Derived PD-Hirsch extensions of filtered crystalline complexes and filtered crysalline dga's

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 $(H_{{\rm zar},{\rm TW}},P)$ and a fundamental filtered complex $(H_{\rm zar},P)$ for a simple normal crossing log scheme $X$ over a family 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 $(H_{{\rm zar},{\rm TW}},P)$ and $(H_{\rm zar},P)$ is canonically isomorphic to Kim and Hain's filtered dga and their filtered complex in [KH], respectively.Comment: 156 pages. arXiv admin note: text overlap with arXiv:1902.0018

Degenerations of log Hodge de Rham spectral sequences, log Kodaira vanishing theorem in characteristic p>0p>0 and log weak Lefschetz conjecture for log crystalline cohomologies

In this article we prove that the log Hodge de Rham spectral sequences of certain proper log smooth schemes of Cartier type in characteristic $p>0$ degenerate at $E_1$. We also prove that the log Kodaira vanishings for them hold when they are projective. We formulate the log weak Lefschetz conjecture for log crystalline cohomologies and prove that it is true in certain cases.Comment: 74 page

Weight-filtered convergent complex

Using log convergent topoi, %In the derived category of filtered complexes of %sheaves of modules over %an isostructure we define two fundamental filtered complexes $(E_{conv},P)$ and $(C_{conv},P)$ for the log scheme obtained by a smooth scheme with a relative simple normal crossing divisor over a scheme of characteristic $p>0$. Using $(C_{conv},P)$, we prove the $p$-adic purity. As a corollary of it, we prove that $(E_{conv},P)$ and $(C_{conv},P)$ are canonically isomorphic. These filtered complexes produce the weight spectral sequence of the log convergent cohomology sheaf of the log scheme. We also give the comparison theorem between the projections of $(E_{conv},P)$ and $(C_{conv},P)$ to the derived category of bounded below filtered complexes of sheaves of modules in the Zariski topos of the log scheme and the weight-filtered isozariskian filtered complex $(E_{zar},P)_{Q}$ of the log scheme defined in our previous book.Comment: 83page