Stability of Hodge bundles and a numerical characterization of Shimura varieties
Stability of Hodge bundles and a numerical characterization of Shimura varieties
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DOI：
10.4310/jdg/1352211224
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发表时间：
2007-06
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影响因子：
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通讯作者：
Martin Moeller;E. Viehweg;K. Zuo
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文献类型：
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作者：
Martin Moeller;E. Viehweg;K. Zuo
考虑拟投射流形U上的g维阿贝尔簇族f：A-> U。设从U到极化阿贝尔簇的模概型的诱导映射是一般有限的，并且存在一个射影流形Y，包含U作为正规交叉因子S的补，使得对数1形成的层是nef，并且它的行列式关于U是充分的。我们的特点是否$U $是一个志村品种的数值数据附加到霍奇结构的变化，而不是由属性的映射从U的模计划或CM点的存在。更准确地说，我们证明了U是一个志村品种，当且仅当两个条件成立。首先，Hodge结构的复权一变差的每个不可约局部子系统V要么是酉的，要么满足Arakelov等式。其次，对于U的泛覆盖中的每个因子M，其切丛的行为类似于复球的切丛，与V相关联的迭代Kodaira-Spencer映射在M方向上具有最小可能长度。
Consider a family f:A --> U of g-dimensional abelian varieties over a quasiprojective manifold U. Suppose that the induced map from U to the moduli scheme of polarized abelian varieties is generically finite and that there is a projective manifold Y, containing U as the complement of a normal crossing divisor S, such that the sheaf of logarithmic one forms is nef and that its determinant 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 from U to the moduli scheme or by the existence of CM points. More precisely, we show that U is a Shimura variety, if and only if two conditions hold. First, each irreducible local subsystem V of the complex weight one variation of Hodge structures is either unitary or satisfies the Arakelov equality. Secondly, 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.