Rohit Nagpal, Andrew Snowden
We classify the subvarieties of infinite dimensional affine space that are stable under the infinite symmetric group. We determine the defining equations and point sets of these varieties as well as the containments between them.
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The Oeljeklaus-Toma (OT-) manifolds are compact, complex, non-Kahler manifolds constructed by Oeljeklaus and Toma, and generalizing the Inoue surfaces. Their construction uses the number-theoretic data: a number field $K$ and a torsion-free subgroup $U$ in the group of units of the ring of integers of $K$, with rank of $U$ equal to the number of real embeddings of $K$. We prove that any complex subvariety of smallest possible positive dimension in an OT-manifold is also flat affine. This is used to show that if all non-trivial elements in $U$ are primitive in $K$, then $X$ contains no proper c…
In this paper we extend the properties of ordinary points of curves [10] to ordinary closed points of one-dimensional affine reduced schemes and then to ordinary subvarieties of codimension one.
We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties of high rank systems of polynomials. In the second, we use these properties to deduce results in Algebraic Geometry, such as an effective Stillman conjecture over algebraically closed fields, an analogue of Nullstellensatz for varieties over finite fields, and a strengthening of a recent result of [5]. We also show that for $k$-varieties $\mathbb X \subset …
We give examples of infinitely extendable (not as cones) arithmetically Cohen-Macaulay and arithmetically Gorenstein subvarieties of projective spaces and which are not complete intersections. The proof uses the computation of the dimension of the Hilbert scheme of codimension $2$ subschemes of projective spaces due to G. Ellingsrud and of arithmetically Gorenstein codimension $3$ subschemes due to J. O. Kleppe and R.-M. Miró-Roig.
We develop criteria for affine varieties to admit uniruled subvarieties of certain dimensions. The measurements are from long exact sequences of versions of symplectic cohomology, which is a Hamiltonian Floer theory for some open symplectic manifolds including affine varieties. Symplectic cohomology is hard to compute, in general. However, certain vanishing and invariance properties of symplectic cohomology can be used to prove that our criteria for finding uniruled subvarieties hold in some cases. We provide applications of the criteria in birational geometry of log pairs in the direction of …
Let $X$ be a quasi-affine algebraic variety isomorphic to the complement of a closed subvariety of dimension at most $n-3$ in $\C^n$. We find some conditions under which an isomorphism of two closed subvarieties of $X$ can be extended to an automorphism of $X$.
We describe the boundary of linear subvarieties in the moduli space of multi-scale differentials. Linear subvarieties are algebraic subvarieties of strata of (possibly) meromorphic differentials that in local period coordinates are given by linear equations. The main example of such are affine invariant submanifolds, that is, closures of $\operatorname{SL}(2,\mathbb{R})$-orbits. We prove that the boundary of any linear subvariety is again given by linear equations in generalized period coordinates of the boundary. Our main technical tool is an asymptotic analysis of periods near the boundary o…
Hurwitz-Hecke Invariant Subvarieties
本文提出一种基于纯组合输入和 Hurwitz 空间理论在平移曲面分层中构造仿射不变子簇的方法，并证明其可用于构造 Bouw-Moeller Teichmueller 曲线及七个 Eskin-McMullen-Mukamel-Wright 秩二轨道闭包。
The paper concerns the affine varieties that are homogeneous with respect to a (non-standard) graduation over the group Z^m. Among the other properties it is shown that every such a variety can be embedded in its Zariski tangent space at the origin, so that it is smooth if and only if it is isomorphic to an affine space. The results directly apply to the study of Hilbert schemes of subvarieties in P^n.
We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of Hessenberg varieties over GL_n(C) and show that they have no odd-dimensional homology. We provide an explicit geometric construction which partitions each Hessenberg variety into pieces homeomorphic to affine space. We characterize these affine pieces by fillings of Young tableaux and show that the dimension of the af…
The trace test in numerical algebraic geometry verifies the completeness of a witness set of an irreducible variety in affine or projective space. We give a brief derivation of the trace test and then consider it for subvarieties of products of projective spaces using multihomogeneous witness sets. We show how a dimension reduction leads to a practical trace test in this case involving a curve in a low-dimensional affine space.
