Degeneration of varieties to simple normal crossings

Question

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?

Answer 1

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.