Horospherical Varieties Research Articles

On orbits of automorphism groups on horospherical varieties

AbstractIn this paper, we describe orbits of the automorphism group on an affine horospherical variety in terms of degrees of homogeneous with respect to natural grading locally nilpotent derivations. In case of (possibly nonnormal) toric varieties, a description of orbits of the automorphism group in terms of corresponding weight monoid is obtained.

Explicit deformation of the horospherical variety of type $\mathrm{G}_2$

We give two simple geometric constructions of a smooth family of projective varieties with central fiber isomorphic to the horospherical variety of type $\mathrm{G}_2$ and all other fibers isomorphic to the isotropic orthogonal Grassmannian $\operatorname{OGr}(2,7)$, and we discuss briefly the derived category of this family. Bibliography: 8 titles.

Equivariant Cobordism of Smooth Projective Spherical Varieties

AbstractWe study the equivariant cobordism rings for the action of a torusTon smooth varieties over an algebraically closed field of characteristic zero. We prove a theorem describing the rationalT-equivariant cobordism rings of smooth projectiveG-spherical varieties with the action of a maximal torusTofG. As an application, we obtain explicit presentations for the rational equivariant cobordism rings of smooth projective horospherical varieties of Picard number one.

Orbits of the Automorphism Group of Horospherical Varieties, and Divisor Class Group

В 2013 г. Бажов доказал критерий того, что две точки полного торического многообразия лежат в одной орбите связной компоненты единицы группы автоморфизмов. Этот критерий сформулирован в терминах группы классов дивизоров. В том же году Аржанцев и Бажов получили аналогичный критерий для аффинного торического многообразия. В работе доказывается необходимое условие, аналогичное этому критерию, в случаях аффинного и проективного орисферических многообразий.

Orbits of the Automorphism Group of Horospherical Varieties, and Divisor Class Group

Orbits of the Automorphism Group of Horospherical Varieties, and Divisor Class Group

Simplicity of tangent bundles of smooth horospherical varieties of Picard number one

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 general points.

Geometry of Horospherical Varieties of Picard Rank One

AbstractWe study the geometry of smooth non-homogeneous horospherical varieties of Picard rank one. These have been classified by Pasquier and include the well-known odd symplectic Grassmannians. We focus our study on quantum cohomology, with a view towards Dubrovin’s conjecture. We start with describing the cohomology groups of smooth horospherical varieties of Picard rank one. We show a Chevalley formula for these and establish that many Gromov–Witten invariants are enumerative. This enables us to prove that in many cases the quantum cohomology is semisimple. We give a presentation of the quantum cohomology ring for odd symplectic Grassmannians. In the last sections, we turn to derived categories of coherent sheaves. We first discuss a general construction of exceptional bundles on horospherical varieties. We work out in detail the case of the horospherical variety associated to the exceptional group $G_2$ and construct a full rectangular Lefschetz exceptional collection in the derived category.

Real Structures on Horospherical Varieties

We study the equivariant real structures on complex horospherical varieties, generalizing classical results known for toric varieties and flag varieties. We obtain a necessary and sufficient condition for the existence of an equivariant real structure on a given horospherical variety, and we determine the number of equivalence classes of equivariant real structures on horospherical homogeneous spaces. We then apply our results to classifying the equivalence classes of equivariant real structures on smooth projective horospherical varieties of Picard rank 1.

Conjecture 𝒪 holds for some horospherical varieties of Picard rank 1

Property 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 𝒪 is a conjecture that property 𝒪 holds for any Fano variety. Pasquier classified the smooth nonhomogeneous horospherical varieties of Picard rank 1 into five classes. Conjecture 𝒪 has already been shown to hold for the odd symplectic Grassmannians, which is one of these classes. We will show that conjecture 𝒪 holds for two more classes and an example in a third class of Pasquier’s list. Perron–Frobenius theory reduces our proofs to be graph-theoretic in nature.

Smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$

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.

Cohomology of line bundles on horospherical varieties

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. The main tool in this article is the machinery of Grothendieck–Cousin complexes, and we also prove a Kunneth-like formula for local cohomology.

