Complete orbit spaces of affine torus actions

Uniformly rational varieties with torus action

A smooth variety is called uniformly rational if every point admits a Zariski open neighborhood isomorphic to a Zariski open subset of the affine space. In this note we show that every smooth and rational affine variety endowed with an algebraic torus action such that the algebraic quotient has dimension 0 or 1 is uniformly rational.

Polyhedral Divisors and Algebraic Torus Actions

We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our theory extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.

Integral affine structures on spheres and torus fibrations of Calabi-Yau toric hypersurfaces I

We describe in purely combinatorial terms dual pairs of integral affine structures on spheres which come from the conjectural metric collapse of mirror families of Calabi-Yau toric hypersurfaces. The same structures arise on the base of a special Lagrangian torus fibration in the Strominger-Yau-Zaslow conjecture. We study the topological torus fibration in the large complex structure limit and show that it coincides with our combinatorial model.

Torsion Points on an Algebraic Subset of an Affine Torus

Work of Laurent and Sarnak, following a conjecture of Lang, shows that the number of torsion points of order n on an algebraic subset of an affine complex torus is polynomial periodic. In this paper, we find bounds on the degree and period of this number as a function of n. Some examples, including the number of n torsion points on Fermat curves, are computed to illustrate the methods.

Polyhedral divisors and torus actions of complexity one over arbitrary fields

We show that the presentation of affine $\mathbb{T}$-varieties of complexity one in terms of polyhedral divisors holds over an arbitrary field. We also describe a class of multigraded algebras over Dedekind domains. We study how the algebra associated to a polyhedral divisor changes when we extend the scalars. As another application, we provide a combinatorial description of affine $\mathbf{G}$-varieties of complexity one over a field, where $\mathbf{G}$ is a (not-nescessary split) torus, by using elementary facts on Galois descent. This class of affine $\mathbf{G}$-varieties is described via …

Compactifying torus fibrations over integral affine manifolds with singularities

This is an announcement of the following construction: given an integral affine manifold $B$ with singularities, we build a topological space $X$ which is a torus fibration over $B$. The main new feature of the fibration $X\to B$ is that it has the discriminant in codimension 2.

Gluing affine torus actions via divisorial fans

Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a ``proper polyhedral divisor'' introduced in earlier work, we develop the concept of a ``divisorial fan'' and show that these objects encode the equivariant gluing of affine varieties with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and we study examples like C*-surfaces and projectivizations of (non-split) vector bundles over tori…

Integral affine structures on spheres and torus fibrations of Calabi-Yau toric hypersurfaces II

This paper is a continuation of our paper math.AG/0205321 where we have built a combinatorial model for the torus fibrations of Calabi-Yau toric hypersurfaces. This part addresses the connection between the model torus fibration and the complex and Kähler geometry of the hypersurfaces.

Quotients of an affine variety by an action of a torus

Let X be an affine T-variety. We study two different quotients for the action of T on X: the toric Chow quotient X/_CT and the toric Hilbert scheme H. We introduce a notion of the main component H_0 of H which parameterizes general T-orbit closures in X and their flat limits. The main component U_0 of the universal family U over H is a preimage of H_0. We define an analogue of a universal family W_X over the main component of the X/_CT. We show that the toric Chow morphism restricted on the main components lifts to a birational projective morphism from U_0 to W_X. The variety W_X also provides…

Smooth varieties with torus actions

In this paper we provide a characterization of smooth algebraic varieties endowed with a faithful algebraic torus action in terms of a combinatorial description given by Altmann and Hausen. Our main result is that such a variety X is smooth if and only if it is locally isomorphic in the étale topology to the affine space endowed with a linear torus action. Furthermore, this is the case if and only if the combinatorial data describing X is locally isomorphic in the étale topology to the combinatorial data describing affine space endowed with a linear torus action. Finally, we provide an effecti…

