Filtrations on the log de Rham complex
Fornasiero, Marianna
Osaka J. Math., Tome 44 (2007) no. 1, p. 285-304 / Harvested from Project Euclid
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 $\tilde{L}^{-\bullet}$.
Publié le : 2007-06-14
Classification:  14FXX
@article{1183667982,
     author = {Fornasiero, Marianna},
     title = {Filtrations on the log de Rham complex},
     journal = {Osaka J. Math.},
     volume = {44},
     number = {1},
     year = {2007},
     pages = { 285-304},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1183667982}
}
Fornasiero, Marianna. Filtrations on the log de Rham complex. Osaka J. Math., Tome 44 (2007) no. 1, pp.  285-304. http://gdmltest.u-ga.fr/item/1183667982/