Olivier Haution
Consider a finite l-group acting on the affine space of dimension n over a field k, whose characteristic differs from l. We prove the existence of a fixed point, rational over k, in the following cases: --- The field k is p-special for some prime p different from its characteristic. --- The field k is perfect and fertile, and n = 3.
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A smooth complex rational affine surface with uncountably many real forms
本文构造了一个光滑复有理仿射曲面，证明其具有不可数多个互不同构的实形式，揭示了实代数几何中实形式的丰富性与复杂性。
We prove that any smooth rational projective surface over the field of complex numbers has an open covering consisting of 3 subsets isomorphic to affine planes.
We describe a method to construct completions of affine spaces into total spaces of $\mathbb{Q}$-factorial terminal Mori fiber spaces over the projective line. As an application we provide families of examples with non-rational, birationally rigid and non-stably rational general fibers.
The goal of this paper is to introduce a new constructive geometric proof of the affine version of Chevalley's Theorem. This proof is algorithmic and a verbatim implementation resulted in an efficient code for computing the constructible image of rational maps between affine varieties. Our approach extends the known descriptions of uniform matrix product states to $\operatorname{uMPS}(2,2,5)$
We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves $B$ in smooth projective surfaces $X$ defined over an algebraically closed field of characteristic zero. We observe that except for two exceptions, these surfaces $X \setminus B$ admit infinitely many families of $\mathbb{A}^1$-fibrations over the projective line with irreducible fibers and a unique singular fiber of arbitrarily large multiplicity. For $\mathbb{A}^1$-fibrations over the affine line, we give a new and essentially self-contained proof that the set of equivalence cla…
Rationality of quotients by linear actions of affine groups
设G是SL_n与C^n的（特殊）仿射群，即SL_n与C^n的半直积.本文研究G的表示论，特别是V/G的合理性问题，其中V是一般自由的G-表示.我们表明，这个问题的答案是肯定的，如果V的维数是足够大的V是不可分解的。如果V是两步扩张0 -> S -> V -> Q -> 0，且S，Q是完全可约的，则我们有一个更精确的定理。
Vector Bundles on Rational Topologically Contractible Affine Threefolds
研究有理连通且维数为3的拓扑可缩光滑仿射复簇上代数向量丛是否平凡的问题，通过证明给出肯定回答，如证明科拉斯 - 拉塞尔三维空间上代数向量丛是平凡的。
In this paper we show that not all affine rational complex surfaces can be parametrized birationally and surjectively. For this purpose, we prove that, if S is an affine complex surface whose projective closure is smooth, a necessary condition for S to admit a birational surjective parametrization from an open subset of the affine complex plane is that the infinity curve of S must contain at least one rational component. As a consequence of this result we provide examples of affine rational surfaces that do not admit birational surjective parametrizations.
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.
Detecting affine equivalences between certain types of parametric curves, in any dimension
本文提出一种计算任意维度特定类型参数曲线间仿射等价性的算法，适用于有理曲线及含亚纯分量且有理逆的曲线，通过二元分解避免多项式系统求解，并在Maple中实现。
Semisimple derivations, rational slice and kernels over affine domains
研究半单导出在仿射域中的性质，重点探讨整数特征值导出的核结构，通过有理切片和局部化描述核的构造方法。
