TITLE: A question on Steenbrink's paper, limit of Hodge structures
QUESTION [5 upvotes]: Steenbrink in his paper "Limit of Hodge Structures", (supplemented by the book "Mixed Hodge Structures" by Peters and Steenbrink) discuss the limit mixed Hodge structures for a fibration over the unit disc $\Delta \subset \mathbb{C}$, i.e. a fibration
$$\pi: X \rightarrow \Delta$$
whose fibers are projective varieties and the only singular fiber is over 0, which is a reduced divisor with smooth normal crossing components.
Question: Does Steenbrink's construction in his paper (or book) work in the general case i.e. for a fibration
$$\pi: X \rightarrow \Delta^n$$
I guess this also involves the issues with "simultaneous semi-stable reduction" of this family, which has been discussed in the paper "A tour of stable reduction with applications".

REPLY [3 votes]: If $\pi$ is semistable, then Fujisawa, Limits of Hodge structures in several variables. Compositio (1999), does this. You might also look at some later papers by the same author for some refinements.