Modular compactifications of moduli spaces for polarized K3 surfaces are constructed using the tools of logarithmic geometry in the sense of Fontaine and Illusie. The relationship between these new moduli spaces and the classical minimal and toroidal compactifications of period spaces is discussed, and it is explained how the techniques of this paper yield models for the latter spaces over number fields. The paper also contains a discussion of Picard functors for log schemes and a logarithmic version of Artin's method for proving representability by an algebraic stack.
"Semistable degenerations and period spaces for polarized K3 surfaces." Duke Math. J. 125 (1) 121 - 203, 1 October 2004. https://doi.org/10.1215/S0012-7094-04-12515-1