We explain an algorithm to calculate Arthur's weighted orbital integral in terms of the number of rational points on the fundamental domain of the associated affine Springer fiber. The strategy is to count the number of rational points of the truncated affine Springer fibers in two ways: by the Arthur-Kottwitz reduction and by the Harder-Narasimhan reduction. A comparison of results obtained from these two approaches gives us recurrence relations between the number of rational points on the fundamental domains of the affine Springer fibers and Arthur's weighted orbital integrals. As an example…
On rational additive group actions
我们刻画了加性群在特征为零的域上定义的代数簇上的有理作用，并给出了加性群与其速度向量场的可积性.这扩展了经典的对应关系的正则作用的加法群仿射代数簇和所谓的局部幂零导子的坐标环。这导致特别是一个完整的表征经常添加剂群作用半仿射品种在其相关的向量场。在其他应用中，我们审查的合理的对应部分的Makar-Limanov不变量的仿射品种的性质，并描述了合理的均匀添加剂群作用复曲面品种的结构。
Algebraic models of the line in the real affine plane
研究实仿射直线到实仿射平面的光滑有理闭嵌入，在双有理微分同胚下分类，发现存在非等价类，区别于经典范畴的唯一性。
Extension of automorphisms of rational smooth affine curves
我们提供的存在性，对于每一个复杂的有理光滑仿射曲线$Γ$，线性作用的$\mathrm{Aut}(Γ)$上的仿射3维空间$\mathbb{A}^3$，连同一个$\mathrm{Aut}(Γ)$ -等变封闭嵌入$Γ$到$\mathbb{A}^3$。不可能减小目标的尺寸，也精确地描述了这种阻塞的原因。
Affine congruences and rational points on a certain cubic surface
我们建立了某些仿射同余的解的个数的估计。然后利用这些估计证明了Q上分裂的三次曲面的Manin猜想，其奇异性类型为D_4。这改善了布朗宁的结果，并回答了Tschinkel提出的问题。
We classify closed curves isomorphic to the affine line in the complement of a smooth rational projective plane conic Q. Over a field of characteristic zero, we show that up to the action of the subgroup of the Cremona group of the plane consisting of birational endomorphisms restricting to biregular auto-morphisms outside Q, there are exactly two such lines: the restriction of a smooth conic osculating Q at a rational point and the restriction of the tangent line to Q at a rational point. In contrast, we give examples illustrating the fact that over fields of positive characteristic, there ex…
We introduce equivariant versions of uniform rationality: given an algebraic group G, a G-variety is called G-uniformly rational (resp. G-linearly uniformly rational) if every point has a G-invariant open neighborhood equivariantly isomorphic to a G-invariant open subset of the affine space endowed with a G-action (resp. linear G-action). We establish a criterion for Gm-uniform rationality of affine variety equipped with hyperbolic Gm-action with a unique fixed point, formulated in term of their Altmann-Hausen presentation. We prove the Gm-uniform rationality of Koras-Russel threefolds of the …
Automorphism groups of certain rational hypersurfaces in complex four-space
罗素三次空间是一个光滑的可收缩的三重仿射复形，它不同构于仿射三空间。在以前的文章中，我们讨论了这个簇的自同构群的结构。在这里，我们回顾了这种结构的一些后果，并推广到其他超曲面的变形的Koras-Russell三倍的一些结果。
Pattern characterization of rationally smooth affine Schubert varieties of type A
有限维旗流形G/P中的Schubert簇是一类研究得很好的以相应的外尔群W的元素为指标的投射簇。特别是这些品种的平滑度和合理平滑度有很多测试。Lakshmibai-Sandhya的一个关键结果是，在类型A中，平滑舒伯特簇恰好是那些通过避免模式4231和3412的排列索引的簇。最近，关于旗流形的无穷维类似物的研究有很多，其中G是Kac-Moody群，W是仿射外尔群或抛物商.本文研究了W是A型仿射外尔群或仿射置换的情形。我们发展了仿射置换的模式避免的概念。我们的主要结果是有理光滑Schubert簇对应于仿射置换的模式4231和3412和扭曲的螺旋置换的一个特征。
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.
Non-complete rational T-varieties of complexity one
本文将复杂度一有理T-簇的Cox环组合方法从完备情形推广到非完备情形，并刻画了相应阶乘分次仿射代数的典范生成元与关系。
The distribution of rational points and polynomial maps on an affine variety over a finite field on average
设$p$是素数，$V/\mathbb{F}_p$是仿射$r$ -空间中的绝对不可约仿射簇.在本文中，我们考虑的问题，如何经常一个盒子$B$将包含预期数量的点。特别是，我们给出了一个下界的体积$B$，保证几乎所有的翻译$B$在$r$ -空间包含预期数量的点。这表明，在“几乎所有”意义上，Weil估计在较小的区域中成立。
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。
We generalize the results of Clemens, Ein, and Voisin regarding rational curves and zero cycles on generic projective complete intersections to the logarithmic setup.
