Arakelov 不等式及其算术性质

本课题主要研究代数曲面纤维化中的 Arakelov 不等式及其在算术几何上的应用。Viehweg, Zuo 及 Möller 证明 Arakelov 不等式等号成立当且仅当底曲线是一条 Shimura 曲线。这将纤维化的相对不变量的研究与著名的 Oort 猜想联系起来: 对于亏格大于等于8，Shimura 曲线不一般的落在 Torelli 轨迹中不存在。我们希望利用纤维化的相对不变量来判断 Arakelov 不等式何时成立进而得到 Torelli 轨迹中何时存在 Shimura 曲线。自然地，我们还试图将 Arakelov 不等式推广到正特征情形，我们的主要研究方法来源于 Langer 教授 (Invent. Math.) 最近在 Higgs 层的 Bogomolov 不等式方面的工作。我们还继续与 Mina Teicher 教授等利用变子群理论合作研究一些一般型曲面的基本群。

In this project, we study the Arakelov inequality for fibered surfaces and its application in arithmetic geometry. Viehweg, Zuo and Möller proved that the equality holds for Arakelov inequality if and only if the base curve is a Shimura curve . Then the relative invariants of fibrations is related to the famous Oort conjecture：For the genus large than 7, there does not exists Shimura curve contained generically in the Torelli locus of curves. We try to use the relative invariants of the fibration to study when the equality holds for Arakelov inequality, in turn, to study the existence of Shimura curve. Naturally, we try to generalize the Arakelov inequality to the positive characteristic. Our main methods comes from the works of Prof. Langer (Invent. Math.) on the Bogomolov inequality for Higgs sheaves. We shall also use braid monodromy theory to study fundamental group of some surfaces of general type with Prof. Mina Teicher，etc.