Mixed Hodge structures on log smooth degenerations

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I succeeded to construct "product" and "the trace map" on the relative log de Rham cohomology groups of a log deformation, which is a typical example of log smooth degeneration. Moreover, I proved that these data provided a polarization on mixed Hodge structures, where the polarization is not a bilinear form on the graded pieces by the weight filtration on the log de Rham cohomology groups, but a bilinear form on the whole log de Rham cohomology groups satisfying the appropriated properties on the graded pieces by the weight filtration.