We study real rational models of the euclidean plane $\mathbb{R}^2$ up to isomorphisms and up to birational diffeomorphisms. The analogous study in the compact case, that is the classification of real rational models of the real projective plane $\mathbb{R}\mathbb{P}^2$ is well known: up to birational diffeomorphisms, there is only one model. A fake real plane is a nonsingular affine surface defined over the reals with homologically trivial complex locus and real locus diffeomorphic to $\mathbb{R}^2$ but which is not isomorphic to the real affine plane. We prove that fake planes exist by givin…
Affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces
设$X$是一个有理齐性空间，$QH^*(X)_{loc}^\times$是$X$的小量子上同调环中局部化在量子参数上的可逆元群.我们推广arXiv：math/0609796的结果，并明确实现arXiv：dg-ga/9511011中描述的映射$π_1({\rm Aut}(X))\to QH^*(X)_{loc}^\times$。我们还证明了这个映射是一个嵌入，并在等变量子上同调环$QH^*_T(X)_{loc}^\times$中实现了它。我们给出了这些元素的乘积的显式公式。证明依赖于一个推广，一个商的等变同调环的仿射格拉斯曼，一个公式证明了彼得马扎尔arXiv：0705.3826。它还使用彼得森的未发表的结果-最近证明的林和Shimozono在arXiv：0705.1386 -之间的比较等变同调环的仿射格拉斯曼和等变量子上同调环。
Let X be an affine real algebraic set . We investigate on the theory of algebraically constructible functions on X and the description of the semi-algebraic subsets of X when we replace the polynomial functions on X by some rational continuous functions on X.
Flexible bundles over rigid affine surfaces
我们构造了一个光滑有理仿射曲面S，该曲面具有有限自同构群，但具有以下性质：柱体SxA^2的自同构自同构群在余维至少为2的闭子集的补集上无限传递。这样的表面S特别是刚性的，但相对于Makar-Limanov不变量不是稳定刚性的。
Affine surfaces with a huge group of automorphisms
我们描述了一族有理仿射曲面S，它具有如下意义的自同构群：Aut（S）的所有代数子群生成的正规子群不是由任何可数的这类子群族生成的，并且Aut（S）与这个子群的商包含一个不可数的生成元集上的自由群.
We investigate surjective parametrizations of rational algebraic varieties, in the vein of recent work by Jorge Caravantes, J. Rafael Sendra, David Sevilla, and Carlos Villarino. In particular, we show how to construct plenty of examples of affine surfaces $S$ not admitting a finite surjective morphism $f: \mathbb{A}^2 \to S$.
Given an affine rational complexity-one $T$-variety $X$, we construct an explicit embedding of $X$ in affine space $\mathbb{A}^n$. We show that this embedding is well-poised, that is, every initial ideal of $I_X$ is a prime ideal, and determine the tropicalization of $X$. We then study valuations of the coordinate ring $R_X$ of $X$ which respect the torus action, showing that for full rank valuations, the natural generators of $R_X$ form a Khovanskii basis. This allows us to determine Newton-Okounkov bodies of rational projective complexity-one $T$-varieties, partially recovering (and generali…
Rationality of moduli of elliptic fibrations with fixed monodromy
证明了椭圆K3曲面和椭圆有理曲面的模空间具有固定单值群的合理性结果。
Degenerate Affine Flag Varieties and Quiver Grassmannians
本文通过循环箭图的箭图格拉斯曼流形研究退化仿射旗簇的有限维逼近，证明了其胞腔分解及不可约分量的正规性与有理奇点性质。
On C-fibrations over projective curves
本文的目标是提出一个修改后的版本（GML）的ML不变，它应该考虑到在一个投影基地的裁决，并允许进一步分层光滑仿射有理曲面。我们提供了一个非平凡的例子，GML不变计算光滑仿射有理表面承认没有C-行动。我们将GML不变量应用于某些三重ML不变量的计算。
