the articles of Ogus ([23]), Danilov ([1]), Ishida ([16]). In this context, we analyze two filtrations on ωx•: the decreasing Ogus filtration L̃•, which is a sort of extension of the Deligne weight filtration W• to log smooth log schemes over Spec ℂ, and an increasing filtration, which we call the Ishida filtration and denote by I•, defined by using the Ishida complex Ω̃X• of X. Moreover, we have the Danilov de Rham complex ΩX•(log D) with logarithmic poles along D = X - Xtriv (Xtriv being the trivial locus for the log structure on X), endowed with an increasing weight filtration (the Danilov weight filtration W•). Then we prove that the Danilov de Rham complex ΩX•(log D) coincides with the log de Rham complex ωX• and the Ishida filtration I• (which is a globalization of the Danilov weight filtration W•) coincides with the opposite Ogus filtration L̃-•.