In this paper we extend the construction of the canonical polarized variation of Hodge structures over tube domain considered by B. Gross in \cite{G} to bounded symmetric domain and introduce a series of invariants of infinitesimal variation of Hodge structures, which we call characteristic subvarieties. We prove that the characteristic subvariety of the canonical polarized variations of Hodge structures over irreducible bounded symmetric domains are identified with the characteristic bundles defined by N. Mok in \cite{M}. We verified the generating property of B. Gross for all irreducible bou…
Matroids and the space of torus-invariant subvarieties of the Grassmannian with given homology class
本文研究了复Grassmannian中具有给定同调类的环面不变代数子簇的刻画问题，利用拟阵理论技术给出了部分解答，并将其应用于计算特定Grassmannian的Euler-Chow级数。
Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This is a far reaching generalization of usual theory of Newton polytopes (which is concerned with toric varieties). As applications we give new, simple and transparent proofs of some well-known theorems in both algebraic geometry (e.g. Hodge Index Theorem) and convex geometry (e.g. Alexandrov-Fenchel inequality). Our main tools are classical Hilbert theory on de…
A note on subvarieties of powers of OT-manifolds
该研究证明OT-流形上有限对有限全纯对应空间离散，结合模型论结果推出无真无限复解析子集的OT-流形的笛卡尔幂无有意义复解析子簇族，证明借助OT-流形的有限非分歧覆盖完成。
Surjective morphisms from affine space to its Zariski open subsets
本文构造了仿射空间以指定 Zariski 开子集为像的满射自同态，并利用 Noether 归一化将其推广到任意同维仿射簇。
An infinite series of Gorenstein local algebras failing the affine homogeneity property
本文研究了一类无限序列的有限维格罗then斯坦局部代数，证明其极大理想中存在非socle的不变子空间，从而不满足仿射同质性性质，并探讨了其在射影超曲面上的加法作用及相关推广的哈塞-茨辛格尔对应。
The goal of this paper is to construct a Frobenius splitting on $G/U$ via the Poisson geometry of $(G/U,π_{G/U})$, where $G$ is a semi-simple algebraic group of classical type defined over an algebraically closed field of characteristic $p > 3$, $U$ is the uniradical of a Borel subgroup of $G$ and $π_{G/U}$ is the standard Poisson structure on $G/U$. We first study the Poisson geometry of $(G/U,π_{G/U})$. Then, we develop a general theory for Frobenius splittings on $\mathbb{T}$-Poisson varieties, where $\mathbb{T}$ is an algebraic torus. In particular, we prove that compatibly split subvariet…
For a connected reductive group G and a finite-dimensional G-module V, we study the invariant Hilbert scheme that parameterizes closed G-stable subschemes of V affording a fixed, multiplicity-finite representation of G in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus T of G, together with an open T-stable subscheme admitting a good quotient. The fibers of the quotient map classify affine G-schemes having a prescribed categorical quotient by a maximal unipotent subgroup of G. We show that V contains only finitely many multiplicity-free G-subva…
This paper is dedicated to the problem of infinite transitivity for algebraically generated automorphism groups of the affine plane. We provide a necessary and sufficient condition of infinite transitivity for a large family of subgroups generated by one-parameter additive groups.
Varieties covered by affine spaces, uniformly rational varieties and their cones
本研究将仿射锥的格罗莫夫椭圆性结论，从旗流形、有理光滑射影曲面推广到逐次爆发后的光滑射球星形簇、一致有理射影簇及稳定一致有理完全簇。
In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.
Affine Pavings of Hessenberg Ideal Fibers
本文定义了旗流形的闭子簇Hessenberg理想纤维并证明其由仿射铺砌，在$G_2$型中显式描述这些纤维，据此完全分类了Weyl群在正则半单Hessenberg簇上同调的Tymoczko点作用。
The main result of this article is to construct infinite families of non-equivalent equivariant real forms of linear C*-actions on affine four-space. We consider the real form of C* whose fixed point is a circle. In [F-MJ] one example of a non-linearizable circle action was constructed. Here, this result is generalized by developing a new approach which allows us to compare different real forms. The constructions of these forms are based on the structure of equivariant O2(C)-vector bundles.