非正规仿射环面曲面的自同构群

Automorphism groups of non-normal affine toric surfaces

研究非正规仿射环面曲面的自同构群，发现当奇异轨迹含一维环面轨道时性质成立，否则可能不成立，并构造反例。

Equivariant oriented homology of the affine Grassmannian

We generalize the property of small-torus equivariant K-homology of the affine Grassmannian to general oriented (co)homology theory in the sense of Levine and Morel. The main tool we use is the formal affine Demazure algebra associated to the affine root system. More precisely, we prove that the small-torus equivariant oriented cohomology of the affine Grassmannian satisfies the GKM condition. We also show that its dual, the small-torus equivariant homology, is isomorphic to the centralizer of the equivariant oriented cohomology of a point in the the formal affine Demazure algebra.

Polyhedral divisors of affine trinomial hypersurfaces

We find polyhedral divisors corresponding to the torus action of complexity one on affine trinomial hypersurfaces. Explicit computations for particular classes of such hypersurfaces including Pham-Brieskorn surfaces and rational trinomial hypersurfaces are given.

Normal singularities with torus actions

We propose a method to compute a desingularization of a normal affine variety X endowed with a torus action in terms of a combinatorial description of such a variety due to Altmann and Hausen. This desingularization allows us to study the structure of the singularities of X. In particular, we give criteria for X to have only rational, (QQ-)factorial, or (QQ-)Gorenstein singularities. We also give partial criteria for X to be Cohen-Macaulay or log-terminal. Finally, we provide a method to construct factorial affine varieties with a torus action. This leads to a full classification of such varie…

On toric ind-varieties and pro-affine semigroups

An ind-variety is an inductive limit of closed embeddings of algebraic varieties and an ind-group is a group object in the category of ind-varieties. These notions were first introduced by Shafarevich in the study of the automorphism group of affine spaces and have been studied by many authors afterwards. An ind-torus is an ind-group obtained as an inductive limit of closed embeddings of algebraic tori that are also algebraic group homomorphisms. In this paper, we introduce the natural definition of toric ind-varieties as ind-varieties having an ind-torus as an open set and such that the actio…

Ring of conditions for affine space

The exponential sum (ES) is a linear combination of characters of an additive group $\mathbb C^n$. The exponential analytic set (EAS) is a set of common zeroes of a finite tuple of ESs. We consider ES and EAS as an analogs of Laurent polynomial and of algebraic variety in complex torus $(\mathbb{C}\setminus0)^n$. Respectively we construct the ring of conditions for $\mathbb C^n$ as an analog of the ring of conditions for $(\mathbb{C}\setminus0)^n$. The construction of this ring is based on the definition of associated to EAS algebraic subvariety of some multidimensional torus and on the applyi…

On (2,3) torus decompositions of QL-configurations

Let $Q$ be an affine quartic which does not intersect transversely with the line at infinity $L_{\infty}$. In this paper, we show the existence of a $(2,3)$ torus decomposition of the defining polynomial of $Q$ and its uniqueness except for one class.

仿射空间商环上的环面作用

Torus Actions on Quotients of Affine Spaces

研究约化复代数群线性作用下的复向量空间GIT商环上环面作用的不动点轨迹，证明在稳定点集上自由作用时，不动点轨迹的分量是Levi子群作用下的线性子空间的GIT商环。

关于仿射环面簇由其自同构群的刻画

On the characterization of affine toric varieties by their automorphism group

本文证明了正规仿射环面簇在特定范畴中由自同构群唯一决定的充要条件，并指出代数环面情形下的维数限制结果是最优的。

Clôture intégrale et opérations de tores algébriques de complexité un dans les variétés affines

We describe explicitly the normalization of affine varieties with an algebraic torus action of complexity one in terms of polyhedral divisors. We also provide a description of homogeneous integrally closed ideals of affine T-varieties of complexity one.

