Differential Geometry of Orbit space of Extended Affine Jacobi Group $A_n$

Splitting mixed Hodge structures over affine invariant manifolds

We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The result is deduced from a generalization of a theorem of Möller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in the torsion of a factor of the Jacobians. This statement can be viewed as a splitting of certain mixed Hodge structures.

The cohomology ring of certain compactified Jacobians

We provide an explicit presentation of the equivariant cohomology ring of the compactified Jacobian $J_{q/p}$ of the rational curve $C_{q/p}$ with planar equation $x^{q}=y^{p}$ for $(p,q)=1$. We also prove analogous results for the closely related affine Springer fiber $Sp_{q/p}$ in the affine flag variety of $SL_{p}$. We show that the perverse filtration on the cohomology of $J_{q/p}$ is multiplicative, and the associated graded ring under the perverse filtration is a degeneration of the ring of functions on a moduli space of maps $\mathbb{P}^{1}\to C_{q/p}$. We also propose several conjectur…

关于仿射空间的自同态与雅可比问题

On endomorphisms of affine spaces and the Jacobian problem

本文研究仿射空间平展自同态的纤维基数、像补集及几何次数，构造反例否证若干猜想，并证明雅可比猜想在特定复合条件下成立。

Euler characteristics of theta divisors of Jacobians for spectral curves

We show how to calculate the Euler characteristic of an affine Jacobi variety of a spectral curve from its defining equations.

Complete orbit spaces of affine torus actions

Given an action of an algebraic torus on a normal affine variety, we describe all open subsets admitting a complete orbit space.

关于一般环面模型的雅可比方程

On the Jacobian syzygies for generic toric models

本文针对仿射环面中的一般超曲面，在射影空间关联超曲面配置，并计算了相应雅可比代数的最小分次分解。

Twisted orbital integrals and irreducible components of affine Deligne-Lusztig varieties

We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and $q$-analogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo the action of the $σ$-centralizer group to the Mirkovic-Vilonen basis of a certain weight space of a representation of the Langlands dual group.

$p$-adic 域上仿射曲面的自同构群的轨道

Orbits of automorphism groups of affine surfaces over $p$-adic fields

本文研究了 $p$-adic 仿射曲面自同构群的轨道闭包和平稳测度。

SL₂的仿射旗 manifold 中 Iwahori 组 codimension 一到四子群的轨道

Orbits of subgroups of codimension one to four of the Iwahori group in the affine flag variety of $\text{SL}_2$

研究SL₂仿射旗 manifold中Iwahori组codimension一到四子群的轨道分解，探讨Schubert细胞的分解方法与贡献。

Affine toric varieties with an open orbit of an $SL_n$ action

We study affine toric varieties with an action of group $SL_n$ with a dense orbit. A characterisation in terms of $SL_n \times Q$-modules is given where $Q$ is a quasitorus. This characterisation is more explicitly expanded in case $n=3$. It is shown that in case $n=3$ the class group rank is not greater than $3$, however it is unbounded when $n\geq 4$.

Classification of affine normal $SL_2$-varieties with a dense orbit

In this note we give a classification of all affine normal $SL_2$-varieties containing an open dense orbit over an algebraically closed field of characteristic zero. Such a classification was first obtained by Popov. Here we provide an alternative approach.

On orbits of the automorphism group on an affine toric variety

Let X be an affine toric variety. The total coordinates on X provide a canonical presentation of X as a quotient of a vector space by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.

仿射曲面的斜驶自同构的轨道的交

Intersection of orbits of loxodromic automorphisms of affine surfaces

该研究证明域上仿射曲面的两个斜驶自同构若轨道有无穷多交点则必有公共迭代，借助前期仿射曲面自同态动力学工作与算术动力学论证，还得到乘积空间中曲面的动力学Mordell-Lang型结果。

Full rank affine invariant submanifolds

We show that every GL(2, R) orbit closure of translation surfaces is either a connected component of a stratum, the hyperelliptic locus, or consists entirely of surfaces whose Jacobians have extra endomorphisms. We use this result to give applications related to polygonal billiards. For example, we exhibit infinitely many rational triangles whose unfoldings have dense GL(2,R) orbit.

On the distribution of orbits in affine varieties

Given an affine variety $X$, a morphism $ϕ:X\to X$, a point $α\in X$, and a Zariski closed subset $V$ of $X$, we show that the forward $ϕ$-orbit of $α$ meets $V$ in at most finitely many infinite arithmetic progressions, and the remaining points lie in a set of Banach density zero. This may be viewed as a weak asymptotic version of the Dynamical Mordell-Lang Conjecture for affine varieties. The results hold in arbitrary characteristic, and the proof uses methods of ergodic theory applied to compact Berkovich spaces.

