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$$p$$-adic Hodge theory in the semi-stable reduction case. (English) Zbl 0937.14011
From the introduction: The $$p$$-adic Hodge theory is an analogue of the Hodge theory for a variety $$X$$ over a $$p$$-adic field $$K$$ (a complete discrete valuation field of mixed characteristic $$(0,p)$$ with perfect residue field) and it compares $$p$$-adic étale cohomology with the action of the Galois group and de Rham cohomology with some additional structures (depending on how good the reduction of the variety is). In the semi-stable reduction case, it was formulated by J.-M. Fontaine and U. Jannsen [cf. J.-M. Fontaine in: Périodes $$p$$-adiques, Sem. Bures-sur-Yvette 1988, Astérisque 223, 69-111 (1994) and 113-183 (1994; Zbl 0865.14009)] as a conjecture (called the semi-stable conjecture or $$C_{\text{st}}$$ for short), and it asserts that $$p$$-adic étale cohomology with the action of the Galois group and de Rham cohomology with the Hodge filtration and certain additional structures coming from log crystalline cohomology of the special fiber can be constructed from each other. Here is given a survey of the statement and the proof [K. Kato, ibid., Astérisque 223, 269-293 (1994; Zbl 0847.14009) and T. Tsuji, Invent. Math. 137, No. 2, 233-411 (1999)] of $$C_{\text{st}}$$, generalizing it to truncated simplicial schemes. Thanks to the alteration of A. J. de Jong [Publ. Math., Inst. Hautes Étud. Sci. 83, 51-93 (1996; Zbl 0916.14005)], this generalization implies that the $$p$$-adic étale cohomology of any proper variety (which may have singularity) is potentially semi-stable. The details of the proof for simplicial schemes will be given elsewhere.

MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F20 Étale and other Grothendieck topologies and (co)homologies 14G20 Local ground fields in algebraic geometry 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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