Journal of Algebraic Geometry

Numerical bounds for semi-stable families of curves or of certain higher-dimensional manifolds

Author(s): Eckart Viehweg; Kang Zuo
Journal: J. Algebraic Geom. 15 (2006), 771-791.
Posted: November 30, 2005
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Abstract: Given an open subset

of a projective curve

and a smooth family $f:V\to U$ of curves, with semi-stable reduction over

, we show that for a subvariation $\mathbb{V}$ of Hodge structures of $R^1f_*\mathbb{C}_V$ with ${\rm rank} (\mathbb{V})>2$ the Arakelov inequality must be strict. For families of

-folds we prove a similar result under the assumption that the

component of the Higgs bundle of $\mathbb{V}$ defines a birational map.

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Eckart Viehweg
Affiliation: Universität Duisburg-Essen, Mathematik, 45117 Essen, Germany
Email: viehweg@uni-essen.de

Kang Zuo
Affiliation: Universität Mainz, Fachbereich 17, Mathematik, 55099 Mainz, Germany
Email: kzuo@mathematik.uni-mainz.de

PII: S 1056-3911(05)00423-6
Received by editor(s): April 26, 2005
Received by editor(s) in revised form: June 21, 2005
Posted: November 30, 2005
Additional Notes: This work has been supported by the ``DFG-Schwerpunktprogramm Globale Methoden in der Komplexen Geometrie'', and by the DFG-Leibniz program

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