Hirsch weight-filtered log crystalline complex and Hirsch weight-filtered log crystalline dga of a proper SNCL scheme in characteristic p>0

Abstract

We construct a theory of the derived PD-Hirsch extension of the log crystalline complex of a log smooth scheme and we construct a fundamental filtered dga $(H_{\mathrm{zar},\mathrm{TW}},P)$ and a fundamental filtered complex $(H_{\mathrm{zar}},P)$ for a simple normal crossing log scheme $X$ over a family of log points by using the log crystalline method in order to overcome obstacles arising from the incompatibility of the p-adic Steenbrink complexes in [M] and [Nak4] with the cup product of the log crystalline complex of $X$. When the base log scheme is the log point of a perfect field of characteristic $p>0$, we prove that $(H_{\mathrm{zar},\mathrm{TW}},P)$ and $(H_{\mathrm{zar}},P)$ is canonically isomorphic to Kim and Hain's filtered dga and their filtered complex in [KH], respectively.

165 pages. arXiv admin note: text overlap with arXiv:1902.00182