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Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. (English) Zbl 0848.14004
A normal crossing variety is a reduced complex analytic space which is locally isomorphic to a normal crossing divisor on a smooth variety. Moreover, it is called a simple normal crossing variety if the irreducible components are smooth. According to R. Friedman [Ann. Math., II. Ser. 118, 75-114 (1983; Zbl 0569.14002)], we can define the concept of \(d\)-semistability for simple normal crossing varieties. For example, the central fiber of a semistable degeneration is a \(d\)-semistable simple normal crossing variety. Friedman’s theorem (loc. cit.) states that for an arbitrary \(d\)-semistable simple normal crossing compact Kähler surface \(X\) with trivial \(K_X\) and \(H^1(X, {\mathcal O}_X)\), there exists a semistable degeneration \(f'\) as above, i.e., the \(d\)-semistable K3 surface \(X\) has a smoothing. The purpose of this paper is to give an alternative easy proof of Friedman’s theorem, and generalize it to higher dimensional varieties.

MSC:
14D15 Formal methods and deformations in algebraic geometry
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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