The fundamental groupoid scheme and applications

The complexity of a flat groupoid

Grothendieck proved that any finite epimorphism of noetherian schemes factors into a finite sequence of effective epimorphisms. We define the complexity of a flat groupoid $R\rightrightarrows X$ with finite stabilizer to be the length of the canonical sequence of the finite map $R\to X\times\_{X/R} X$, where $X/R$ is the Keel-Mori geometric quotient. For groupoids of complexity at most 1, we prove a theorem of descent along the quotient $X\to X/R$ and a theorem of quotient of a groupoid by a normal subgroupoid. We expect that the complexity could play an important role in the finer study of qu…

Galois symmetries of fundamental groupoids and noncommutative geometry

We define motivic iterated integrals on the affine line, and give a simple proof of the formula for the coproduct in the Hopf algebra of they make. We show that it encodes the group law in the automorphism group of certain non-commutative variety. We relate the coproduct with the coproduct in the Hopf algebra of decorated rooted planar trivalent trees - a planar decorated version of the Hopf algebra defined by Connes and Kreimer. As an application we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. We give a criteria for a motivic iterated integral…

The fundamental group of binoid varieties

Binoid schemes generalise monoid schemes, which in turn enable us to generalise toric varieties. Let $X$ be a binoid scheme. The aim of this paper is to calculate the topological fundamental group of $KX$, where $K=\mathbb{C}$ or $\mathbb{R}$. For the latter, we will give an explicit way of calculating the fundamental group using methods from 2-category theory. Indeed, we will calculate the more general fundamental groupoid. As a specialisation, we will also look at the Stanley Reisner Rings. Our method simplifies in this case, allowing us to describe the fundamental groupoid in terms of the s…

群胚作用在簇上的范畴商

Categorical quotients for actions of groupoids on varieties

本文证明不可数基域上Weyl群胚对仿射簇作用的几何商在k-概形范畴中万有，通过改编Mumford等人的结果并利用闭点和Jacobson概形。

Gauß-Manin connection and Tannaka duality

In analogy with the étale fundamental groups, we express the Gauß-Manin connection for $H^1$ in Tannaka terms. One difficulty is that unlike for fundamental groups, the Tannaka group scheme of relative connections, and the groupoid schemes of absolute connections and of connections on the base do not form an exact sequence. We added a theorem due to Pierre Deligne which improves the main result of the article. It allows to compare the Tannaka group scheme of relative connections with the kernel of the map from the groupoid scheme of absolute connections to the one of the base.

通过李群oids分类化等变变形 I：对数奇点

Categorifying isomonodromic deformations via Lie groupoids I: Logarithmic singularities

本文通过李群oids理论将等变变形提升为函子，研究了具有对数奇点的平坦连接的模堆映射及更高同调信息的几何实现。

Derived deformations of schemes

We introduce a new approach to constructing derived deformation groupoids, by considering them as parameter spaces for strong homotopy bialgebras. This allows them to be constructed for all classical deformation problems, such as deformations of an arbitrary scheme, in any characteristic.

塔恩卡伊对偶与曲线族的高斯-曼金连接

Tannakian duality and Gauss-Manin connections for a family of curves

本文研究了光滑曲线族的微分基本群方案与高斯-曼金连接之间的关系，证明了在高斯-曼金连接中几何相对基本群的上同调映射是同构的，从而将高斯-曼金连接解释为微分基本群的上同调。

关于 Artin 栈与 Lie 群胚的 étale 图册、Poisson 结构与量子化的注记

A note on étale atlases for Artin stacks and Lie groupoids, Poisson structures and quantisation

本文解释了 Artin 栈扩张为非负分次交换上链代数函子的方法，利用 stacky 仿射概形的 étale 图册构建小 étale 景，从而高效表述偏移 Poisson 结构与量子化，并将类似构造推广至 Lie 群胚与可微及解析栈。