Projective normality of finite group quotients and EGZ theorem

Projective normality of quotient varieties modulo finite groups

In this note, we prove that for any finite dimensional vector space $V$ over an algebraically closed field $k$, and for any finite subgroup $G$ of $GL(V)$ which is either solvable or is generated by pseudo reflections such that the $|G|$ is a unit in $k$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |G|}$.

Projective normality of Weyl group quotients

In this note, we prove that for the standard representation $V$of the Weyl group $W$ of a semi-simple algebraic group of type $A_n, B_n, C_n, D_n, F_4$ and $G_2$ over $\mathbb C$, the projective variety $\mathbb P(V^m)/W$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |W|}$, where $V^m$ denote the direct sum of $m$ copies of $V$. We also prove that for any finite group $G$ and for any finite dimentional representation $V$ over $\mathbb C$, the projective variety $P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes n!}$ as a conse…

Affine hyperplane arrangements and Jordan classes

We study the geometry of the stratification induced by an affine hyperplane arrangement H on the quotient of a complex affine space by the action of a discrete group preserving H. We give conditions ensuring normality or normality in codimension 1 of strata. As an application, we provide the list of those categorical quotients of closures of Jordan classes and of sheets in all complex simple algebraic groups that are normal. In the simply connected case, we show that normality of such a quotient is equivalent to its smoothness.

On torus quotients of Schubert varieties in orthogonal Grassmannian-II

Let $G=SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n+2}$. In this article, we prove projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle.

Normality of orbit closures in the enhanced nilpotent cone

We continue the study of the closures of $GL(V)$-orbits in the enhanced nilpotent cone $V\times\cN$ begun by the first two authors. We prove that each closure is an invariant-theoretic quotient of a suitably-defined enhanced quiver variety. We conjecture, and prove in special cases, that these enhanced quiver varieties are normal complete intersections, implying that the enhanced nilpotent orbit closures are also normal.

格拉斯曼流形在一维环面作用下的GIT商

On the GIT quotient of Grassmannians by one dimensional torus

本文研究格拉斯曼流形在对应单根的一维环面作用下的GIT商结构，利用外尔群组合刻画半稳定轨迹，分析轨道结构并计算相关几何不变量。