Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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


Authors: Eckart Viehweg and Kang Zuo
Journal: J. Algebraic Geom. 15 (2006), 771-791
DOI: https://doi.org/10.1090/S1056-3911-05-00423-6
Published electronically: November 30, 2005
MathSciNet review: 2237270
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Abstract | References | Additional Information

Abstract: Given an open subset $ U$ of a projective curve $ Y$ and a smooth family $ f:V\to U$ of curves, with semi-stable reduction over $ Y$, 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 $ n$-folds we prove a similar result under the assumption that the $ (n,0)$ component of the Higgs bundle of $ \mathbb{V}$ defines a birational map.


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Additional Information

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

DOI: https://doi.org/10.1090/S1056-3911-05-00423-6
Received by editor(s): April 26, 2005
Received by editor(s) in revised form: June 21, 2005
Published electronically: 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

American Mathematical Society