高维簇上的几何阿贝尔截面
Geometrically Abelian Sections on Higher-Dimensional Varieties
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中文总结 AI 辅助
该研究探讨高维簇的算术信息检测，通过几何阿贝尔基本群截面，在p进域、有限生成域上推导相关结论，还给出Esnault与Wittenberg定理强化版本的新证明。
中文摘要 AI 辅助
我们研究可通过充分小的开子簇的几何阿贝尔基本群的截面，检测到一个簇的多少算术信息。在p进域上，此类截面确定了阿尔巴内塞挠子上的一个有理点；在一个猜想假设下，若该簇的指数为1，则此点由原簇上次数为1的0-闭链表示，对于曲线，后一结论无条件成立。在有限生成域上，此类截面仍迫使相对布饶尔群在特征外消失。关键输入是将附属于截面的陈类障碍与局部塔特对偶产生的障碍等同起来，作为副产品，我们给出了Esnault与Wittenberg定理的开子簇强化版本的新证明。
英文摘要
We investigate how much arithmetic information about a variety can be detected by sections of the geometrically abelian fundamental groups of sufficiently small open subvarieties. Over a \(p\)-adic field, such a section determines a rational point on the Albanese torsor, and, under a conjectural assumption, this point is represented by a \(0\)-cycle of degree \(1\) on the original variety if the variety has index \(1\). For curves, the latter statement holds unconditionally. Over finitely generated fields, such sections still force the relative Brauer group to vanish away from the characteristic. The key input is an identification of the Chern class obstruction attached to a section with the obstruction arising from local Tate duality. As a byproduct, we give a new proof of an open-subvariety strengthening of a theorem of Esnault and Wittenberg.
发表机构
- University of Pennsylvania（宾夕法尼亚大学）
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