Workshop Horospherical Varieties - LMA Poitiers June 2026

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

The talk will discuss the classification of projective homogeneous varieties over an arbitrary field $k$. Based on the previous talk, we will show that every such variety is a product of an abelian torsor and a projective variety $Z$, homogeneous under a semisimple algebraic group. We will then show that if $Z$ has a $k$-point, then it is a rational variety in a strong sense : every $k$-point has an open neighborhood isomorphic to an affine space. The proof will illustrate new phenomena arising in positive characteristics.

Stéphanie Cupit-Foutou

On a construction of non-Kähler compact complex manifolds

I will present a construction of non-algebraic complex manifolds as quotient of spherical varieties. This construction extends that of Hopf manifolds and more generally of LVMB-manifolds (acronym for López de Medrano-Verjovsky-Meersseman-Bosio). The manifolds we obtained fiber over non-algebraic stacks. In my talk, I will focus on the horospherical case, which was considered jointly with T. Waedt.

Thibaut Delcroix

Chow stability of (horo)spherical varieties

I will discuss joint work in progress with King Leung Lee and Naoto Yotsutani on Chow stability and secondary polytopes for projective spherical embeddings. This is the GIT stability notion corresponding to the Chow point of the embedding, and when considered for Kodaira embeddings of a higher and higher power of an ample line bundle, this is closely related with K-stability. I will present the problem, our target results and the general strategy of the proof, which relies of the Cayley method of Gelfand-Kapranov-Sturmfels-Zelevinsky and which is closer to completion in the horospherical case.

Johannes Hofscheier

The Mukai conjecture for horospherical log Fano pairs

The Mukai conjecture concerns the geography of Fano varieties, predicting a relationship between the Picard rank and the divisibility of the anti-canonical divisor. Kento Fujita proposed a generalisation to log Fano pairs $(X, \Delta)$ (that is, $-(K_X + \Delta)$ is ample), explicitly formulating the conjecture to allow for the presence of singularities, and proved it in the toric case without assuming that $X$ is $\mathbb{Q}$-factorial. In this talk, I will present a proof of Fujita's generalised Mukai conjecture for log Fano pairs where the underlying variety is horospherical, extending his toric result to this broader setting. A key ingredient in the proof is a generalisation of Eikelberg's description of $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisors from the toric to the spherical setting, which allows us to handle the non-$\mathbb{Q}$-factorial singularities that arise in this context. This is joint work in progress with Heath Pearson.

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

Towards the Classification of Horospherical Divisorial Contractions

We classify all equivariant divisorial contractions of rank-two locally factorial horospherical varieties, and show that such a classification no longer holds starting from rank three.

Matilde Maccan

Projective homogeneous varieties over a field I

We begin with the natural problem of classifying smooth projective homogeneous algebraic varieties over an algebraically closed field. Such varieties decompose as direct products of abelian varieties and quotients of the form $G/P$, where $G$ is a simple simply connected group and $P$ contains a Borel subgroup. Over the complex numbers, these objects are classically known as flag varieties, and their classification is well understood. However, in characteristic $p>0$, the situation becomes more subtle: when p is at least five, these varieties were described by Wenzel, Haboush, and Lauritzen in terms of root systems and Frobenius kernels. In this talk, I will present their classification together with my contribution, which consists in extending it (in as uniform a way as possible) to small characteristics. This will prepare for Part II, joint work in progress with Brion and Srinivasan, where the ground field will be arbitrary.

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

Some results on automorphisms of simple derivations

Let $D$ be a simple derivation of the polynomial ring $\mathbb{C}[x_1,\dots,x_n]$, that is a derivation preserving only the trivial ideals. One expects the automorphism group of such a derivation to be rather small. In this talk, I will present some results in this direction. I will first prove that the identity component $\mathrm{Aut}^0(D)$is a unipotent algebraic group of dimension at most $(n-2)$, and that this bound is optimal. As an application, in dimension $3$ one obtains a dichotomy: either $\mathrm{Aut}(D)$ is discrete, or it is isomorphic to the additive group acting by translations. I will also discuss more recent results concerning the extremal case $\dim \mathrm{Aut}^0(D)= n-2$. {Joint work with Ivan Pan and Alvaro Rittatore.}

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

On the geometry of smooth horospherical varieties of Picard rank one

Smooth horospherical varieties of Picard rank one were classified by Pasquier. I will review commun geometric features of (non-homogeneous) horospherical varieties of Picard rank one with a special view towards enumerative geometry and moduli spaces of rational curves.

Monday	Tuesday	Wednesday
	9h30 - 10h10 Kim	9h30 - 10h10 Monahan
	Coffee	Coffee
	10h40 - 11h20 Delcroix	10h40 - 11h20 Lapuyade
	11h40 -12h20 Kwon	11h40 -12h20 Moser-Jauslin
	Lunch
14h - 14h40 Perrin	14h - 14h40 Hwang
15h - 15h40 Barban	15h - 15h40 Hofscheier
Coffee	Coffee
16h10 - 16h50 Maccan	16h20 - 17h00 Cupit-Foutou
17h00 - 17h40 Brion	17h20 - 18h00 Montagard
	19h30 - ... Social Diner

Workshop

Horospherical Varieties

Speakers