A. A'Campo-Neuen, J. Hausen
Consider an algebraic torus of small dimension acting on an open subset of a complex vector space, or more generally on a quasiaffine variety such that a separated orbit space exists. We discuss under which conditions this orbit space is quasiprojective. One of our counterexamples provides a toric variety with enough effective invariant Cartier divisors that is not embeddable into a smooth toric variety.
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Toric varieties admitting an action of a unipotent group with a finite number of orbits
本研究刻画了容许有限轨道数幂零群作用的完备单纯环面簇，并给出了二维情形下这类环面簇的完整列表，为相关代数几何研究提供了分类结果。
Let X be a complete toric variety and Aut(X) be the automorphism group. We give an explit description of Aut(X)-orbits on X. In particular, we show that Aut(X) acts on X transitively if and only if X is a product of projective spaces.
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 $n\geq 4$.
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.
A subset X of a vector space V is said to have the "Separation Property" if it separates linear forms in the following sense: given a pair (a, b) of linearly independent forms on V there is a point x on X such that a(x)=0 and b(x) is not equal to 0. A more geometric way to express this is the following: every homogeneous hyperplane H in V is linearly spanned by its intersection with X. The separation property was first asked for conjugacy classes in simple Lie algebras. We give an answer for orbit closures in representation spaces of an algebraic torus. We consider also the strong and the weak…
We address the distribution properties of points of small height on proper toric varieties and applications to the related Bogomolov property. We introduce the notion of monocritical toric metrized divisor and we prove that equidistribution occurs for every generic, small sequence with respect to a toric metrized divisor, for every place if and only if the divisor is monocritical. Furthermore, when this is the case, the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of the ambient toric variety. We also study the $v$-adic modulus distrib…
The Nash problem on arc families is affirmatively answered for a toric variety by Ishii and Kollar's paper which also shows the negative answer for general case. The Nash problem is one of questions about the relation between arc families and valuations. In this paper, the relation is described clearly for a toric variety. The arc space of a toric variety admits an action of the group scheme determined by the torus. Each orbit on the arc space corresponds to a lattice point in the cone and therefore corresponds to a toric valuation. The dominant relation among the orbits is described in terms …
Orbit spaces of Weyl groups acting on compact tori: a unified and explicit polynomial description
本文提出了一种统一且明确的多项式描述方法，用于描述韦尔群作用于紧致环面的轨道空间，并应用于具有晶体对称性的三角多项式优化。
The purpose of this short note is to prove a formula for the Chern-Mather classes of a toric variety in terms of its orbits and the local Euler obstructions at general points of each orbit (Theorem 2). We use the general definition of the Chern-Schwartz-MacPherson classes (see [7]) and their special expression in case of a toric variety (see [2]). As a corollary, we obtain a formula by Matsui-Takeuchi [8, Corollary 1.6]. Alternatively, one could deduce the formula of Theorem 2 from the Matsui-Takeuchi formula, by using our general result [11, Theoreme 3] for the degree of the polar varieties i…
We write the equivariant Todd class of a general complete toric variety as an explicit combination of the orbit closures, the coefficients being analytic functions on the Lie algebra of the torus which satisfy Danilov's requirement.
Equivariant unirationality of tori in small dimensions
本文研究有限群在小维数环面上作用的等变单有理性质，聚焦该代数几何问题的基础分析。
In 2013 Bazhov proved a criterium for two points on a complete toric variety to lie in the same orbit of the neutral component of automorphism group. This criterium is in terms of divisor class group. Arzhantsev-Bazhov (2013) obtained a similar criterium for affine toric varieties. We prove a necessary condition similar this criteria to the cases of affine and projective horospherical varieties.
We compute the motivic Donaldson-Thomas theory of small crepant resolutions of toric Calabi-Yau 3-folds.
Radiant toric varieties and unipotent group actions
该研究针对具有开轨道的极大幂零子群作用的完备环面簇，刻画其性质，分析对应扇的Demazure根、幂零子群结构及相关正则子群，推进环面簇与群作用领域的研究。
We classify smooth Fano threefolds that admit degenerations to toric Fano threefolds with ordinary double points.