In this paper we construct free resolutions of certain class of closed subvarieties of affine space of symmetric matrices (of a given size). Our class covers the symmetric determinantal varieties (i.e., determinantal varieties in the space of symmetric matrices), whose resolutions were first constructed by Józefiak-Pragacz-Weyman. Our approach follows the techniques developed in Free Resolutions of Some Schubert Singularities by Kummini-Lakshmibai-Sastry-Seshadri, and uses the geometry of Schubert varieties.
We say that a group G acts infinitely transitively on a set X if for every integer m the induced diagonal action of G is transitive on the cartesian mth power of X with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of affine cones over flag varieties, the second of non-degenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups of a reinforced type.
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。
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。
We initiate the study of affine Deligne-Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. We prove that for GLn and its inner forms, Lusztig's semi-infinite Deligne-Lusztig construction is isomorphic to an affine Deligne-Lusztig variety at infinite level. We prove that their homology groups give geometric realizations of the local Langlands and Jacquet--Langlands correspondences in the setting that the Weil parameter is induced from a character of an unramified field extension. In particular, we resolve Lusztig's 1979 conjecture in this set…
Equivariant algebraic and semi-algebraic geometry of infinite affine space
本文研究了无限仿射空间中Sym(∞)作用下的代数与半代数几何性质，证明了n=1时的消去量词结果，但该结果在n>1时不成立。
On the infinite Frobenius action on de Rham fundamental groups of affine curves
本文研究仿射曲线de Rham基本群上的无穷Frobenius作用，计算相关实混合Hodge结构的扩张类，并在模曲线情形下将其与Rankin-Selberg L-函数特殊值联系，证明扩张非分裂。
Tangent cones to Schubert varieties for Kac--Moody groups
针对$\tilde{A}_{n-1}$型仿射Kac–Moody群的旗簇，证明不同Weyl群对合对应的Schubert子簇在单位陪集点处的切锥不重合，推广了有限维情形的相关结果，所用技术工具涉及Weyl群嵌入组合学与幂么根基余伴随轨道。
The goal of this work is to construct a perverse t-structure on the infinity-category of l-adic LG-equivariant sheaves on the loop Lie algebra Lg and to show that the affine Grothendieck-Springer sheaf S is perverse. Moreover, S is an intermediate extension of its restriction to the locus of ``compact" elements with regular semi-simple reduction. Note that classical methods do not apply in our situation because LG and Lg are infinite-dimensional ind-schemes.
The Lie algebra of polynomial vector fields on the affine space is $2$-generated
该研究证明了仿射空间上多项式向量场的李代数由两个特定元素生成。
Zero velocity Lagrangians in the Hitchin system
本文在 Higgs 丛模空间中构造了一类由零速度条件定义的新拉格朗日子流形，证明了其与 Hitchin 映射通用纤维的交点数公式，并构造了对偶模空间上的向量丛，猜想其对应镜像超全纯丛。
Quasi-Compactness in Infinite Dimension
本文研究了无限维仿射空间及素谱逆极限中开集的准紧性特征，通过弱稳定性、反紧性和柱集等条件给出等价判定方法，并举例说明非准紧的仿射空间。
Topological simplicity of the group of automorphisms of the affine plane
本文证明任意无限域k上无限维代数群SAut_k(𝔸²)是拓扑简单的，还指出高维闭正规子群含所有tame自同构，并讨论了差异显著的有限域情形。
Infinite transitivity of tame groups of automorphisms of affine spaces
该研究针对有限域上n维仿射空间的驯服自同构群，探究其对m个点有序子集的作用，构造可映射有序序列的驯服自同构并证明其复杂度的上下界，运用多数学分支方法获得相关定性与定量结果。
On the semi-infinite Deligne--Lusztig varieties for $\mathrm{GSp}$
本文证明了GSp的半无限Deligne–Lusztig变种与无限水平仿射Deligne–Lusztig变种同构，并重新诠释了X_h的表示构造方法。
In infinite-dimensional Lie theory, the affine Kac-Moody Lie algebras and groups play a distinguished role due to their many applications to various areas of mathematics and physics. Underlying these infinite-dimensional objects there are closely related group schemes and Lie algebras of finite type over Laurent polynomial rings. The language of SGA3 is perfectly suited to describe such objects. The purpose of this short article is to provide a natural description of the affine Kac-Moody groups and Lie algebras using this language.
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