摘要翻译：
Hodge相关子是由赋给光滑复曲线的某些积分所给出的复数。证明了它们是一个Feynman积分的相关子，并描述了曲线基本群的前幂完备上的实混合Hodge结构。我们引入了模体相关子，它是模体李代数的元素，其周期是Hodge相关子。它们描述了曲线的基组。我们通过一定的联系来描述实混合Hodge结构在一个变体上的变化，在变体的乘积上通过一条细小的直线。我们称它们为扭曲连接。推广这一点，我们提出了光滑上同调Saito的Hodge复形子范畴的DG增强。我们表明，当曲线变化时，Hodge相关器是描述实际MHS相应变化的扭连接系数。Hodge相关器的例子包括经典多对数和椭圆多对数，以及它们的推广。模曲线上最简单的Hodge相关子是Rankin-Selberg积分。动相关子的例子包括动上同调中的Beilinson元素，例如在模曲线上传递Beilinson-Kato Euler系统的元素。
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英文标题：
《Hodge correlators》
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作者：
A.B. Goncharov
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最新提交年份：
2016
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics        数学
二级分类：Number Theory        数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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英文摘要：
  Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by an afine line. We call them twistor connections. Generalising this, we suggest a DG enhancement of the subcategory of Saito's Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin-Selberg integrals. Examples of the motivic correlators include Beilinson's elements in the motivic cohomology, e.g. the ones delivering the Beilinson - Kato Euler system on modular curves.
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PDF链接：
https://arxiv.org/pdf/0803.0297