On the field of differential rational invariants of a subgroup of affine group (Partial differential case)
本文考虑特征为零的微分域$(F;\partial_1,...,\partial_m)$，仿射群$ GL(n,C)\propto C^n$的子群$H$关于它在$F^n$中的恒等表示，以及$x=(x_1,x_2,...,x_n)$列向量中的两个微分有理函数域$$C< x, \partial >^H=\{f^{\partial}< x> \in C< x, \partial> : f^{\partial}< hx+ h_0> = f^{\partial}< x> {whenever} (h,h_0)\in H \},$$ $$C< x, \partial>^{(GL^{\partial}(m,F),H)}=\{f^{\partial}< x> \in C< x, \partial> : f^{g^{-1}\partial}< hx+ h_0> = f^{\partial}< x> {whenever} g\in GL^{\partial}(m,F) {and} (h,h_0)\in H \}$$，其中$C$是$(F,\partial)$的常数域，$C< x, \partial>$是$\partial$-$x_1,x_2,...,x_n$中的$C$和$$GL^{\partial}(m,F)= \{g=(g_{jk})_{j,k=\bar{1,m}}\in GL(m,F): \partial…
Fields of Parametrization and Optimal Affine Reparametrization of Rational Curves
在本文中，我们提出了三个相关的结果参数化领域的主题。设C是特征为零的域上的有理曲线。设K是在Q上生成的域，使得它是C的定义域但不是参数化域。已知存在K的二次扩张来参数化C。首先，我们证明了存在无穷多个K的二次扩张是C的参数化域。因此，我们证明了证人品种，出现在上下文中的参数Weil的下降方法，总是一个特殊的曲线相关的代数扩张，称为超圆。证明簇可能不是给定扩张的超圆，而是另一个超圆。我们使用这两个事实，提出了一个算法来解决以下的最佳重新参数化问题。给定曲线C的双有理参数化f（t），计算在+b处的仿射重新参数化，使得f（at+b）在尽可能小的域上具有系数。该算法的主要优点是不需要计算曲线上的任何有理点。
Affine complements in the projective space with the trivial group of automorphisms
该研究构造了一类新的仿射补集，证明当 $M$ 为不小于5的奇数且 $S$ 为有理超曲面时，其自同构群为平凡群。
Approximate Parametrization of Plane Algebraic Curves by Linear Systems of Curves
众所周知，不可约代数曲线是有理的（即参数的）当且仅当它的亏格为零。本文给出了一个公差$ε>0$和一个$ε$ -不可约代数仿射平面曲线$\mathcal C$的正次数$d$，引入了$ε$ -有理性的概念，并给出了一个利用$(d-2)$ -次数曲线的线性系来参数化近似仿射$ε$ -有理平面曲线的算法.该算法输出次数至多为$d$的有理曲线$\bar{\mathcal C}$的有理参数化，该有理曲线在无穷远处具有与$\mathcal C$相同的点。此外，虽然我们没有提供理论分析，但我们的实证分析表明，$\bar{\mathcal C}$和$\mathcal C$在实践中是接近的。
On the field of differential rational invariants of a subgroup of affine group (Ordinary differential case)
考虑了特征为零的常微分域$(F,d)$，仿射群$ GL(n,C)\propto C^n$的子群$H$关于它在$F^n$中的恒等表示，以及$x=(x_1,x_2,...,x_n)$列向量中的两个微分有理函数域$$C< x, d>^H=\{f^d< x> \in C< x, d> : f^d< hx+ h_0> = f^d< x> {for any} (h,h_0)\in H \},$$ $$C< x, d>^{(F^*,H)}=\{f^d< x> \in C< x, d> : f^{g^{-1}d}< hx+ h_0> = f^d< x> {for any} g\in F^* {and} (h,h_0)\in H \}$$，其中$C$是$(F,d)$的常数域，$C< x, d>$是$C$上的$x_1,x_2,...,x_n$中的微分有理函数域.域$C< x, d>^H$（$C< x, d>^{(F^*,H)}$）是研究路径等价问题的重要工具。曲线）相对于运动组$H$的微分几何。在本文中，一个纯代数的方法来描述这些领域。研究了场$C< x, d>^{(F^*,H)}$及其与$C< x, d>^H$的关系。它还表明，$C< x, d>^H$可以从$H$的一些代数（无导数）不变量。关键词：微分域，微分有理函数，不变量，微分超越度。2000年数学学科分类：12H05、53A04、53A55
We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group $\mathbb{S}^1$ up to equivariant isomorphisms. As an application, we show that every compact differentiable surface endowed with an action of the circle $S^1$ admits a unique smooth rational real quasi-projective model up to $\mathbb{S}^1$-equivariant birational diffeomorphism.
Fixed points of group actions and rational maps
本文的目的是对Reichstein和Youssin（Math.AG/9903162）关于有限群作用在有理映射下的不动点的性质的一些结果给出简单的证明。我们的证明适用于任何特性。我们还给出了西村引理的一个简短证明。
We investigate geometry of D-affine varieties. Our main result is that a D-affine rational projective surface over an algebraically closed field is a generalised flag variety of a reductive group.
