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On the monodromy of families of algebraic surfaces. (Russian) Zbl 0566.14018

Konstr. Algebraicheskaya Geom., Sb. Nauchn. Tr. Yarosl. Pedagog. Inst. Im. K. D. Ushinskogo 194, 58-78 (1981).
From the authors’ introduction: This paper is a variation of the authors’ report at the Winter Mathematical School in Albegraic Geometry, Yaroslav 1979, and also of reports at the seminar on algebraic geometry at the Steklov Mathematical Institute, 1979. It contains an exposition of the results from the following papers: V. S. Kulikov, ”The exact Clemens-Schmid sequence and its application to degenerations of surfaces”, deposited in VINITI, No.2072-78 dep. (1978; see the preceding review); Vik. S. Kulikov, Math. USSR, Izv. 11, 957-989 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 41, 1008-1042 (1977; Zbl 0367.14014); U. Persson, Mem. Amer. Math. Soc. 189 (1977; Zbl 0368.14008), which demonstrate applications of the method of mixed Hodge structures to the study of degenerations of surfaces. Let \(\pi: X\to \Delta\) be a proper morphism from the complex manifold X to the unit disc \(\Delta =\{t\in {\mathbb{C}}| | t| <1\}\). Suppose that the fibres \(X_ t=\pi^{-1}(t)\), \(t\neq 0\), of this family are smooth n-dimensional manifolds. The fibre \(X_ 0=\pi^{-1}(0)\) (the so-called degenerate fiber) may have singularities. For the description of a degenerate fiber it is very important to know the connection between the numerical characteristics of the smooth and degenerate fibers. - In § 1 the authors build a sequence which connects the cohomology of the smooth and the degenerate fibers (the so-called Clemens-Schmid sequence). If X is a Kähler manifold, then the morphisms in this sequence are morphisms of mixed Hodge structures and the sequence is exact. In § 2 the authors recall how to define a mixed Hodge structure in terms of the Clemens- Schmid sequence and apply it to finding a connection between the numerical characters of the smooth and the degenerate fibers. § 4 contains a specialisation of this results in case of K 3 surfaces. All types of degenerate fibers and their connection with the monodromy are described.
Reviewer: V.Iliev

MSC:

14J10 Families, moduli, classification: algebraic theory
14D15 Formal methods and deformations in algebraic geometry
14D20 Algebraic moduli problems, moduli of vector bundles