# 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 --> *A*_{g} is generically finite and there is a compactification Y with complement S = Y \ U a normal crossing divisor such that Ω_{Y}^{1} (log S) is nef and ω_{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 --> *A*_{g} 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 R^{1}f_{*}C_{A} 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 *A*_{g} of Hodge type.