Weight filtrations on log crystalline cohomologies of families of open smooth varieties
type: Item
instance of: scholarly article
title: Weight filtrations on log crystalline cohomologies of families of open smooth varieties
zbMATH Open document ID: 1187.14002
DOI: 10.1007/978-3-540-70565-9
author: Yukiyoshi Nakkajima
author: Atsushi Shiho
published in: Lecture Notes in Mathematics
publication date: 2008-08-28T00:00:00Z
review text: The main goal of this book is to construct a theory of weights for the log crystalline cohomology of families of open smooth varieties in characteristic \(p>0\), analogous to Deligne's theory of mixed Hodge structures assigned to families of open smooth varieties over the complex number field. As their basic results the authors list: \(p\)-adic purity, functoriality, the weight-filtered base change theorem, the weight filtered Künneth formula, convergence of the weight filtration, weight filtered Poincaré duality and the \(E_2\)-degeneration of \(p\)-adic weight spectral sequences. Let \(U\) be a smooth variety over the complex number field \({\mathbb C}\). Let \(X\) be a smooth variety over \({\mathbb C}\) with a simple normal crossings divisor \(D\) such that \(U=X-D\). Let \(j:U\to X\) be the natural open immersion. Set \(D^{(0)}=X\) and, for a positive integer \(k\), let \(D^{(k)}\) be the disjoint union of all \(k\)-fold intersections of the different irreducible components of \(D\). Consider the sheaf \(\Omega_{X/ {\mathbb C}}^i(\log(D))\) of differential \(i\)-forms on \(X\) with logarithmic poles along \(D\). Counting the number of logarithmic poles of local sections of \(\Omega_{X/ {\mathbb C}}^i(\log(D))\) defines the weight filtration \(P=\{P_k\}_{k\in{\mathbb Z}}\) on \(\Omega_{X/ {\mathbb C}}^i(\log(D))\). Let \(a^{(k)}:D^{(k)}\to X\) for \(k\in{\mathbb Z}_{\geq0}\) be the natural morphism of schemes over \({\mathbb C}\). Then we have the well known Poincaré isomorphism \[ \text{Res}:\text{gr}_k^P\Omega_{X/ {\mathbb C}}^{\bullet}(\log(D))\cong a_*^{(k)}(\Omega_{D^{(k)}/ {\mathbb C}}^{\bullet}\otimes \omega(D^{(k)}/{\mathbb C})(-k)) \] where \(\omega(D^{(k)}/{\mathbb C})\) is the orientation sheaf of \(D^{(k)}/ {\mathbb C}\). It induces the weight spectral sequence \[ E_1^{-k,h+k}=H^{h-k}(D^{(k)},\Omega_{D^{(k)}/ {\mathbb C}}^{\bullet}\otimes \omega(D^{(k)}/{\mathbb C}))(-k)\Rightarrow H^h(X,\Omega_{X/ {\mathbb C}}^{\bullet}(\log(D))). \] Moreover, in th…
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14F40
zbMATH DE Number: 5317211
zbMATH Keywords: log crystalline cohomology
zbMATH Keywords: weight filtration
reviewed by: Elmar Große-Klönne
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-540-70565-9
OpenAlex ID: W2504906153
description: scientific article; zbMATH DE number 5317211