Hodge结构变分的Arakelov-Nevanlinna不等式及其应用
Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications
摘要：
我们证明了一个第二主要定理型不等式的任何对数光滑投影对$(X,D)$，使$X\setminus D$支持一个复杂的极化变化的霍奇结构。这可以看作是一个Nevanlinna理论模拟的Arakelov不等式的变化的霍奇结构由于Deligne，彼得斯和Jost-Zuo。作为一个应用程序，我们得到在这种情况下，我们使用的一个著名的双曲性结果的纳德尔推导出一个巨大的推广的双曲性的标准。我们的证明的第一个成分是第二主要定理类型不等式的任何对数光滑投影对$(X,D)$，使$X\setminus D$支持的度量，其全纯截面曲率是有界的从上面的一个负常数。我们的证明的第二个成分是一个明确的约束上的全纯截面曲率的Griffiths-Schmid度量构建从一个变化的霍奇结构。作为我们的方法的副产品，我们还建立了一个第二主要定理类型的不等式对$(X,D)$，使$X\setminus D$是双曲嵌入在$X$。
英文摘要：
We prove a Second Main Theorem type inequality for any log-smooth projective pair $(X,D)$ such that $X\setminus D$ supports a complex polarized variation of Hodge structures. This can be viewed as a Nevanlinna theoretic analogue of the Arakelov inequalities for variations of Hodge structures due to Deligne, Peters and Jost-Zuo. As an application, we obtain in this context a criterion of hyperbolicity that we use to derive a vast generalization of a well-known hyperbolicity result of Nadel. The first ingredient of our proof is a Second Main Theorem type inequality for any log-smooth projective pair $(X,D)$ such that $X\setminus D$ supports a metric whose holomorphic sectional curvature is bounded from above by a negative constant. The second ingredient of our proof is an explicit bound on the holomorphic sectional curvature of the Griffiths-Schmid metric constructed from a variation of Hodge structures. As a byproduct of our approach, we also establish a Second Main Theorem type inequality for pairs $(X,D)$ such that $X\setminus D$ is hyperbolically embedded in $X$.
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