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Stability of Hodge bundles and a numerical characterization of Shimura varieties

Martin Möller, Eckart Viehweg, Kang Zuo

Number 10
Authors Martin Möller
Eckart Viehweg
Kang Zuo
Year 2007

Let U be a connected quasi-projective manifold and f : A --> U a family of abelian varieties of dimension g. Suppose that the induced map U --> Ag is generically finite and there is a compactification Y with complement S = Y \ U a normal crossing divisor such that ΩY1 (log S) is nef and ωY(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 --> Ag or by the existence of CM points. More precisely, we show that f : A --> U is a Kuga fibre space, if and only if two conditions hold. First, each irreducible local subsystem V of R1f*CA either unitary or satisfies the Arakelov equality. Second, for each factor M in the universal cover of U whose tangent bundle behaves like the one 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 --> U is rigid, it will be a connected Shimura subvariety of Ag of Hodge type.

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