Stringy invariants for horospherical varieties of complexity one

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×G/H, where C is a smooth projective curve and the closed subgroup H contains a maximal unipotent subgroup of G. The simplest example of such a variety is a normal surface with a non-trivial ?*-action. Our formula extends the results of Batyrev-Moreau [BM13] on stringy invariants of horospherical embeddings. The proof involves the study of the arc space of a horospherical variety of complexity one and a combinatorial description of its orbits. In contrast to [BM13], the number of orbits is no longer countable, which adds significant difficulties to the problem. As a corollary of our main theorem, we obtain a smoothness criterion using a comparison of the stringy and usual Euler characteristics.

Flexibility of normal affine horospherical varieties

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 with only constant invertible functions is flexible.

Kähler geometry of horosymmetric varieties, and application to Mabuchi’s K-energy functional

Abstract We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop Kähler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin–Abreu–Donaldson geometry of toric varieties. Namely we associate convex functions with Hermitian metrics on line bundles, and express the curvature form in terms of this function, as well as the corresponding Monge–Ampère volume form and scalar curvature. We provide an expression for the Mabuchi functional and derive as an application a combinatorial sufficient condition of properness similar to one obtained by Li, Zhou and Zhu on group compactifications. This finally translates to a sufficient criterion of existence of constant scalar curvature Kähler metrics thanks to the recent work of Chen and Cheng. It yields infinitely many new examples of explicit Kähler classes admitting cscK metrics.

Standard Embeddings of Smooth Schubert Varieties in Rational Homogeneous Manifolds of Picard Number 1

Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomogeneous smooth Schubert varieties in symplectic Grassmannians and in the 20-dimensional $F_4$-homogeneous manifold associated to a short simple root.

The Log Minimal Model Program for Horospherical Varieties Via Moment Polytopes

In a previous work, we described the Minimal Model Program in the family of ℚ-Gorenstein projective horospherical varieties, by studying certain continuous changes of moment polytopes of polarized horospherical varieties. Here, we summarize the results of the previous work and we explain how to generalize them in order to describe the Log Minimal Model Program for pairs (X, Δ) when X is a projective horospherical variety.

Flexibility of affine horospherical varieties of semisimple groups

Let be an algebraically closed field of characteristic zero and the additive group of . An algebraic variety is said to be flexible if the tangent space at every regular point of is generated by the tangent vectors to orbits of various regular actions of . In 1972, Vinberg and Popov introduced the class of affine -varieties which are also known as affine horospherical varieties. These are varieties on which a connected algebraic group acts with an open orbit in such a way that the stationary subgroup of each point in the orbit contains a maximal unipotent subgroup of . In this paper the flexibility of affine horospherical varieties of semisimple groups is proved. Bibliography: 9 titles.

GEOMETRIC STRUCTURES MODELED ON SMOOTH PROJECTIVE HOROSPHERICAL VARIETIES OF PICARD NUMBER ONE

Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric structures modeled on horospherical varieties. Using Cartan geometry, we prove that a geometric structure modeled on a smooth projective horospherical variety of Picard number one is locally equivalent to the standard geometric structure when the geometric structure is defined on a Fano manifold of Picard number one.

Smooth projective horospherical varieties with nef tangent bundles

We show that smooth projective horospherical varieties with nef tangent bundles are rational homogeneous spaces.

Гибкость $S$-многообразий полупростых групп

Пусть $k$ - алгебраически замкнутое поле характеристики нуль, и пусть $\mathbb{G}_a=(k,+)$ - его аддитивная группа. Алгебраическое многообразие $X$ называется гибким, если касательное пространство в каждой его регулярной точке порождено касательными векторами к орбитам различных регулярных действий группы $\mathbb{G}_a$. В 1972 г. Э. Б. Винберг и В. Л. Попов ввели класс аффинных $S$-многообразий, т.е. таких многообразий, на которых действует связная алгебраическая группа с открытой орбитой, причем стационарная подгруппа любой точки этой орбиты содержит максимальную унипотентную подгруппу группы $G$. В этой статье мы докажем гибкость $S$-многообразий полупростых групп. Библиография: 9 названий.