For $G$ a connected reductive group, $γ\in \kg(F)$ semisimple regular integral, we introduce a fundamental domain $F_γ$ for the affine Springer fibers $\xx_γ$. There is a beautiful way to reduce the purity conjecture of $\xx_γ$ to that of $F_γ$, we call it the Arthur-Kottwitz reduction. When restricted to the unramified case, it turns out that these fundamental domains behave well in family. We formulate a rationality conjecture about a generating series of their Poincaré polynomials. We then study them in detail for the group $\gl_{3}$. In particular, we pave them in affine spaces and we prov…
We introduce a characterization for affine equivalence of two surfaces of translation defined by either rational or meromorphic generators. In turn, this induces a similar characterization for minimal surfaces. In the rational case, our results provide algorithms for detecting affine equivalence of these surfaces, and therefore, in particular, the symmetries of a surface of translation or a minimal surface of the considered types. Additionally, we apply our results to designing surfaces of translation and minimal surfaces with symmetries, and to computing the symmetries of the higher-order Enn…
Affine Demazure modules and $T$-fixed point subschemes in the affine Grassmannian
设$G$是定义在$\C$上的$A$或$D$型单代数群，$T$是$G$的极大环面.对于$G$的支配余权$λ$，仿射格拉斯曼$Gr_G$中舒伯特簇$\bar{Gr}_G^λ$的$T$ -不动点子格式$(\bar{Gr}_G^λ)^T$是有限格式.证明了$λ$的一级仿射Demazure模的对偶与$(\bar{Gr}_G^λ)^T$的函数环（由$Gr_G$上的某个线丛扭曲）之间存在自然同构.利用这一事实给出了$A,D,E$型仿射代数的基本表示与格顶点代数之间的Frenkel-Kac-Segal同构的几何证明。
Generic Bernstein-Sato polynomial on an irreducible affine scheme
给出了在包含$\mathbb{Q}$的交换酉整环$\mathcal{C}$中系数为$p$多项式的情形，定义了不可约仿射概型$V \subset \text{Spec}(\mathcal{C})$上的一般Bernstein-Sato多项式的概念.我们证明了这样一个非零有理多项式的存在性，它覆盖和推广了H.比奥斯卡当$\mathcal{C}$是代数或解析空间的环时，我们推导出参数空间的分层，使得在每一层上，存在一个非零有理多项式，该多项式对该层的任何一点都是Bernstein-Sato多项式.这推广了A. Leykin在病例$p=1$中获得。
For G a semisimple algebraic group, we revisit the description of the components of the affne Springer fiber given by ts, with s a regular semisimple element. We then compute the fixed points of each component of a particular affne Springer fiber for Type A
1-flat irreducible G-structures, equivalently, irreducible G-structures admitting torsion-free affine connections, have been studied extensively in differential geometry, especially in connection with the theory of affine holonomy groups. We propose to study them in a setting in algebraic geometry, where they arise from varieties of minimal rational tangents (VMRT) associated to families of minimal rational curves on uniruled projective manifolds. We prove that such a structure is locally symmetric when the dimension of the uniruled projective manifold is at least 5. By the classification resu…
A cusp singularity is an elliptic surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In 1981, Looijenga proved that whenever a cusp singularity is smoothable, the minimal resolution of the dual cusp is an anticanonical divisor of some smooth rational surface. He conjectured the converse. Recent work of Gross, Hacking, and Keel has proven Looijenga's conjecture using methods from mirror symmetry. This paper provides an alternative proof of Looijenga's conjecture based on a combinatorial criteri…