具有有限个自同构群轨道的复杂度一环面作用簇

Varieties with a torus action of complexity one having a finite number of automorphism group orbits

该研究针对复杂度为一的环面作用簇，推导得出其自同构群轨道数目有限的充分条件，为代数几何领域相关簇的结构研究提供了理论支撑。

The moduli stack of affine stable toric varieties

Let X be an irreducible affine T-variety. We consider families of affine stable toric T-varieties over X and give a description of the corresponding moduli space as the quotient stack of an open subscheme in a certain toric Hilbert scheme under the action of a torus.

On the Brauer group of the product of a torus and a semisimple algebraic group

Let T be a torus (not assumed to be split) over a field F, and denote by $_n{H^{2}_{et}(X,Gm)}$ the subgroup of elements of exponent dividing n in the cohomological Brauer group of a scheme X over the field F. We provide conditions on X and n for which the pull-back homomorphism $_n{H^2_{et}(T,Gm)}\to_n{H^2_{et}(X\times T,Gm)}$ is an isomorphism. We apply this to compute the Brauer group of some reductive groups and of non singular affine quadrics. Apart from this, we investigate the p-torsion of the Azumaya algebra defined Brauer group of a regular affine scheme over a field F of characterist…

Horizontal Ga-actions on affine T-varieties of complexity one

We classify the $\mathbb{G}_{a}$-actions on normal affine varieties defined over any field that are horizontal with respect to a torus action of complexity one. This generalizes previous results that were available for perfect ground fields (cf. [Flenner-Zaidenberg2005, Liendo2010, L-Liendo2016]).

极大交换幂幺子群与仿射球面簇的特征刻画

Maximal commutative unipotent subgroups and a characterization of affine spherical varieties

本文描述了不可约仿射簇自同构群的极大交换幂幺子群，证明群同构保持幂幺元素，并利用此结果证明自同构群能检测球面性与权幺半群，进而刻画了仿射环面簇与光滑仿射球面簇由其自同构群唯一确定的性质。

任意域上的恰当多面体除子与环面作用

Proper polyhedral divisors and torus actions over arbitrary fields

本文对任意域上带代数环面有效作用的几何整正规仿射簇，借助带伽罗瓦半线性作用的恰当多面体除子给出代数几何组合描述。

Affine algebraic groups with periodic components

A connected component of an affine algebraic group is called periodic if all its elements have finite order. We give a characterization of periodic components in terms of automorphisms with finite number of fixed points. It is also discussed which connected groups have finite extensions with periodic components. The results are applied to the study of the normalizer of a maximal torus in a simple algebraic group.

孤立环面不变量与刚性簇的自同构群

Isolated torus invariants and automorphism groups of rigid varieties

该研究针对刚性仿射簇自同构群单位分支为环面的猜想，在环面簇与复杂度1环面作用簇上完成证明，给出刚性簇上m-悬垂的刚性判别准则并研究对应自同构群。

Ramified automorphic induction and zero-dimensional affine Deligne-Lusztig varieties

To any connected reductive group G over a non-archimedean local field F (of characteristic p > 0) and to any maximal torus T of G, we attach a family of extended affine Deligne-Lusztig varieties (and families of torsors over them) over the residue field of F. This construction generalizes affine Deligne-Lusztig varieties of Rapoport, which are attached only to unramified tori of G. Via this construction, we can attach to any maximal torus T of G and any character of T a representation of G. This procedure should conjecturally realize the automorphic induction from T to G. For G = GL_2, we prov…

Characterizing quasi-affine spherical varieties via the automorphism group

Let $G$ be a connected reductive algebraic group. In this note we prove that for a quasi-affine $G$-spherical variety the weight monoid is determined by the weights of its non-trivial $\mathbb{G}_a$-actions that are homogeneous with respect to a Borel subgroup of $G$. As an application we get that a smooth affine $G$-spherical variety that is non-isomorphic to a torus is determined by its automorphism group inside the category of smooth affine irreducible varieties.

