Given a connected reductive algebraic group $G$ and a spherical $G$-variety $X$, a $B$-root subgroup on $X$ is a one-parameter additive group of automorphisms of $X$ normalized by a Borel subgroup $B \subset G$. We obtain a complete description of all $B$-root subgroups on a certain open subset of $X$. When $X$ is horospherical, we extend the construction of standard $B$-root subgroups introduced earlier by Arzhantsev and Avdeev for affine $X$ and obtain a complete description of all standard $B$-root subgroups, which naturally generalizes the well-known description of root subgroups on toric varieties. As an application, for horospherical $X$ that is either complete or contains a unique closed $G$-orbit, we determine all $G$-stable prime divisors in $X$ that can be connected with the open $G$-orbit via the action of a suitable $B$-root subgroup. For horospherical $X$, we also find sufficient conditions for the existence of $B$-root subgroups on $X$ that preserve the open $B$-orbit in $X$. Finally, when $G$ is of semisimple rank $1$ and $X$ is horospherical and complete, we determine all $B$-root subgroups on $X$, which enables us to describe the Lie algebra of the connected automorphism group of $X$.
A horospherical variety is a normal algebraic variety where a connected reductive algebraic group acts with an open orbit isomorphic to a torus bundle over a flag variety. In this article we study the cohomology of line bundles on complete horospherical varieties.
Let X and Y be horospherical Mori fibre spaces which are birational equivariantly with respect to the group action. Then, there is a horospherical Sarkisov program from X/S to Y /T .
We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only constant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety is flexible.
In this paper we describe orbits of automorphism group on a horospherical variety in terms of degrees of homogeneous with respect to natural grading locally nilpotent derivations. In case of (may be non-normal) toric varieties a description of orbits of automorphism group in terms of corresponding weight monoid is obtained.
We classify all smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$ and we give a first description of their geometry via the Log Minimal Model Program.
In 2013 Bazhov proved a criterium for two points on a complete toric variety to lie in the same orbit of the neutral component of automorphism group. This criterium is in terms of divisor class group. Arzhantsev-Bazhov (2013) obtained a similar criterium for affine toric varieties. We prove a necessary condition similar this criteria to the cases of affine and projective horospherical varieties.
Recently, Kanemitsu has discovered a counterexample to the long-standing conjecture that the tangent bundle of a Fano manifold of Picard number one is (semi)stable. His counterexample is a smooth horospherical variety. There is a weaker conjecture that the tangent bundle of a Fano manifold of Picard number one is simple. We prove that this weaker conjecture is valid for smooth horospherical varieties of Picard number one. Our proof follows from the existence of an irreducible family of unbendable rational curves whose tangent vectors span the tangent spaces of the horospherical variety at gene…
We present geometric realizations of horospherical two-orbit varieties, by showing that their blow-up along the unique closed-invariant orbit is the zero locus of a general section of a homogeneous vector bundle over some auxiliary variety. As an application, we compute the cohomology ring of the $G_2$-variety, including its quantum version. We also consider the Spin$_7$-variety, which deserves a different treatment.
Automorphism groups of toroidal horospherical varieties
Center for Complex Geometry, Institute for Basic Science (IBS)（基础科学研究院复杂几何中心）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学天元基金中心）
Property $\mathcal{O}$ for an arbitrary complex, Fano manifold $X$, is a statement about the eigenvalues of the linear operator obtained from the quantum multiplication of the anticanonical class of $X$. Conjecture $\mathcal{O}$ is a conjecture that Property $\mathcal{O}$ holds for any Fano variety. Pasquier listed the smooth non-homogeneous horospherical varieties of Picard rank 1 into five classes. Conjecture $\mathcal{O}$ has already been shown to hold for the odd symplectic Grassmannians which is one of these classes. We will show that Conjecture $\mathcal{O}$ holds for two more classes an…
We study the equivariant real structures on complex horospherical varieties, generalizing classical results known for toric varieties and flag varieties. In particular, we obtain a necessary and sufficient condition for the existence of such real structures and determine the number of equivalence classes. We then apply our results to classify the equivariant real structures on smooth projective horospherical varieties of Picard rank 1.
