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In this paper, we prove compatibilities of various definitions of relatively unipotent log de Rham fundamental groups for certain proper log smooth integral morphisms of fine log schemes of characteristic zero. Our proofs are purely algebraic. As an application, we give a purely algebraic calculation of the monodromy action on the unipotent log de Rham fundamental group of a stable log curve. As a corollary we give a purely algebraic proof to the transcendental part of Andreatta–Iovita–Kim’s article: obtaining in this way a complete algebraic criterion for good reduction for curves.

The crystalline Chern classes of the value of a locally free crystal vanish on a smooth variety defined over a perfect field. Out of this we conclude new cases of de Jong’s conjecture relating the geometric étale fundamental group of a smooth projective variety defined over an algebraically closed field and the constancy of its category of isocrystals. We also discuss the case of the Gauß–Manin convergent isocrystal.

We correct some statements and proofs of K. S. Kedlaya [ Local and global structure of connections on nonarchimedean curves , Compos. Math. 151 (2015), 1096–1156]. To summarize, Proposition 1.1.2 is false as written, and we provide here a corrected statement and proof (and a corresponding modification of Remark 1.1.3); the proofs of Theorem 2.3.17 and Theorem 3.8.16, which rely on Proposition 1.1.2, are corrected accordingly; some missing details in the proofs of Theorem 3.4.20 and Theorem 3.4.22 are filled in; and a few much more minor corrections are recorded.

We construct a log algebraic version of the homotopy sequence for a normal crossing log variety over a log point of characteristic zero and prove some exactness properties of it. Our proofs are purely algebraic.

In this paper, we prove the blow-up invariance for Hodge-Witt sheaves with modulus, which is a generalization of a result of Koizumi for Witt sheaves and that of Kelly-Miyazaki and Koizumi for Hodge sheaves. As a consequence, we obtain the representability of Hodge-Witt sheaves with modulus in the category of motives with modulus under the assumption of resolution of singularities.

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.

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.

For open and singular varieties in positive characteristic p we study the existence of an integral p-adic cohomology theory which is finitely generated, compatible with log crystalline cohomology and rationally compatible with rigid cohomology. We develop such a theory under certain assumptions of resolution of singularities in positive characteristic, by using cdp- and cdh-topologies. Without resolution of singularities in positive characteristic, we prove the existence of a good p-adic cohomology theory for open and singular varieties in cohomological degree 1, by using split proper generically étale hypercoverings. This is a slight generalisation of a result due to Andreatta--Barbieri-Viale. We also prove that this approach does not work for higher cohomological degrees.

For open and singular varieties in positive characteristic p we study the existence of an integral p-adic cohomology theory which is finitely generated, compatible with log crystalline cohomology and rationally compatible with rigid cohomology. We develop such a theory under certain assumptions of resolution of singularities in positive characteristic, by using cdp- and cdh-topologies. Without resolution of singularities in positive characteristic, we prove the existence of a good p-adic cohomology theory for open and singular varieties in cohomological degree 1, by using split proper generically étale hypercoverings. This is a slight generalisation of a result due to Andreatta--Barbieri-Viale. We also prove that this approach does not work for higher cohomological degrees.