On compact analytic threefolds with non-trivial Albanese tori
Let \(f: V\to W\) be a fibre space (proper surjective morphism with connected fibres) of a n-dimensional compact connected manifold V over a m-dimensional compact connected manifold W and let denote by \(C_{n,m}\) the following conjecture: \(\kappa (V)\geq \kappa (W)+ \kappa (V_ x)\) where \(V_ x\) is a general fibre of f and \(\kappa\) ( ) means Kodaira dimension. The author proves conjecture \(C_{n,n-1}\) for fibre spaces of curves and conjecture \(C_{3,1}\) for fibre spaces of surfaces with positive Kodaira dimensions over a curve. For fibre spaces of surfaces with Kodaira dimension 0 the conjecture \(C_{3,1}\) is proved by using the theory of variation of Hodge structure except for the case when smooth fibres of f are isomorphic to complex tori of K3 surfaces and V is not in the class C (i.e. V is not bimeromorphically equivalent with a Kähler manifold). The results above are used to study the structure of the Albanese mappings of compact analytic threefolds with nonpositive Kodaira dimension. Let V be an analytic threefold and let \(\alpha\) : \(V\to A(V)\) be the Albanese mapping. Suppose \(\kappa (V)=0\). If \(\alpha\) is not surjective then \(\alpha\) (V) is a nonsingular curve of genus \(\geq 2\) and all the fibres of \(\alpha\) are connected. Smooth fibres are bimeromorphic to complex tori or K3 surfaces and almost all fibres of \(\alpha\) are nonalgebraic; V does not belong to \(class\quad C\). In case \(\alpha\) is surjective if \(\dim A(V)=3,\) then \(\alpha\) is bimeromorphic, if dim A(v)\(=2\) then \(\alpha\) is an elliptic threefold with constant moduli, if \(\dim A(V)=1\) then \(\alpha\) has connected fibres, a general fibre has Kodaira dimension 0 and if V belongs to the class C, then \(\alpha\) is bimeromorphic to an analytic fibre bundle. In the case \(\kappa (V)=- \infty\) let \(\beta\) : \(V\to W\), \(\gamma\) : \(W\to A(V)\) be the Stein factorization of \(\alpha\). Then \(\dim W\leq 2.\) If \(\dim W=2,\) the general fibres of \(\beta\) are \({\mathbb{P}}^ 1\). If dim \(W=1\), then \(\alpha\) (V) is a nonsingular curve and \(W=\alpha (V)\). General fibres of \(\beta\) are surfaces with \(\kappa =-\infty\) or bimeromorphic to complex tori or K3 surfaces. Moreover, if V belongs to the class C, general fibres of \(\beta\) are ruled surfaces. The author gives examples of analytic threefolds which show that each case above actually occur.
Albanese mappings of compact analytic threefolds
scientific article; zbMATH DE number 4021600
scientific article