Mixed Hodge structures on log smooth degenerations

Abstract

We introduce the notion of a log smooth degeneration, which is a logarithmic analogue of the central fiber of some kind of degenerations of complex manifolds over polydiscs. Under suitable conditions, we construct a natural cohomological mixed Hodge complex on the reduction of a compact log smooth degeneration. In particular, we obtain mixed Hodge structures on the log de Rham cohomologies and $E_1$-degeneration of the log Hodge to de Rham spectral sequence for a certain kind of compact reduced log smooth degenerations.