I am reading a paper ( Kawamata, Y.; Namikawa, Y. Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Invent. math. 1994, 118, 395–409.) I have a question about some affirmations:
Definition A normal crossing variety is a reduced complex analytic space which is locally isomorphic to a normal crossing divisor on a smooth variety. It is called simple normal crossing variety if the irreducible components are smooth.
Friedman defined the concept of d-semistability for simple normal crossing varieties.
Then we have this note:
Note In general, a simple normal crossing variety $X$ admits a logarithmic structure $\mathcal U$ if and only if it is $d$-semistable.
My question is: Is not true that for every normal crossing divisor we can define a logarithmic structure? (http://en.wikipedia.org/wiki/Log_structure) Then to a simple normal crossing variety $X$ we can give always a logarithmic structure right? (or only locally?) Then how come we have a characterization in terms of d-semistability? What I am missing?