On the isotriviality of families of projective manifolds over curves

By E. Viehweg, K. Zuo. 2000

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Let Y be a projective non-singular curve of genus g, X a projective manifold, both defined over the field of complex numbers, and let f:X ---> Y be a surjective morphism with general fibre F. If the Kodaira dimension of X is non-negative, and if Y is the projective line we show that f has at least 3 singular fibres. In general, for non-isotrivial morphisms f, one expects that the number of singular fibres is at least 3, if g=0, or at least 1, if g=1. Using the strong additivity of the Kodaira dimension, this is verified, if either F is of general type, or if F has a minimal model with a semi-ample canonical divisor. The corresponding result has been obtained by Migliorini and Kovacs, for families of surfaces of general type and for families of canonically polarized manifolds, and by Oguiso-Viehweg for families of elliptic surfaces. As a byproduct we obtain explicit bounds for the degree of the direct image of powers of the dualizing sheaf, generalizing those obtained by Bedulev-Viehweg for families of surfaces of general type.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7b4c79f296914a22090b8684ab09f30fff38dc05

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Dynamics of automorphisms on projective complex manifolds (2008, methodology): “Proof. By [ 54 ] Theorem 0.2, f has at least three singular fibres lying over a set of points of P 1 on which g|P 1 permutes.”
Building blocks of étale endomorphisms of complex projective manifolds (2009, background): “3 follows from [54], since π has at least three singular fibers preserved by g.”
Kodaira dimension and zeros of holomorphic one-forms (2012, methodology): “In particular, there are no nontrivial smooth morphisms from a variety of general type to an abelian variety; this was proved by Viehweg and Zuo [VZ01] when the base is an elliptic curve and then by Hacon and Kov´acs [HK05] in general.”
Families of canonically polarized varieties over surfaces (2005, background): “Families over a curve have been studied by several authors in recent years and they are now fairly well understood—the strongest results known were obtained in [20,21], and [14].”