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We construct a theory of the derived PD-Hirsch extension of the logcrystalline complex of a log smooth scheme and we construct a fundamentalfiltered dga $(H_{{\rm zar},{\rm TW}},P)$ and a fundamental filtered complex$(H_{\rm zar},P)$ for a simple normal crossing log scheme $X$ over a family oflog points by using the log crystalline method in order to overcome obstaclesarising 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 thebase log scheme is the log point of a perfect field of characteristic $p>0$, weprove that $(H_{{\rm zar},{\rm TW}},P)$ and $(H_{\rm zar},P)$ is canonicallyisomorphic to Kim and Hain's filtered dga and their filtered complex in [KH],respectively.

Hirsch weight-filtered crystalline complex and Hirsch weight-filtered crystalline dga of a proper SNCL scheme