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**Global smoothings of varieties with normal crossings.**
*(English)*
Zbl 0569.14002

The present paper considers varieties X with normal crossings and such that they can occur as the central fibre of a 1-parameter degeneration of smooth algebraic varieties \(X_ t\) (also called a smoothing of X). Let D be the double locus of X: the author shows that the sheaf \({\mathcal O}_ D(X)\) can be defined intrinsically without using the degeneration. Hence the triviality of such sheaf (d-semistability, in the terminology of the author) is a necessary condition for X to be smoothable. The author also shows how the limit Hodge structure of Schmid and Steenbrink can be defined intrinsically on these varieties (as a subsheaf of a logarithmic sheaf on the normalization of X). After some generalities, the author studies the deformation theory of these varieties, paying special attention to the deformations of X which give a smoothing [some of these d-semistable varieties have later been shown by U. Persson and H. Pinkham, Duke Math. J. 50, 477-486 (1983; Zbl 0529.14007) not to be smoothable].

A note worthy application is to the study of degenerations of K 3 surfaces where all the possibilities which could occur by the work of Kulikov-Persson-Pinkham are shown to occur effectively. In particular the author gives an explicit description of the versal deformation space: it is a normal crossing of two smooth varieties \(N_ 1\), \(N_ 2\), with \(N_ 1\cap N_ 2\) a divisor in \(N_ 2\), parametrizing equisingular d- semi-stable deformations; whereas the points of \(N_ 2-N_ 1\) parametrize smooth K 3’s, and \(N_ 1\) gives equisingular deformations.

A note worthy application is to the study of degenerations of K 3 surfaces where all the possibilities which could occur by the work of Kulikov-Persson-Pinkham are shown to occur effectively. In particular the author gives an explicit description of the versal deformation space: it is a normal crossing of two smooth varieties \(N_ 1\), \(N_ 2\), with \(N_ 1\cap N_ 2\) a divisor in \(N_ 2\), parametrizing equisingular d- semi-stable deformations; whereas the points of \(N_ 2-N_ 1\) parametrize smooth K 3’s, and \(N_ 1\) gives equisingular deformations.

Reviewer: F.Catanese

### MSC:

14D15 | Formal methods and deformations in algebraic geometry |

14J15 | Moduli, classification: analytic theory; relations with modular forms |

32G13 | Complex-analytic moduli problems |