Let $G$ be a simple algebraic group over an algebraically closed field $K$ and let $N = N_G(T)$ be the normalizer of a fixed maximal torus $T\leq G$. Further, let $U$ be the unipotent radical of a fixed Borel subgroup $B$ that contains $T$ and let $U^-$ be the unipotent radical of the opposite Borel subgroup $B^-$. The Bruhat decomposition implies the decomposition $G = NU^-UN$. The Zariski closed subset $U^-U\subset G$ is isomorphic to the affine space $A_K^m$ where $m = \dim G -\dim T$ is the number of roots in the corresponding root system. Here we construct a subgroup $\mathcal{N}\leq \ope…
By an additive action on an algebraic variety $X$ we mean a regular effective action $\mathbb{G}_a^n\times X\to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. In this paper, we give a classification of additive actions on complete toric surfaces.
An infinite type toric variety is a normal toric variety given by a fan with infinitely many cones. We construct examples in this paper coming from representation theory of loop groups. The fans that appear are cones on Voronoi tilings on a vector space equipped with an inner product. We also explain the affine analogue of the connection between a generic torus orbit closure in a flag variety and the closure a maximal torus in the wonderful compactification.
Rational Curves in Projective Toric Varieties
本文研究了射影托利丘变种中有理曲线的构造与性质，提出度数为d的凯莱结构，并建立了希尔伯特方案与最大光滑凯莱结构之间的对应关系。
We prove that the intersection homology Poincare' polynomial P(X) of an affine toric variety X is bounded below by the product P(Y)P(X/Y), where Y is the closure of any orbit in X and X/Y is a slice transverse to the orbit. This proves a combinatorial conjecture of Kalai for rational polytopes.
By an additive action on an algebraic variety $X$ we mean a regular effective action $\mathbb{G}_a^n\times X\to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. In this paper, we give a uniqueness criterion for additive action on a complete toric variety.
Let \X be an affine toric variety under a torus \T and let T be a subtorus. The general T-orbit closures in \X and their flat limits are parametrized by the main component H_0 of the toric Hilbert scheme. Further, the quotient torus \T/T acts on H_0 with a dense orbit. We describe the fan of this toric variety; this leads us to an integral analogue of the fiber polytope of Billera and Sturmfels. We also describe the relation of H_0 to the main component of the inverse limit of GIT quotients of \X by T.
We show that the dual of the cone of divisors on a complete Q-factorial toric variety X whose stable base loci have dimension less than k is generated by curves on small modifications that move in families sweeping out the birational transforms of k-dimensional subvarieties of X. We give an example showing that it does not suffice to consider curves on X itself.
Perverse sheaves on smooth toric varieties and stacks
本文将光滑环面簇及其轨道分层上的反常层范畴显式刻画为某代数上的有限维模范畴，并推广到等变情形以描述环面轨形上的反常层，同时比较了相关导出范畴。
By an additive action on an algebraic variety $X$ of dimension $n$ we mean a regular action $\mathbb{G}_a^n \times X \to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. We prove that if a complete toric variety $X$ admits an additive action, then it admits an additive action normalized by the acting torus. Normalized additive actions on a toric variety $X$ are in bijection with complete collections of Demazure roots of the fan of $X$. Moreover, any two normalized additive actions on $X$ are isomorphic.
We study the enumerative geometry of orbits of multidimensional toric action on projective algebraic varieties and develop a new cyclic differential-graded operad, conjecturally governing the real version of the enumerative geometry of these toric orbits.
In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference ideals, and group actions by difference tori. Connections between toric difference varieties and affine N[x]-semimodules are established by proving the correspondence between the irreducible invariant difference subvarieties and the faces of the N[x]-submodules and the orbit-face correspondence. Finally, an algorithm is given to decide whether a binomial di…
The K-rings of non-singular complex pro jective varieties as well as quasi- toric manifolds were described in terms of generators and relations in an earlier work of the author with V. Uma. In this paper we obtain a similar description for complete non-singular toric varieties. Indeed, our approach enables us to obtain such a description for the more general class of torus manifolds with locally standard torus action and orbit space a homology polytope.
