Pink’s theory of Hodge structures and the Hodge conjecture over function fields
Pink’s theory of Hodge structures and the Hodge conjecture over function fields
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Pink 的 Hodge 结构理论和函数域上的 Hodge 猜想
DOI：
10.4171/198-1/2
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Ann
中科院分区：
文献类型：
--
作者：
U. Hartl;Ann
1997年，理查德·平克（Richard Pink）阐明了正特征函数场上的霍奇结构（Hodge structures）的概念，今天称为霍奇-平克结构（Hodge-Pink structures）。它们在底层函数域上形成一个中性的Tannakian范畴。他定义了霍奇实现函子从统一阿贝尔$t$-模块和$t$-动机的格雷格安德森霍奇粉红色的结构。这允许一个关联与每个uniformizable $t$动机一个霍奇粉红集团，类似于芒福德泰特群的一个光滑的投影品种的复数。它进一步使粉红色证明类似的芒福德-泰特猜想德林费尔德模块。此外，基于Pink和第一作者未发表的工作，第二作者在她的毕业论文中证明了Hodge-Pink群等于Papanikolas和Taelman定义的$t$-motive的motivic Galois群。这产生了著名的霍奇猜想的精确模拟，霍奇猜想是复数上的簇的一个杰出的公开问题。
在这份报告中，我们解释粉红色的结果霍奇结构和证明的功能领域模拟霍奇猜想。$t$-动机理论在双重$t$-动机理论中有一个变体。我们阐明了$t$-模、对偶$t$-模和$t$-模之间的关系。我们还构造了交换$t$-模和（对偶）$t$-模的上同调实现和它们之间的比较同构，推广了Drinfeld模的Gekeler的de Rham同构。
In 1997 Richard Pink has clarified the concept of Hodge structures over function fields in positive characteristic, which today are called Hodge-Pink structures. They form a neutral Tannakian category over the underlying function field. He has defined Hodge realization functors from the uniformizable abelian $t$-modules and $t$-motives of Greg Anderson to Hodge-Pink structures. This allows one to associate with each uniformizable $t$-motive a Hodge-Pink group, analogous to the Mumford-Tate group of a smooth projective variety over the complex numbers. It further enabled Pink to prove the analog of the Mumford-Tate Conjecture for Drinfeld modules. Moreover, based on unpublished work of Pink and the first author, the second author proved in her Diploma thesis that the Hodge-Pink group equals the motivic Galois group of the $t$-motive as defined by Papanikolas and Taelman. This yields a precise analog of the famous Hodge Conjecture, which is an outstanding open problem for varieties over the complex numbers.
In this report we explain Pink's results on Hodge structures and the proof of the function field analog of the Hodge conjecture. The theory of $t$-motives has a variant in the theory of dual $t$-motives. We clarify the relation between $t$-motives, dual $t$-motives and $t$-modules. We also construct cohomology realizations of abelian $t$-modules and (dual) $t$-motives and comparison isomorphisms between them generalizing Gekeler's de Rham isomorphism for Drinfeld modules.
DOI：
--
发表时间：
2006
期刊：
Integrable systems, geometry, and topology, AMS/IP Studies of Advanced Mathematics, American Mathematical Society 36
影响因子：
--
作者：
FURUYA;Jun;Martin Guest
通讯作者：
Martin Guest