Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations. Appendix by Arnaud Beauville. (English) Zbl 0743.32025

Catanese, Fabrizio

Invent. Math. 104, No. 2, 263-289 (1991); Appendix 289 (1991).

The central concern of this paper is the development of a higher dimensional analogue of the remarkable fact that the existence of a pencil of genus \(\geq 2\) on a compact complex surface \(S\) is a topological property of \(S\). More precisely, let \(X\) be an irregular compact Kähler manifold of dimension n and \(\alpha: X\to\text{Alb}(X)\) the Albanese morphism. The author defines \(X\) to be of Albanese general type if the image of \(\alpha\) has dimension n but \(\alpha\) is not surjective; it is easy to see that this property depends only on the complex cohomology of \(X\). The main result concerns morphisms \(f: X\to Y\), where \(X\) is an irregular compact Kähler manifold and \(Y\) is normal, of dimension \(k<n\), and has a smooth model which is of Albanese general type; it states that there is a bijection between the set of such morphisms and a certain set of real subspaces of \(H^ 1(X;\mathbb{C})\) (namely the saturated \(2k\)-wedge subspaces).
In later sections, the author gives applications to moduli of algebraic surfaces and considers the problem of stability under deformations of the existence of irrational pencils and higher dimensional analogues.
{For a fuller discussion of the motivation, ideas and methods of this interesting paper, see the author’s excellent introduction.}.

Reviewer: P.E.Newstead (Liverpool)

MSC:

32J27	Compact Kähler manifolds: generalizations, classification
32G13	Complex-analytic moduli problems
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
53C55	Global differential geometry of Hermitian and Kählerian manifolds
14D20	Algebraic moduli problems, moduli of vector bundles
14J15	Moduli, classification: analytic theory; relations with modular forms

Keywords:

Hodge theory; Albanese variety; Kähler manifold; moduli of algebraic surfaces; deformations; irrational pencils

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