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。

Uniqueness of Embeddings of the Affine Line into Algebraic Groups

Let $Y$ be the underlying variety of a connected affine algebraic group. We prove that two embeddings of the affine line $\mathbb{C}$ into $Y$ are the same up to an automorphism of $Y$ provided that $Y$ is not isomorphic to a product of a torus $(\mathbb{C}^\ast)^k$ and one of the three varieties $\mathbb{C}^3$, $\operatorname{SL}_2$, and $\operatorname{PSL}_2$.

Lagrangian 3-torus fibrations

We give a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. The main result of this article is the proof that the topological Calabi-Yau compactifications of M. Gross' "Topological Mirror Symmetry", can be made into symplectic compactifications. As an example, we obtain a pair of compact symplectic 6-manifolds together with Lagrangian fibrations whose underlying affine structures are dual. The symplectic manifolds obtained are homeomorphic to a smooth quintic Calabi-Yau 3-fold and its mirro…

格拉斯曼流形\(gr(2,k)\)中一般点的环面轨道的非平凡奇数维等变上同调

Non-trivial Odd Dimensional Equivariant Cohomology of Torus Orbits of Generic Points in $gr(2,k)$

研究格拉斯曼流形\(gr(2,k)\)中一般点环面轨道的非平凡奇数维等变上同调，构造了有此类上同调的环面簇，还表明对小例子能用\(\mathbb{Q}\)系数在\(H^3_T\)中明确计算等变上同调。

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 $n\geq 4$.

On restriction of roots on affine T-varieties

Let X be a normal affine algebraic variety with regular action of a torus \TT and T\subset\TT be a subtorus. We prove that each root of X with respect to T can be obtained by restriction of some root of X with respect to \TT. This allows to get an elementary proof of the description of roots of the affine Cremona group. Several results on restriction of roots in the case of subtorus action on an affine toric variety are obtained.

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.

k-Double Schur functions and equivariant (co)homology of the affine Grassmannian

The Schubert bases of the torus-equivariant homology and cohomology rings of the affine Grassmannian of the special linear group are realized by new families of symmetric functions called k-double Schur functions and affine double Schur functions.

The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus

Let $U$ be an affine log Calabi-Yau variety containing an open algebraic torus. We show that the naive counts of rational curves in $U$ uniquely determine a commutative associative algebra equipped with a compatible multilinear form. This proves a variant of the Frobenius structure conjecture by Gross-Hacking-Keel in mirror symmetry, and the spectrum of this algebra is supposed to give the hypothetical mirror family. Although the statement of our theorem involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constan…

equivariant K-理论，仿射Grassmannian和完美性

Equivariant $K$-theory, affine Grassmannian and perfection

研究仿射Schubert变在完美仿射Grassmannian上torus-equivariant代数K-理论，并将其与完美复叠的torus-equivariantHochschild同调比较，证明在F_p线性下为同构，提出新的K-理论环计算方法。

The affine part of the Picard scheme

We describe the maximal torus and maximal unipotent subgroup of the Picard variety of a proper scheme over a perfect field.

Generic torus orbit closures in flag Bott manifolds

In this article the generic torus orbit closure in a flag Bott manifold is shown to be a non-singular toric variety, and its fan structure is explicitly calculated.

环面作用、Morse 同调与仿射空间上点的 Hilbert 概形

Torus actions, Morse homology, and the Hilbert scheme of points on affine space

本文在 motivic 同伦论中提出乘法群作用的猜想，并利用奇异簇的 Morse 理论证明部分结果，进而得出仿射空间上点的 Hilbert 概形与原点处支持的子空间同伦等价。

Unramified affine Springer fibers and isospectral Hilbert schemes

For any connected reductive group $G$ over $\mathbb{C}$, we revisit Goresky-Kottwitz-MacPherson's description of the torus equivariant Borel-Moore homology of affine Springer fibers $\mathrm{Sp}_γ\subset \mathrm{Gr}_G$, where $γ=at^d$, and $a$ is a regular semisimple element in the Lie algebra of $G$. In the case $G = GL_n$, we relate the equivariant cohomology of $\mathrm{Sp}_γ$ to Haiman's work on the isospectral Hilbert scheme of points on the plane. We also explain the connection to the HOMFLY homology of $(n, dn)$-torus links, and formulate a conjecture describing the homology of the Hilb…

