Arakelov不等式与$\mathcal{A}_g$中全测地球商的特征刻画
Arakelov inequalities and characterization of totally geodesic ball quotients in $\mathcal{A}_g$
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中文总结 AI 辅助
本文建立主极化复阿贝尔簇族的Arakelov不等式，在等式情形给出阿贝尔簇模空间中全测地球商的数值刻画，去除了Möller-Viehweg-Zuo的强正性条件，并在紧致基与拟射影曲面情形分别给出新证明。
中文摘要 AI 辅助
我们建立了主极化复阿贝尔簇族所对应的Hodge变分结构上的Arakelov不等式。在等式成立的情形下，这导致了阿贝尔簇模空间内某些全测地球商的一个数值特征刻画。我们的结果通过去除Möller-Viehweg-Zuo工作中所施加的强正性条件，推广了他们的工作。对于紧致基空间上的族，我们的证明涉及表明与阿贝尔簇族相关联的周期映射可分解为某些MMP（极小模型纲领）操作，然后将Möller、Viehweg和Zuo的结果推广到适当的奇异情形。在拟射影曲面情形中，我们实施了一种新方法，该方法不经过Miyaoka-Yau型一致化定理，而是利用可追溯至Mok的一个思想、对称空间理论以及半稳定性考虑。
英文摘要
We establish an Arakelov inequality for variations of Hodge structures underlying families of principally polarized complex Abelian varieties. In the equality case, this leads to a numerical characterization of certain totally geodesic ball quotients inside the moduli space of Abelian varieties. Our result extends work of Möller-Viehweg-Zuo by removing the strong positivity conditions imposed in their statement. For families over compact base spaces, our proof involves showing that the period map associated with a family of Abelian varieties factors through certain MMP operations and then generalizing the results of Möller, Viehweg, and Zuo to an appropriate singular setting. In the quasiprojective surface case, we implement a new approach that does not pass through Miyaoka-Yau-type uniformisation theorems but uses an argument going back to an idea of Mok, symmetric space theory and semistability considerations instead.
发表机构
- Universität Duisburg-Essen（杜伊斯堡-埃森大学）
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