Let $\mathcal{X}\to\Delta$, $\Delta \subset \mathbb{C}$ is the unit disk, be a smooth family of varieties whose fibers over $t\neq 0$ are smooth and the central fiber $\mathcal{X}\_0$ is a nice simple normal crossing divisor (in $\mathcal{X}$).


Let $\mathcal{X}\_0=\cup X\_i$, and define $X\_I=\cap\_{i\in I} X\_i$.


This assumption imposes some relations between the $N\_{X\_{I}}^{X\_{J}}$, $J\subset I$.
For example if $\mathcal{X}\_0=X\_1\cup\_D X\_2$ ($D=X\_{12}$), then $N\_D^{X\_1}\otimes N\_{D}^{X\_2}=\mathcal{O}\_D$ is trivial, and conversely if these condition holds, then there is a smooth one parameter family realizing that.


Question:


Is it known under what conditions, a simple normal crossing space can be realized as the central fiber of a smooth family? Is it known if those relations are enough for the existence of such family (similar to example above)


In general, is this probelm studied in literature or not? 



Degeneration of varieties to simple normal crossings
The more modern approach to the question adressed by Friedman is via
logarithmic geometry. Most relevant for your question is the paper of
Kawamata and Nammikawa, "Logarithmic deformations of normal crossing
varieties and smoothing of degenerate Calabi-Yau varieties," Invent. Math.,
118, (1994) 395-409. However, even that paper did not use the language
of log geometry as thoroughly as possible.


Briefly put, suppose we are given a normal crossings variety $X$. The
first step is to understand when one can put a log structure on $X$ of
the correct sort,
(I'll write the log scheme as $X^{\dagger}$) along with a log smooth
morphism $X^{\dagger}\rightarrow {\rm Spec} k^{\dagger}$, where the
latter is the "standard log point", i.e., a point with associated monoid
$k^{\times}\oplus {\bf N}$, where ${\bf N}$ denote the natural
numbers. Kawamata and Namikawa show that this can be done precisely
when Friedman's d-semistability condition holds, i.e., when the local
$T^1$ sheaf
$N\_D:={\mathcal Ext}^1(\Omega^1\_X,{\mathcal O}\_X)$ is the structure sheaf
of the singular locus of $X$.


One then applies log deformation theory, which was sketched out by
K. Kato in his original paper on logarithmic geometry, "Logarithmic structures
of Fontaine-Illusie," and fleshed out by F. Kato in
<http://arxiv.org/abs/alg-geom/9406004>


Kawamata and Namikawa in fact show the log deformation theory of a
normal crossings Calabi-Yau is unobstructed, using similar techniques
for proving the Bogomolov-Tian-Todorov theorem. So for d-semistable Calabi-Yau
varieties, the statement you want is true.


The advantage of using log deformation theory is that if one uses ordinary
deformation theory as Friedman did, it is not likely that the deformation
space will be unobstructed. Typically a normal crossings variety has
many locally trivial deformations which give a large irreducible component
of the deformation space, but most of these locally trivial deformations
do not smooth because they don't carry log structures. Log deformation theory
does not see these bad deformations.