Over a field $k$, we study rational UFDs of finite transcendence degree $n$ over $k$. We classify such UFDs $B$ when $n=2$, $k$ is algebraically closed, and $B$ admits a positive $\mathbb{Z}$-grading, showing in particular that $B$ is affine over $k$. We also consider the Russell cubic threefold over $\mathbb{C}$, and the Asanuma threefolds over a field of positive characterstic, showing that these threefolds admit no elliptic $\mathbb{G}_m$-action. Finally, we show that, if $X$ is an affine $k$-variety and $X\times\mathbb{A}^m_k\cong_k\mathbb{A}^{n+m}_k$, then $X\cong_k\mathbb{A}^n_k$ if and …
Symplectic log Kodaira dimension $-\infty$, affine-ruledness and unicuspidal rational curves
本文研究了辛对数柯尔加雷维度为-∞的辛对 (X, ω, D) 的性质，引入了辛仿射被覆性的概念，并探讨了无 cusps 理性曲线在该结构中的作用。
Bass' triangulability problem
研究了仿射Cremona群的幂幺代数子群的Bass可三角化问题，证明了一个可三角化准则，证明了Cremona群的不可三角化连通可解仿射代数子群的存在性，以及这类子群的稳定可三角化性，特别地，在稳定范围内，我们回答了Bass可三角化问题是肯定的.为此，我们证明了一个定理的不变子域的$1$ -扩张。我们还得到了一般的建设，所有合理的三角化子群的Cremona集团，并作为应用，分类合理的三角化连接一维幂幺仿射代数子群的Cremona集团共轭。
Relating decision and search algorithms for rational points on curves of higher genus
对于定义在亏格至少为2的有理数上的仿射平面曲线，我们证明了解的存在性的判定算法也产生了所有解的搜索算法。
Let $G$ be a split connected reductive group over a finite field of characteristic $p > 2$ such that $G_\text{der}$ is absolutely almost simple. We give a geometric construction of perverse $\mathbb{F}_p$-sheaves on the Iwahori affine flag variety of $G$ which are central with respect to the convolution product. We deduce an explicit formula for an isomorphism from the spherical mod $p$ Hecke algebra to the center of the Iwahori mod $p$ Hecke algebra. We also give a formula for the central integral Bernstein elements in the Iwahori mod $p$ Hecke algebra. To accomplish these goals we construct …
We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the $\tilde{C}_n$ version of the construction to construct a flat noncommutative deformation of the $n$th symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.
Smooth and palindromic Schubert varieties in affine Grassmannians
设G是复数上的单连通单紧李群。仿射Grassmannian是一个投射的ind-簇，同伦等价于G的闭路空间，并且非常类似于经典Grassmannian流形的极大旗簇。它有一个Schubert胞腔剖分，其指标为coroot格或等价地为仿射外尔群模G的外尔群的最小长度陪集表示.仿射舒伯特胞腔的闭包是一个有限维的射影簇，我们称之为仿射舒伯特簇。在本文中，我们完全确定光滑和回文（理性光滑）仿射舒伯特簇。
The Geometry of Fixed Point Varieties on Affine Flag Manifolds
设$G$是代数闭域$k$上的半单单连通代数群，其李代数为$\frak g$ .本文研究了$\frak g\otimes k((π))$中含有不动点零椭圆元的仿射子代数空间，即仿射旗流形上的不动点簇。我们定义了一个自然类的$k^*$ -仿射旗流形上的行动，推广的Lusztig和Smelt介绍的行动。我们在由$N\in\frak{g}\otimes k((π))$和指定类型的$k^*$ -作用$f$组成的$(N,f)$对上给出一个条件，该条件保证$f$诱导包含$N$的对位子代数簇上的作用.对于特殊的线性群和辛群，我们刻画了所有正则半单和零椭圆共轭类，它们包含一个其不动点簇允许这样一个作用的代表。然后，我们使用这些行动找到简单的公式的欧拉特征的那些品种的$k^*$ -不动点是有限的。我们还得到了包含$\frak g$的某些幂零共轭类的给定元素的抛物子代数空间的欧拉特征的组合描述。
In a previous paper Affine symmetries of the equivariant quantum cohomology of rational homogeneous spaces, a general formula was given for the multiplication by some special Schubert classes in the quantum cohomology of any homogeneous space. Although this formula is correct in the non equivariant setting, the stated equivariant version was wrong. We provide corrections for the equivariant formula, thus giving a correct argument for the non equivariant formula. We also give new formulas in the equivariant homology of the affine grassmannian that could lead to Pieri type formulas.
Derived Fundamental Groups for Tate Motives
我们构造了域上连通光滑概形上Tate动机的导出基本群格式。我们发现，存在一个亲仿射派生群计划的有理数，使其类别的完美表示模型的三角形类的合理混合泰特动机。根据一个假设，这是弱于积分版本的Beilinson-Soule消失猜想，我们表明，有一个仿射导出群计划的整数，使其完美的表示模型泰特动机与积分系数。例如，对于数字字段满足该假设。这推广了以前的非衍生结构的基本群计划的泰特动机与合理的系数。
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