Semistable Reduction in Characteristic Zero for Families of Surfaces and Threefolds

We consider the problem of extending the semistable reduction theorem of [KKMS] from the case of one-parameter families of varieties to families over a base of arbitrary dimension. Following [KKMS], semistable reduction of such families can be reduced to a problem in the combinatorics of polyhedral complexes [AK]. In this paper we solve it in the case when the relative dimension of the morphism is at most three, i.e., for families of surfaces and threefolds.