This book is a long research paper. The aim of our research in this paper is to construct a theory of weights for the log crystalline cohomology sheaf of a proper or projective simple normal crossing log scheme in characteristic p > 0. This theory includes a p-adic analogue of the theory of the limit mixed Hodge structure on the cohomology of a proper or projective simple normal crossing log variety over the log point of the complex number field developed by J Steenbrink, T Fujisawa and C Nakayama. In this paper we focus on the compatibility of the cup product of the log crystalline cohomology sheaf of the log scheme with the weight filtration on it and we show the compatibility in the level of a new filtered complex. As an application of this compatibility, we prove the existence of the nondegenerate bilinear form on the primitive part of the log isocrystalline cohomology sheaf which is compatible with the weight filtration under the p-adic variational filtered log hard Lefschetz conjecture. The filtered derived PD-Hirsch extension associated to the log crystalline complex of the log scheme is a key technical ingredient in this paper. This filtered complex is the derived version of the filtered PD-Hirsch extension associated to a log de Rham-Witt complex defined by Kim-Hain.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets
