Existence of logarithmic structures and d-semistability

Question

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?

Answer 1

It is true that if $X\subseteq Y$ is a normal crossings divisor, then $Y$ has a log structure whose sheaf of monoids is the sheaf of regular functions invertible outside of $X$. It is also true that this log structure can then be restricted to $X$, giving a log structure on $X$.

On the other hand, if we start with $X$ a normal crossings variety, we do not get a log structure on it by this construction. Indeed, we are not presented with $X$ as a normal crossings divisor inside a smooth variety $Y$. If, however, there is a variety $Y$ with a normal crossings morphism $\pi:Y\rightarrow S$, with $S$ a smooth dimension $1$ variety and $X$ a fibre of $\pi$, then $X$ does have a log structure by the above construction. This is exactly the log structure that Kawamata and Namikawa construct. Indeed, the condition of $d$-stability is necessary for there to exist such a $Y$.

In fact, $d$-semistability implies more than the existence of a log structure on $X$, as there are many non-$d$-semistable $X$ which carry log structures (e.g., any normal crossings $X\subseteq Y$ for $Y$ an arbitrary smooth variety). More importantly, $d$-semistability implies $X$ is log smooth over the standard log point, $Spec\, k$ with the monoid $k^* \times{\mathbb N}$. I don't have Kawamata and Namikawa's article in front of me, but if I recall they did not use the most recent language for log structures. The complete statement should be:

$X$ is $d$-semistable if and only if there is a log structure on $X$ along with a log morphism to the standard log point which is log smooth.

You can see a more precise discussion in Fumiharu Kato's paper "Log smooth deformation theory." In particular, the statement is Theorem 11.2 in the version on the arxiv.