仿射Deligne-Lusztig簇通过双Bruhat图I：半无限轨道

Affine Deligne-Lusztig varieties via the double Bruhat graph I: Semi-infinite orbits

本文引入双Bruhat图新语言，通过半无限轨道研究仿射Deligne-Lusztig簇的几何，克服了正则性条件限制。

计算仿射平面中轨道 Zariski 闭包的算法

Algorithm to Compute Orbit Zariski Closure in Affine Plane

本文提出一种算法，通过判断有限生成群是否包含于代数群，并计算单个三角自同构的不变子簇，进而计算仿射平面上有限生成群轨道的 Zariski 闭包。

仿射建筑中的烟囱收缩编码仿射旗簇中的轨道

Chimney retractions in affine buildings encode orbits in affine flag varieties

本文通过引入带标签的折叠画廊并刻画烟囱收缩的原像，在任意仿射建筑中建立了收缩几何与折叠画廊组合学之间的统一联系，从而推广了仿射旗簇中双陪集相交的研究。

Automorphisms of the Lie algebra of vector fields on affine n-space

We show that every Lie algebra automorphisms of the vector fields $Vec(A^n)$ of affine n-space $A^n$, of the vector fields $Vec^c(A^n)$ with constant divergence, and of the vector fields $Vec^0(A^n)$ with divergence zero is induced by an automorphism of $A^n$. This generalizes results of the second author obtained in dimension 2. The case of $Vec(A^n)$ is due to Vladimir Bavula. As an immediate consequence, we get the following result which due to Viktor Kulikov. If every injective endomorphism of the simple Lie algebra $Vec(A^n)$ is an automorphism, then the Jacobian Conjecture holds in dimen…

Affine embeddings of homogeneous spaces

Let G be a reductive algebraic group and let H be a reductive subgroup of G. We describe all pairs (G,H) such that for any affine G-variety X with a dense G-orbit isomorphic to G/H the number of G-orbits in X is finite. The maximal number of parameters in families of G-orbits in all affine embeddings of G/H is computed.

The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane

In this paper we prove the following theorem. Let $f:\mathbb{A}^2\rightarrow \mathbb{A}^2$ be a dominate polynomial endomorphisms defined over an algebraically closed field $k$ of characteristic $0$. If there are no nonconstant rational function $g:\mathbb{A}^2-rightarrow \mathbb{P}^1$ satisfying $g\circ f=g$, then there exists a point $p\in \mathbb{A}^2(k)$ whose orbit under $f$ is Zariski dense in $\mathbb{A}^2$. This result gives us a positive answer to a conjecture of Amerik, Bogomolov and Rovinsky ( and Zhang) for polynomial endomorphisms on the affine plane.

Locally nilpotent derivations on affine surfaces with a $\C^*$-action

We give a classification of normal affine surfaces admitting an algebraic group action with an open orbit. In particular an explicit algebraic description of the affine coordinate rings and the defining equations of such varieties is given. By our methods we recover many known results, e.g. the classification of normal affine surfaces with a `big' open orbit of Gizatullin and Popov or some of the classification results of Danilov-Gizatullin, Bertin and others.

仿射旗簇中的半无限轨道与仿射 Springer 纤维的同调

Semi-infinite orbits in affine flag varieties and homology of affine Springer fibers

本文研究仿射 Springer 纤维由仿射 Schubert 子簇给出的有限并的同调，证明在足够正则条件下自然包含映射在所有次数上单射，并将其解释为加权轨道积分的范畴化，同时刻画仿射 Schubert 簇为半无限轨道闭包的交。

高度为2的幂零轨道与仿射Weyl群中的对合

Nilpotent orbits of height 2 and involutions in the affine Weyl group

本文研究特征零域上几乎单群的Borel子群在高度至多为2的幂零元素簇上的有限轨道作用，用两两正交根集合参数化B-轨道，并用仿射Weyl群中对合的Bruhat序完整刻画轨道闭包间的包含关系。

层的仿射松木对应

Affine Matsuki correspondence for sheaves

本文通过莫尔斯理论将仿射格拉斯曼流形上的仿射松木对应提升为层的导出范畴等价，并利用融合结构得到与舒伯特可构层导出范畴的进一步等价。

A combinatorial study of affine Schubert varieties in affine Grassmannian

Let $\overline{\mathtt{X}}_λ$ be the closure of the $\mathtt{I}$-orbit $\mathtt{X}_λ$ in the affine Grassmanian $\mathtt{Gr}$ of a simple algebraic group $G$ of adjoint type, where $\mathtt{I}$ is the Iwahori group and $λ$ is a coweight of $G$. We find a simple algorithm which describes the set $Ψ(λ)$ of all $\mathtt{I}$-orbits in $\overline{\mathtt{X}}_λ$ in terms of coweights. We introduce $R$-operators (associated to positive roots) on the coweight lattice of $G$, which exactly describe the closure relation of $\mathtt{I}$-orbits. These operators satisfy Braid relations generically on the c…

On The Zariski Topology Of Automorphism Groups Of Affine Spaces And Algebras

We study the Zariski topology of the ind-groups of polynomial and free associative algebras $\Aut(K[x_1,...,x_n])$ (which is equivalent to the automorphism group of the affine space $\Aut(K^n))$) and $\Aut(K< x_1,..., x_n>$ via $\Ind$-schemes, toric varieties, approximations and singularities. We obtain some nice properties of $\Aut(\Aut(A))$, where $A$ is polynomial or free associative algebra over a field $K$. We prove that all $\Ind$-scheme automorphisms of $\Aut(K[x_1,...,x_n])$ are inner for $n\ge 3$, and all $\Ind$-scheme automorphisms of $\Aut(K< x_1,..., x_n>)$ are semi-inner. We also …