Pseudo-effective and nef cones on spherical varieties
我们证明了光滑完备球面簇上的nef圈类是有效的，并且nef圈类的乘积也是nef的。设X是光滑射影球面簇，使得其余维k的有效圈类为nef，其中1<= k <= dim（X）-1。我们研究了X的性质。证明了若X是环面簇，则X同构于某些射影空间的乘积，若X是环面簇，则X同构于有理齐性空间，若X是次球面簇，dim（X）≥ 3，k=2，则X上的有效因子是nef;如果X是次球面的且X上的有效因子是nef，则存在从X到有理齐性空间的态射，使得每个纤维同构于Picard数1的某些次球面簇的乘积。
In this paper we determine the stringy motivic volume of log terminal horospherical $G$-varieties of complexity one, where $G$ is a connected reductive linear algebraic group. The stringy motivic volume of a log terminal variety is an invariant of singularities which was introduced by Batyrev and plays an important role in mirror symmetry for Calabi--Yau varieties. A horospherical $G$-variety of complexity one is a normal $G$-variety which is equivariantly birational to a product $C \times G/H$, where $C$ is a smooth projective curve and the closed subgroup $H$ contains a maximal unipotent sub…
Levi subgroup actions on Schubert varieties are studied. In the case of partial flag varieties, the horospherical actions are determined. This leads to a characterization of the toroidal and horospherical partial flag varieties with Picard number 1. In the more general case, we provide a set of necessary conditions for the action of a Levi subgroup on a Schubert variety to be toroidal. The singular locus of a (co)minuscule Schubert variety is shown to contain all the $L_{max}$-stable Schubert subvarieties, where $L_{max}$ is the standard Levi subgroup of the maximal parabolic which acts on the…
Pasquier and Perrin discovered that the ${\rm G}_2$-horospherical manifold ${\bf X}$ of Picard number 1 can be realized as a smooth specialization of the rational homogeneous space parameterizing the lines on the 5-dimensional hyperquadric, in other words, it can be deformed nontrivially to the rational homogeneous space. We show that ${\bf X}$ is the only smooth projective variety with this property. This is obtained as a consequence of our main result that ${\bf X}$ can be recognized by its VMRT, namely, a Fano manifold of Picard number 1 is biregular to ${\bf X}$ if and only if its VMRT at …
The affine closure of cotangent bundles of horospherical spaces
State Key Laboratory of Mathematical Sciences, Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Algebraic hyperbolicity of adjoint linear systems on spherical varieties
Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Department of Mathematical Sciences, KAIST（韩国科学技术院数学科学系）；Center for Complex Geometry, Institute for Basic Science (IBS)（基础科学研究院复杂几何中心）
For a semi-stable abelian variety A_K over a complete discrete valuation field K, we show that every finite subgroup scheme of A_K extends to a log finite flat group scheme over the valuation ring of K endowed with the canonical log structure. To achieve this, we first prove that every weak log abelian variety over an fs log scheme with its underlying scheme locally noetherian, is a sheaf for the Kummer flat topology, which answers a question of Chikara Nakayama. We also give several equivalent conditions defining isogenies of log abelian varieties.
Given a connected reductive algebraic group $G$ and a Borel subgroup $B \subseteq G$, we study $B$-normalized one-parameter additive group actions on affine spherical $G$-varieties. We establish basic properties of such actions and their weights and discuss many examples exhibiting various features. We propose a construction of such actions that generalizes the well-known construction of normalized one-parameter additive group actions on affine toric varieties. Using this construction, for every affine horospherical $G$-variety $X$ we obtain a complete description of all $G$-normalized one-par…
Let $X$ be an irreducible affine algebraic variety that is spherical with respect to an action of a connected reductive group $G$. In this paper we provide sufficient conditions, formulated in terms of weight combinatorics, for the existence of one-parameter additive actions on $X$ normalized by a Borel subgroup $B \subset G$. As an application, we prove that every $G$-stable prime divisor in $X$ can be connected with the open $G$-orbit by means of a suitable $B$-normalized one-parameter additive action.
We study finite $p$-subgroups of birational automorphism groups. By virtue of boundedness theorem of Fano varieties, we prove that there exists a constant $R(n)$ such that a rationally connected variety of dimension $n$ over an algebraically closed field is rational if its birational automorphism group contains a $p$-subgroups of maximal rank for $p > R(n)$. Some related applications on Jordan property are discussed.
For a variety $X$, a big $\mathbb{Q}$-divisor $L$ and a closed connected subgroup $G \subset \mathrm{Aut}(X, L)$ we define a $G$-invariant version of the $δ$-threshold. We prove that for a Fano variety $(X, -K_X)$ and a connected subgroup $G \subset \mathrm{Aut}(X)$ this invariant characterizes $G$-equivariant uniform $K$-stability. We also use this invariant to investigate $G$-equivariant $K$-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of $G$ being a finite group.
