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\(d\)-semistable Calabi-Yau threefolds of type III. (English) Zbl 1436.14075

This note presents new methods of constructing Calabi-Yau threefolds from normal crossing varieties and smoothing. More precisely, this note considers normal crossing varieties, smoothable to Calabi-Yau threefolds, whose dual complexes are two-dimensional. A \(d\)-semistable normal crossing variety \(Y\) is called a Calabi-Yau threefold of type III if it has a trivial dualizing sheaf, it is the central fiber of a semistable degeration of Calabi-Yau threefolds, and its dual complex is two-dimensional.
The main tool for constructing Calabi-Yau threefolds is the smoothing theorem of Y. Kawamata and Y. Namikawa [Invent. Math. 118, No. 3, 395–409 (1994; Zbl 0848.14004)]. Then focusing on normal crossing varieties with three components, together with \(d\)-semistable (also called as logarithmic structure) condition, more than fifty examples of Calabi-Yau threefolds are constructed, some of which are new.
Consider a normal crossing variety \(Y=Y_1\cup Y_2\cup Y_3\). The problem is to smooth \(Y\) to a Calabi-Yau manifold using the theorem of Kawamata-Namikawa. This is done under the \(d\)-semistability conidtion that the normal crossing variety \(Y\) is the central fiber in semistable degeneration. A prototypical result may be formulated as follows.
Proposition: The triviality of the collective normal class of \(Y\) implies the \(d\)-stability of \(Y\). Therefore, if the collective nomral class of \(Y\) is trivial, then \(Y\) is a Calabi-Yau manifold of type III.
Applying this proposition to more general normal crossing varieties produce new examples of Calabi-Yau threefolds of type III. Then homological invariants such as the Hodge numbers and Euler characteristic are computed. Some of these Calabi-Yau threefolds of type III are new not in the list of Calabi-Yau threefolds by toric constructions.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
32G20 Period matrices, variation of Hodge structure; degenerations

Citations:

Zbl 0848.14004
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References:

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