Ivan V. Arzhantsev, Natalia A. Tennova
Affinely closed homogeneous spaces G/H, i.e., affine homogeneous spaces that admit only the trivial affine embedding, are characterized for any affine algebraic group G. As a corollary, a description of affine G-algebras with finitely generated invariant subalgebras is obtained.
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On Affinely Homogeneous Submanifolds: The Power Series Method of Equivalence
本文利用 Cartan 增强的幂级数等价方法，分类了 $S^2\subset\mathbb{R}^4$ 的全部仿射齐性模型及其模空间分支。
Let $M$ be complete flat pseudo-Riemannian homogeneous manifold and $Γ\subset\Iso(\RR^n_s)$ its fundamental group. We show that $M$ is a trivial fiber bundle $G/Γ\to M\to\RR^{n-k}$, where $G$ is the Zariski closure of $Γ$ in $\Iso(\RR^n_s)$. Moreover, we show that the $G$-orbits in $\RR^n_s$ are affinely diffeomorphic to $G$ endowed with the (0)-connection. If the induced metric on the $G$-orbits is non-degenerate, then $G$ (and hence $Γ$) has linear abelian holonomy. If additionally $G$ is not abelian, then $G$ contains a certain subgroup of dimension 6. In particular, for non-abelian $G$ orb…
Let A be a commutative associative integrally closed k-algebra without zero divisors effectively graded by a lattice. We obtain a criterion of local nilpotency of the sum of two homogeneous locally nilpotent derivations (LNDs) of fiber type on A in terms of their degrees. The same problem is solved for commutators of two homogeneous LNDs.
We give a new proof of the theorem stating that for any connected linear algebraic group G over an algebraically closed field k of characteristic 0 and for any closed connected subgroup H of G, the unramified Brauer group of G/H vanishes.
Let G be a reductive group over an algebraically closed field of characteristic p>0. We study properties of embeddings of spherical homogeneous G-spaces. We look at Frobenius splittings, canonical or by a (p-1)-th power, compatible with certain subvarieties. We also look at cohomology vanishing and show the existence of rational G-equivariant resolutions by toroidal embeddings. We show that the class of homogeneous spaces for which our results hold contains the symmetric homogeneous spaces in characteristic not 2 and is closed under parabolic induction.
Generic vanishing on homogeneous spaces in arbitrary characteristic
本文在任意特征下研究同构空间上的通用消失现象，扩展了Schürmann-Simpson-Wang在特征零情况下的定理，并在有限域上给出了迹函数恒等式和Lang-Weil估计。
Given a connected algebraic group G over an algebraically closed field and a G-homogeneous space X, we describe the Chow ring of G and the rational Chow ring of X, with special attention to the Picard group. Also, we investigate the Albanese and the "anti-affine" fibrations of G and X.
Let G be a connected linear algebraic group over an algebraically closed field k, and let H be a connected closed subgroup of G. We prove that the homogeneous variety G/H is a rational variety over k whenever H is solvable, or when dim(G/H) < 11 and characteristic(k) = 0. When H is of maximal rank in G, we also prove that G/H is rational if the maximal semisimple quotient of G is isogenous to a product of almost-simple groups of type A, type C (when characteristic(k) is not 2), or type B_3 or G_2 (when characteristic(k) = 0).
The strength of a homogeneous polynomial (or form) is the smallest length of an additive decomposition expressing it whose summands are reducible forms. Using polynomial functors, we show that the set of forms with bounded strength is not always Zariski-closed. More specifically, if the ground field is algebraically closed, we prove that the set of quartics with strength $\leq3$ is not Zariski-closed for a large number of variables.
We show the vanishing of the second homotopy group of the étale homotopy type of a smooth connected algebraic group over a separably closed field, completed away from the characteristic. This is an algebraic analogue of a classical theorem of Elie Cartan. Based on this result, we establish an explicit formula for the similarly completed second homotopy group of a homogeneous space.
Algebraic and analytic Brauer groups of homogeneous spaces
本文研究了在连通、单连通、半单复代数群作用下，稳定子群闭合且连通的同调空间的代数和解析Brauer群的计算方法与主要贡献。
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.
Let $G$ be a connected reductive group. We find a necessary and sufficient condition for a quasiaffine homogeneous space of $G$ to be embeddable into an irreducible $G$-module. In addition, for an affine homogeneous space we find a criterium for a closed embedding to exist
Let $B$ be a curve defined over an algebraically closed field $k$ and let $X\to B$ be an elliptic surface with base curve $B$. We investigate the geometry of everywhere locally trivial principal homogeneous spaces for $X$, i.e. elements of the Tate-Shafarevich group. If $Y$ is such a principal homogeneous space of order $n$, we find strong restrictions on the $\mathbb{P}^{n-1}$ bundle over $B$ into which $Y$ embeds. Examples for small values of $n$ show that, in at least some cases, these restrictions are sharp. Finally, we determine these bundles in case $k$ has characteristic zero, $B = \mat…
We study the local symplectic algebra of parameterized curves introduced by V. I. Arnold. We use the method of algebraic restrictions to classify symplectic singularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed 2-forms to the germ of an analytic curve is a finite dimensional vector space. We also show that the action of local diffeomorphisms preserving the curve on this vector space is determined by the infinitesimal action of liftable vector fields. We apply these results to obtain the complete symplectic classification of curves with the semig…
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