Algebraic (Volume) Density Property for Affine Homogeneous Spaces

Let $X$ be a connected affine homogenous space of a linear algebraic group $G$ over $\C$. (1) If $X$ is different from a line or a torus we show that the space of all algebraic vector fields on $X$ coincides with the Lie algebra generated by complete algebraic vector fields on $X$. (2) Suppose that $X$ has a $G$-invariant volume form $ω$. We prove that the space of all divergence-free (with respect to $ω$) algebraic vector fields on $X$ coincides with the Lie algebra generated by divergence-free complete algebraic vector fields on $X$ (including the cases when $X$ is a line or a torus). The pr…

On the coproduct in affine Schubert calculus

The cohomology of the affine flag variety of a complex reductive group is a comodule over the cohomology of the affine Grassmannian. We give positive formulae for the coproduct of an affine Schubert class in terms of affine Stanley classes and finite Schubert classes, in (torus-equivariant) cohomology and K-theory. As an application, we deduce monomial positivity for the affine Schubert polynomials of the second author.

Automorphism Groups of Affine Varieties and a Characterization of Affine n-Space

We show that the automorphism group of affine n-space $A^n$ determines $A^n$ up to isomorphism: If $X$ is a connected affine variety such that $Aut(X)$ is isomorphic to $Aut(A^n)$ as ind-groups, then $X$ is isomorphic to $A^n$ as a variety. We also show that every finite group and every torus appears as $Aut(X)$ for a suitable affine variety $X$, but that $Aut(X)$ cannot be isomorphic to a semisimple group. In fact, if $Aut(X)$ is finite dimensional and $X$ not isomorphic to the affine line $A^1$, then the connected component $Aut(X)^0$ is a torus. Concerning the structure of $Aut(A^n)$ we pro…

环面轨道闭包与无1条带的表

Torus orbit closures and 1-strip-less tableaux

本文比较Grassmannian上一般环面轨道闭包类的Klyachko与Berget-Fink公式，并引入自然产生的半标准组合对象——无1条带表。

Poincare polynomials of generic torus orbit closures in Schubert varieties

The closure of a generic torus orbit in the flag variety $G/B$ of type $A$ is known to be a permutohedral variety and its Poincare polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincare polynomial of a generic torus orbit closure in a Schubert variety in $G/B$. When the generic torus orbit closure in a Schubert variety is smooth, its Poincare polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.

$p$-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。

Torus orbits on homogeneous varieties and Kac polynomials of quivers

In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients.

Demushkin's Theorem in Codimension One

Demushkin's Theorem says that any two toric structures on an affine variety X are conjugate in the automorphism group of X. We provide the following extension: Let an (n-1)-dimensional torus T act effectively on an n-dimensional affine toric variety X. Then T is conjugate in the automorphism group of X to a subtorus of the big torus of X.

Orbit spaces of torus actions on Hessenberg varieties

We consider effective actions of a compact torus $T^{n-1}$ on an even-dimensional smooth manifold $M^{2n}$ with isolated fixed points. We prove that under certain conditions on weights of tangent representations, the orbit space is a manifold with corners. Given that the action is Hamiltonian, the orbit space is homeomorphic to $S^{n+1} \setminus (U_1 \sqcup \ldots \sqcup U_l)$ where $S^{n+1}$ is the $(n+1)$--sphere and $U_1, \ldots, U_l$ are open domains. We apply the results to regular Hessenberg varieties and manifolds of isospectral Hermitian matrices of staircase form.

The Torus-Equivariant Cohomology of Nilpotent Orbits

We consider aspects of the geometry and topology of nilpotent orbits in finite-dimensional complex simple Lie algebras. In particular, we give the equivariant cohomologies of the regular and minimal nilpotent orbits with respect to the action of a maximal compact torus of the overall group in question.