From the introduction: There are two things that quickly become clear in surveying the work done on the subject of Calabi-Yau threefolds, by which we mean a projective threefold \(X\) (with some specified class of possible singularities) with \(K_x=0\) and \(h^1({\mathcal O}_X)=0\). First, there are a huge number of such threefolds, even non-singular. Second, one reason there appears to be so many is this: Suppose you degenerate a non-singular Calabi-Yau \(X\) to a threefold \(X'\) with canonical singularities. If \(X'\) has a crepant desingularization \(\widetilde X\), then \(\widetilde X\) will not, in general, be in the same deformation family as \(X\) or even be diffeomorphic to \(X\).
Classification of these degenerations seems to be a hopeless problem. All we know, of course, is that there are only a finite number of families of such degenerate quintics. So, if we want to try to simplify the task of Calabi-Yau classification, it might help to concentrate on Calabi-Yau threefolds which do not arise as crepant resolutions of degenerations of other Calabi-Yau threefolds. This motivates:
Definition: A non-singular Calabi-Yau threefold \(\widetilde X\) is primitive if there is no birational contraction \(\widetilde X\to X\) with \(X\) smoothable to a Calabi-Yau threefold which is not deformation equivalent to \(\widetilde X\). Given a Calabi-Yau threefold \(\widetilde X\), when can we find a birational contraction morphism \(\widetilde X\to X\) such that \(X\) is smoothable? Now, any birational contraction will yield a threefold \(X\) with at worst canonical singularities. So if we want to begin to understand the smoothability of such threefolds, we first need to understand if they have obstructed deformation theory or not. Thus we have:
Question. Given a Calabi-Yau threefold \(X\) with canonical singularities, is \(\text{Def} (X)\) non-singular? If not, can we get some reasonable dimension estimates for components of \(\text{Def}(X)\)?
If \(X\) has canonical singularities \(\text{Def}(X)\) can indeed be singular. However, as we shall show in this paper, we can still control the dimension of components of \(\text{Def}(X)\) if \(X\) has canonical singularities. We have:
Theorem 3.8. Let \(\widetilde X\) be a non-singular Calabi-Yau threefold, and \(\pi:\widetilde X\to X\) be a birational contraction morphism, such that \(X\) has isolated complete intersection singularities. Then there is a deformation of \(X\) which smooths all singular points of \(X\) except possibly the ordinary double points of \(X\).
In the case that the isolated singularity is not a complete intersection, we have a much weaker statement. We include a rather artificial hypothesis on the singularities we will consider (see definition 4.2) called good.
Theorem 4.3. Let \(\widetilde X\) be a non-singular Calabi-Yau threefold and \(\pi: \widetilde X\to X\) a birational contraction such that \(X\) is \(\mathbb{Q}\)-factorial and for each \(P\in\text{Sing}(X)\), the \(\text{germ}(X,P)\) is good. Then \(X\) is smoothable.
Recall that a birational projective contraction \(\pi:\widetilde X\to X\) is primitive if it cannot be factored in the projective category. One application of theorem 4.3 given in section 5 is theorem 5.8:
Let \(\pi: \widetilde X\to X\) be a primitive contraction contracting a divisor \(E\) to a point. Then \(X\) is smoothable unless \(E\cong \mathbb{P}^2\) or \(F_1\).