摘要翻译：
考虑拟射影流形U上G维阿贝尔簇F:a-->U。假定从U到极化阿贝尔簇模格式的诱导映射是泛型有限的，且存在射影流形Y,包含U作为一个法交因子S的补数，使得对数1形式的簇是nef并且它的行列式相对于U是充足的。我们用Hodge结构的变化所附的数值数据来刻画$U$是否是Shimura变体，而不是用从U到模格式的映射的性质或CM点的存在性来刻画。更确切地说，我们证明U是Shimura变体，当且仅当两个条件成立。首先，Hodge结构的复权一变分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，对于切丛为复球的U的泛覆盖中的每个因子M，与V相关的迭代Kodaira-Spencer映射在M方向上具有最小的可能长度。
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英文标题：
《Stability of Hodge bundles and a numerical characterization of Shimura
  varieties》
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作者：
Martin Moeller, Eckart Viehweg, Kang Zuo
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最新提交年份：
2009
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics        数学
二级分类：Complex Variables        复变数
分类描述：Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves
全纯函数，自守群作用与形式，伪凸性，复几何，解析空间，解析束
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英文摘要：
  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.
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PDF链接：
https://arxiv.org/pdf/0706.3462