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".


1 Answer 1


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.

  • $\begingroup$ Thank you very much! In his paper Fujisawa has assumed semi-stable of the family. For base space of dimension greater than one, (currently) there is no semi-stable reduction theorem like the dimension one case. I am curious what are the obstructions? $\endgroup$
    – Wenzhe
    May 14, 2018 at 11:24
  • $\begingroup$ Fujisawa has a sequel to the Compositio paper: arxiv.org/abs/1506.02271 $\endgroup$
    – Joe Berner
    Sep 4, 2019 at 13:06

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