Horospherical Varieties

Lorenzo Barban

Chow quotient of flag varieties

The Chow quotient of a smooth projective variety under a group action was introduced by Kapranov, Sturmfels and Zelevinsky with the aim to construct an intrinsic notion of quotient, which would not depend on a particular choice of a linearization as in the case of the Mumford’s GIT quotient. In this talk we report on a joint work with L. E. Solá Conde and G. Occhetta, where we explicitly compute the Chow quotient of the complete flag variety of vector subspaces of the $4$-dimensional affine space under the action of the maximal torus of $\mathrm{PGL}(4)$. We also discuss its birational geometry, and the relation with the Chow quotients of partial flag varieties.Time permitting, we survey related works in progress.

Michel Brion

Projective homogeneous varieties over a field II

(Based on joint work with Matilde Maccan and Srimathy Srinivasan)

Stéphanie Cupit-Foutou

Thibaut Delcroix

Johannes Hofscheier

Dongseon Hwang

Automorphism groups of toroidal horospherical varieties

We present our recent work on the structure of the connected component of the automorphism group of a smooth, complete, toroidal horospherical variety, by generalizing the notion of Demazure roots using the toric bundle structure. In particular, we provide a criterion for the reductivity of $\mathrm{Aut}^0(X)$ in terms of an analogous notion of Demazure roots for such toric bundles, i.e., projective toric bundles over rational homogeneous spaces. As an application, we prove the K-unstability of certain $\mathbb{P}^1$-bundles over rational homogeneous spaces. This is joint work with Lorenzo Barban and Minseong Kwon.

Shin-young Kim

Characterization of smooth projective horospherical varieties of Picard number one

In this talk, I will explain characterization of smooth projective horospherical varieties of Picard number one through its variety of minimal rational tangents. That is, we let X be a smooth projective horospherical variety of Picard number one and we show that a uniruled projective manifold of Picard number one is biholomorphic to X if its variety of minimal rational tangents at a general point is projectively equivalent to that of X. To get a local flatness of the geometric structure arising from the variety of minimal rational tangents, we apply the methods of W normal complete step prolongations. We compute the associated Lie algebra cohomology space of degree two and show the vanishing of holomorphic sections of the vector bundle having this cohomology space as a fiber. This is joint work with J. Hong.

Minseong Kwon

Algebraic hyperbolicity of adjoint linear systems on horospherical varieties

A well-known principle is that given a smooth projective variety, very general hypersurfaces of sufficiently high degree are algebraically hyperbolic. Recently, Moraga and Yeong conjectured an optimal lower bound for degree of an adjoint linear system that guarantees algebraic hyperbolicity of very general elements, and proved it for toric varieties. In this talk, I will give an affirmative answer to the conjecture of Moraga and Yeong for a large class of spherical varieties, including horospherical varieties. This is a joint work with Haesong Seo.

André Lapuyade

Matilde Maccan

Projective homogeneous varieties over a field I

(Based on joint work with Michel Brion and Srimathy Srinivasan)

Sean Monahan

Horospherical varieties with quotient singularities

I will present a combinatorial characterization of when horospherical varieties have (at worst) quotient singularities. Combining this characterization with the Cox quotient construction, we can deduce that having quotient singularities is a global (not just local) property for horospherical varieties.

Pierre-Louis Montagard

Lucy Moser-Jauslin

A survey on equivariant real forms of horospherical varieties

In this talk, we will discuss the classification of real forms of almost homogeneous complex varieties under the action of a reductive group. . We will concentrate on the special case of horospherical varieties. In this case, the results can be described using a combinatorial description based on the classification of horospherical subgroups by B. Pasquier, and on Luna-Vust theory for spherical varieties. We will also discuss briefly generalizations to forms over perfect fields. This is joint work with R. Terpereau.

Nicolas Perrin