M. Jablonski
Let G be a real or complex linear algebraic reductive group. Let H and F be reductive subgroups. We study the natural H action on G/F. The main theorem of this note shows that generic H orbits are closed. This theorem is then applied to study representations of G, representations of H which are induced from representations of G, and intersections of reductive subgroups of G.
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Effective good divisibility of rational homogeneous varieties
本文计算了有理齐性簇的有效好整除性，推广了Naldi和Occhetta关于复Grassmann流形的结果，并由此证明了到经典型Lie群有理齐性簇的非常值态射不存在。
For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies some simple condition.
In 2011, Khurana, Lam and Wang define the following property. (*)A commutative unital ring A satisfies the property ''power stable range one'' if for all a, b $\in$ A with aA + bA = A there are an integer N = N (a, b) $\ge$ 1 and $λ$ = $λ$(a, b) $\in$ A such that b N + $λ$a $\in$ A x , the unit group of A. In 2019, Berman and Erman consider rings with the following property (**) A commutative unital ring A has enough homogeneous polynomials if for any k $\ge$ 1 and set S := {p 1 , p 2 , ..., p k } , of primitive points in A n and any n $\ge$ 2, there exists an homogeneous polynomial P (X 1 , X…
Let $A$ be an abelian variety over a field. The homogeneous (or translation-invariant) vector bundles over $A$ form an abelian category ${\rm HVec}_A$; the Fourier-Mukai transform yields an equivalence of ${\rm HVec}_A$ with the category of coherent sheaves with finite support on the dual abelian variety. In this paper, we develop an alternative approach to homogeneous vector bundles, based on the equivalence of ${\rm HVec}_A$ with the category of finite-dimensional representations of a commutative affine group scheme (the "affine fundamental group" of $A$). This displays remarkable analogies …
We consider those projective bundles (or Brauer-Severi varieties) over an abelian variety that are homogeneous, i.e., invariant under translation. We describe the structure of these bundles in terms of projective representations of commutative algebraic groups; the irreducible bundles correspond to Heisenberg groups and their standard representations. Our results extend those of Mukai on semi-homogeneous vector bundles, and yield a geometric view of the Brauer group of abelian varieties.
We give a characterization of connected solvable groups in terms of the existence of representations with certain geometric properties. The existence of such representations for the group of upper triangular matrices played an important role in the proof of the authors' equivariant Riemann-Roch theorem.
We give a stratification of the GIT quotient of the Grassmannian $G_{2,n}$ modulo the normaliser of a maximal torus of $SL_{n}(k)$ with respect to the ample generator of the Picard group of $G_{2,n}$. We also prove that the flag variety $GL_{n}(k)/B_{n}$ can be obtained as a GIT quotient of $GL_{n+1}(k)/B_{n+1}$ modulo a maximal torus of $SL_{n+1}(k)$ for a suitable choice of an ample line bundle on $GL_{n+1}(k)/B_{n+1}$.
We study the stability of some homogeneous bundles on P^3 by using their representations of the quiver associated to the homgeneous bundles on P^3. In particular we show that homogeneous bundles on P^3 whose support of the quiver representation is a parallelepiped are stable, for instance the bundles E whose minimal free resolution is of the kind 0 --> S^{l_1, l_2, l_3} V (t) --> S^{l_1 +s, l_2, l_3} V (t+s) --> E --> 0 are stable.
Finite $F$-representation type for homogeneous coordinate rings of non-Fano varieties
本文通过微分算子与扭转全局截面的联系，证明了一大类正特征齐次坐标环不具有有限$F$-表示类型，包括阿贝尔簇、多数Calabi-Yau簇和一般型完全交。
Let H be a closed subgroup of a linear algebraic group G defined over a field F. There is an equivalence of categories between the category of linear finite-dimensional representations of H and the category of finite rank G-homogeneous vectorbundles on G/H. In this short note we study this correspondence for the sheaves of principal parts (=jetbundles) of homogeneous linebundles on projective space in characteristic zero. We describe the representation corresponding to the principal parts of a homogeneous linebundle and as a corollary we obtain known formulas on the splitting type of the princ…
An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer-Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstr…
Root stack valuative criterion for good moduli spaces
本文在剩余特征的温和假设下，证明了好模空间映射与约化群gerbe的根叠赋值判别法，并给出了其在挠子parahoric扩张、叠有理点、gerbe与齐性空间及纤维丛几何中的应用。
For all spherical homogeneous spaces G/H, where G is a simply connected semisimple algebraic group and H a connected solvable subgroup of G, we compute the spectra of the representations of G on spaces of regular sections of homogeneous line bundles over G/H.
Given an algebraic variety $X\subset\mathbb{P}^N$ with stabilizer $H$, the quotient $PGL_{N+1}/H$ can be interpreted a parameter space for all $PGL_{N+1}$-translates of $X$. We define $X$ to be a $\textit{homogeneous variety}$ if $H$ acts on it transitively, and satisfies a few other properties, such as $H$ being semisimple. Some examples of homogeneous varieties are quadric hypersurfaces, rational normal curves, and Veronese and Segre embeddings. In this case, we construct new compactifications of the parameter spaces $PGL_{N+1}/H$, obtained compactifying $PGL_{N+1}$ to the classically known …
In this paper, we show that the (admissible) character stack, which is a stack version of the character variety, is an open substack of the Teichmüller stack of homogeneous spaces of SL(2,C). We show that the tautological family over the representation variety, given by deforming the holonomy, is always a complete family. This is a generalisation of the work of E. Ghys about deformations of complex structures these homogeneous spaces.
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