Conjugacy classes of groups of prime order in $\mathrm{PGL}_{k+1}(\mathbb{C})$

特征≤7下休眠算子的广义超几何方程与二维TQFT

Generalized hypergeometric equations and $2$d TQFT for dormant opers in characteristic $\leq 7$

该研究针对素特征p下的广义超几何微分方程导出的PGLₙ-算子，证明其在休眠算子类中具刚性，并结合已有计数几何成果，给出p≤7时计算休眠PGLₙ-算子数量的二维TQFT的完整显式描述。

A classification of degree $2$ semi-stable rational maps $\mathbb{P}^2\to\mathbb{P}^2$ with large finite dynamical automorphism group

Let $K$ be an algebraically closed field of characteristic $0$. In this paper we classify the $\text{PGL}_3(K)$-conjugacy classes of semi-stable dominant degree $2$ rational maps $f:{\mathbb P}^2_K\dashrightarrow{\mathbb P}^2_K$ whose automorphism group $$\text{Aut}(f):=\{ϕ\in\text{PGL}_3(K): ϕ^{-1}\circ f\circϕ=f\}$$ is finite and of order at least $3$. In particular, we prove that $\#\text{Aut}(f)\le24$ in general, that $\#\text{Aut}(f)\le21$ for morphisms, and that $\#\text{Aut}(f)\le6$ for all but finitely many conjugacy classes of $f$.

The Verlinde formula for PGL(p)

Let G be a complex semi-simple group, X a Riemann surface, M_G the moduli space of principal G-bundles on X. When G is simply-connected, there exists a closed formula expressing the dimension of the space H^0(M_G,L) for any line bundle L on M_G (this is usually called among mathematicians, somewhat incorrectly, the "Verlinde formula").In this paper we find an analogous formula for G = PGL(r) where r is prime.

Prime order isometries of unimodular lattices and automorphisms of ihs manifolds

We characterize conjugacy classes of isometries of odd prime order in unimodular Z-lattices. This is applied to give a complete classification of odd prime order non-symplectic automorphisms of irreducible holomorphic symplectic manifolds up to deformation and birational conjugacy.

中心同源与约化群中的共轭类

Central isogenies and conjugacy classes in reductive groups

将Steinberg关于半单元素中心化子分量群的描述推广到解释当通用覆盖非平展时幂单元素中心化子非约化的现象，并应用于L-参数模空间和通用变形环的特殊纤维中的重数计算。

Topology and purity for torsors

We study the homotopy theory of the classifying space of the complex projective linear groups to prove that purity fails for $PGL_p$-torsors on regular noetherian schemes when $p$ is a prime. Extending our previous work when $p=2$, we obtain a negative answer to a question of Colliot-Thélène and Sansuc, for all $PGL_p$. We also give a new example of the failure of purity for the cohomological filtration on the Witt group, which is the first example of this kind of a variety over an algebraically closed field.

Three chapters on Cremona groups

In the first part of this article, we answer a question of I. Dolgachev, which is related to the following problem: given a birational map $f\in\mathrm{Bir}(\mathbb{P}^m_\mathbf{k})$ and a linear projective map $A\in\mathrm{PGL}_{m+1}(\mathbf{k})$, when is $A\circ f$ regularizable? Dolgachev's initial question is whether this may happen for all $A$ in $\mathrm{PGL}_{m+1}(\mathbf{k})$, and the answer is negative. We then look at the sequence $n\mapsto\mathrm{deg}\, f^n$, $f\in\mathrm{Bir}(\mathbb{P}^2_\mathbf{k})$. We show that there is no constraint on the sequence $n\mapsto\mathrm{deg}\, f^n-…

p-elementary subgroups of the Cremona group

We classify, up to conjugacy, the subgroups of the Cremona group isomorphic to (Z/p)^r, where p is prime and r is maximal.

The essential dimension of the normalizer of a maximal torus in the projective linear group

Let p be a prime, k be a field of characteristic different from p containing a primitive p-th root of unity and N be the normalizer of the maximal torus in the projective linear group PGLn. We compute the exact value of the essential dimension ed(N;p) of N at p for every n.

Some torsion classes in the Chow ring and cohomology of $BPGL_n$

In the integral cohomology ring of the classifying space of the projective linear group $PGL_n$ (over $\mathbb{C}$), we find a collection of $p$-torsions $y_{p,k}$ of degree $2(p^{k+1}+1)$ for any odd prime divisor $p$ of $n$, and $k\geq 0$. If in addition, $p^2\nmid n$, there are $p$-torsion classes $ρ_{p,k}$ of degree $p^{k+1}+1$ in the Chow ring of the classifying stack of $PGL_n$, such that the cycle class map takes $ρ_{p,k}$ to $y_{p,k}$. We present an application of the above classes regarding Chern subrings.

On unramified automorphic forms over the projective line

Let $q$ be a prime power and $\mathbb{F}_q$ be the finite field with $q$ elements. In this article we investigate the space of unramified automorphic forms for $\mathrm{PGL}_n$ over the rational function field defined over $\mathbb{F}_q$ (i.e.\ for $\mathbb{P}^1$ defined over $\mathbb{F}_q$). In particular, we prove that the space of unramified cusp form is trivial and (for $n=3$) that the space of eigenforms is one dimensional. Moreover, we show that there are no nontrivial unramified toroidal forms for $\mathrm{PGL}_3$ over $\mathbb{P}^1$ and conjecture that the space of all toroidal automor…

On planar Cremona maps of prime order

This paper contains a new proof of the classification of elements of prime order in the Cremona group Bir(P^2), up to conjugation. In addition, we give explicit geometric constructions of these Cremona transformations, and provide a parameterization of their conjugacy classes. Analogous constructions in higher dimensions are also discussed.

关于 $\mathrm{GL}_n(\mathbb{C})$ 和 $\mathrm{PGL}_n(\mathbb{C})$ 的有限不可约子群

On the finite irreducible subgroups of $\mathrm{GL}_n(\mathbb{C})$ and $\mathrm{PGL}_n(\mathbb{C})$

本文是一篇面向非专业读者的综述，梳理了$\mathrm{GL}_n(\mathbb{C})$和$\mathrm{PGL}_n(\mathbb{C})$有限不可约子群的分类成果，给出了不同维数下的分类结论与群列表，并开发程序提升相关结果的可获取性。