Open Access
September 2012 Stability of Hodge bundles and a numerical characterization of Shimura varieties
Martin Möller, Eckart Viehweg, Kang Zuo
J. Differential Geom. 92(1): 71-151 (September 2012). DOI: 10.4310/jdg/1352211224


Let $U$ be a connected non-singular quasi-projective variety and $f : A \to U$ a family of abelian varieties of dimension $g$. Suppose that the induced map $U \to \mathcal{A}_g$ is generically finite and there is a compactification $Y$ with complement $S = Y \backslash U$ a normal crossing divisor such that $\Omega_Y^1 (\log S)$ is nef and $\omega_Y (S)$ is ample with respect to $U.

We characterize whether $U$ is a Shimura variety by numerical data attached to the variation of Hodge structures, rather than by properties of the map $U \to \mathcal{A}_g$ or by the existence of CM points.

More precisely, we show that $f : A \to U$ is a Kuga fibre space, if and only if two conditions hold. First, each irreducible local subsystem $\mathbb{V}$ of $R_1 f_* \mathbb{C}_A$ is either unitary or satisfies the Arakelov equality. Second, for each factor $M$ in the universal cover of $U$ whose tangent bundle behaves like that of a complex ball, an iterated Kodaira-Spencer map associated with $V$ has minimal possible length in the direction of $M$. If in addition $f : A \to U$ is rigid, it is a connected Shimura subvariety of $\mathcal{A}_g$ of Hodge type.


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Martin Möller. Eckart Viehweg. Kang Zuo. "Stability of Hodge bundles and a numerical characterization of Shimura varieties." J. Differential Geom. 92 (1) 71 - 151, September 2012.


Published: September 2012
First available in Project Euclid: 6 November 2012

zbMATH: 06130614
MathSciNet: MR3003876
Digital Object Identifier: 10.4310/jdg/1352211224

Rights: Copyright © 2012 Lehigh University

Vol.92 • No. 1 • September 2012
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