Let $G$ be a semisimple algebraic group over $\mathbb{C}$. For a reduced word $\bf i$ of the longest element in the Weyl group of $G$ and a dominant integral weight $λ$, one can construct the string polytope $Δ_{\bf i}(λ)$, whose lattice points encode the character of the irreducible representation $V_λ$. The string polytope $Δ_{\bf i}(λ)$ is singular in general and combinatorics of string polytopes heavily depends on the choice of $\mathbf i$. In this paper, we study combinatorics of string polytopes when $G = SL_{n+1}(\mathbb{C})$, and present a sufficient condition on $\mathbf i$ such that …
Delta-matroids and toric degenerations in OG(n,2n+1)
本文在极大正交Grassmann流形OG(n,2n+1)中构造了环面轨道闭包到Richardson簇并集的显式退化，由此得到上同调类公式，并揭示退化片段对应delta-拟阵基多面体对单位超立方体的分解。
We show that Fourier transforms on the Weyl algebras have a geometric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential operators on regular toric varieties, whose fans are related to each other by reflections of one-dimensional cones. The simplest class of examples is provided by the toric varieties related by such reflections to projective spaces. It includes the blow-up at a point in affine space and resolution of singularities of varieties appearing in the study of the minimal orbit of sl(n+1).
Toric Richardson Varieties and Slice Links
本文证明类型A辫子簇中标准环面有稠密轨道当且仅当相关链环光滑可切片，并将光滑切片亏格与轨道余维数联系，刻画代数环面，证明positroid链环的切片性等价于平凡性。
We first construct a derived equivalence between a small crepant resolution of an affine toric Calabi-Yau 3-fold and a certain quiver with a superpotential. Under this derived equivalence we establish a wall-crossing formula for the generating function of the counting invariants of perverse coherent systems. As an application we provide certain equations on Donaldson-Thomas, Pandeharipande-Thomas and Szendroi's invariants. Finally, we show that moduli spaces associated with a quiver given by successive mutations are realized as the moduli spaces associated the original quiver by changing the s…
We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of connected components of refined open Bruhat cells in Grassmanians G(k,n) over real numbers.
We determine the integral homology of the orbit space of a maximal compact torus action on the Grassmannian Gr(2,5). Our approach uses the well-known Geometric Invariant Theory of the maximal algebraic torus action on this Grassmannian.
The cohomological rigidity problem for toric manifolds asks whether toric manifolds are diffeomorphic (or homeomorphic) if their integral cohomology rings are isomorphic. Many affirmative partial solutions to the problem have been obtained and no counterexample is known. In this paper, we study the diffeomorphism classification of toric Fano $d$-folds with $d=3,4$ or with Picard number $\ge 2d-2$. In particular, we show that those manifolds except for two toric Fano $4$-folds are diffeomorphic if their integral cohomology rings are isomorphic. The exceptional two toric Fano $4$-folds (their ID…
Functoriality of toric coherent-constructible correspondence
本文证明了环面簇商栈上的 ind-相干层范畴与实环面覆盖上奇异支撑在 FLTZ 骨架中的层范畴等价，推广了 Kuwagaki 的非等变对应，并研究了环面态射和轨道闭包包含下的函子性。
Given a reductive group $G$ and a parabolic subgroup $P\subset G$, with maximaltorus $T$, we consider (following Dabrowski's work) the closure $X$ of a generic $T$-orbit in $G/P$, and determine in combinatorial termswhen the toric variety $X$ is $\mathbb{Q}$-Gorenstein Fano, extending in this way the classification of smooth Fano generic closures given by Voskresenski\uı and Klyachko. As an application, we apply the well known correspondence between Gorenstein Fano toric varieties and reflexive polytopes in order to exhibit which reflexive polytopes correspond to generic closures -- this list …
We classify stably/retract rational norm one tori in dimension $n-1$ for $n=2^e$ $(e\geq 1)$ is a power of $2$ and $n=12, 14, 15$. Retract non-rationality of norm one tori for primitive $G\leq S_{2p}$ where $p$ is a prime number and for the five Mathieu groups $M_n\leq S_n$ $(n=11,12,22,23,24)$ is also given.
Minimal Nilpotent Orbits and Toric Varieties
本文研究了 $\mathfrak{sl}_{n+1}(\mathbb C)$ 中极小幂零轨道的概形论交集，证明其坐标环可平坦退化至 $C_n$ 型环面簇的等变上同调环，并证明了该交集及其轨道簇的既约性与 Cohen-Macaulay 性质。
We establish connections between invariant theory and maximum likelihood estimation for discrete statistical models. We show that norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We use notions of stability under a torus action to characterize the existence of the maximum likelihood estimate, and discuss connections to scaling algorithms.
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