arXiv subjects
Publications and source records attributed to Boris Pasquier.
We study the local rigidity of projective smooth horospherical varieties of rank one and Picard number two. These varieties have been already considered by the second author in a work where their automorphism groups are computed. The results given here are a natural continuation of this work.
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 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.
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 a previous work, we described the Minimal Model Program in the family of $\Qbb$-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 $(X,\D)$ when $X$ is a projective horospherical variety.
We list combinatorial criteria of some singularities, which appear in the Minimal Model Program or in the study of (singular) Fano varieties, for spherical varieties. Most of the results of this paper are already known or are quite easy corollary of known results. We collect these results, we precise some proofs and add few results to get a coherent and complete survey.
Let $X$ be a horospherical $G$-variety and let $D$ be an effective $\mathbb{Q}$-divisor of $X$ that is stable under the action of a Borel subgroup $B$ of $G$ and such that $D+K\_X$ is $\mathbb{Q}$-Cartier. We prove, using Bott-Samelson resolutions, that the pair $(X,D)$ is klt if and only if $\lfloor D\rfloor=0$
The main results of this paper are already known (V.V. Shokurov, the non-vanishing theorem, 1985). Moreover, the non-$\mathbb{Q}$-factorial MMP was more recently considered by O~Fujino, in the case of toric varieties (Equivariant completions of toric contraction morphisms, 2006), for klt pairs (Special termination and reduction to pl flips, 2007) and more generally for log-canonical pairs (Foundation of the minimal model program, 2014). Here we rewrite the proofs of some of these results, by following the proofs given by Y. Kawamata, K. Matsuda, and K. Matsuki (Introduction to the minimal model problem, 1985) of the same results in $\mathbb{Q}$-factorial MMP. And, in the family of $\mathbb{Q}$-Gorenstein spherical varieties, we answer positively to the questions of existence of flips and of finiteness of sequences of flips. I apologize for the first version of this paper, which I wrote without knowing that these results already exist.
We describe the Minimal Model Program in the family of $\mathbb{Q}$-Gorenstein projective horospherical varieties, by studying a family of polytopes defined from the moment polytope of a Cartier divisor of the variety we begin with. In particular, we generalize the results on MMP in toric varieties due to M. Reid, and we complete the results on MMP in spherical varieties due to M. Brion in the case of horospherical varieties.
For a few pairs $G\subset \hat G$ of reductive groups, we study the decomposition of irreducible $\hait G$-modules into $G$-modules. In particular, we observe the saturation property for all of these pairs.
Let $G\subset\hat{G}$ be two complex connected reductive groups. We deals with the hard problem of finding sub-$G$-modules of a given irreducible $\hat{G}$-module. In the case where $G$ is diagonally embedded in $\hat{G}=G\times G$, S. Kumar and O. Mathieu found some of them, proving the PRV conjecture. Recently, the authors generalized the PRV conjecture on the one hand to the case where $\hat{G}/G$ is spherical of minimal rank, and on the other hand giving more sub-$G$-modules in the classical case $G\subset G\times G$. In this paper, these two recent generalizations are combined in a same more general result.
Let $X$ be a minuscule homogeneous space, an odd quadric, or an adjoint homogenous space of type different from $A$ and $G_2$. Le $C$ be an elliptic curve. In this paper, we prove that for $d$ large enough, the scheme of degree $d$ morphisms from $C$ to $X$ is irreducible, giving an explicit lower bound for $d$ which is optimal in many cases.
Let G be a complex connected reductive group. The PRV conjecture, which was proved independently by S. Kumar and O. Mathieu in 1989, gives explicit irreducible submodules of the tensor product of two irreducible G-modules. This paper has three aims. First, we simplify the proof of the PRV conjecture, then we generalize it to other branching problems. Finally, we find other irreducible components of the tensor product of two irreducible G-modules that appear for "the same reason" as the PRV ones.
We use the toric degeneration of Bott-Samelson varieties and the description of cohomolgy of line bundles on toric varieties to deduce vanishings results for the cohomology of lines bundles on Bott-Samelson varieties.
For a $G$-variety $X$ with an open orbit, we define its boundary $\partial X$ as the complement of the open orbit. The action sheaf $S_X$ is the subsheaf of the tangent sheaf made of vector fields tangent to $\partial X$. We prove, for a large family of smooth spherical varieties, the vanishing of the cohomology groups $H^i(X,S_X)$ for $i>0$, extending results of F. Bien and M. Brion. We apply these results to study the local rigidity of the smooth projective varieties with Picard number one classified in a previous paper of the first author.
We prove a conjecture of L.Bonavero, C. Casagrande, O. Debarre and S. Druel, on the pseudo-index of smooth Fano varieties, in the special case of horospherical varieties.
We classify all smooth projective horospherical varieties with Picard number 1. We prove that the automorphism group of any such variety X acts with at most two orbits and that this group still acts with only two orbits on X blown up at the closed orbit. We characterize all smooth projective two-orbits varieties with Picard number 1 that satisfy this latter property.
We describe smooth projective horospherical varieties with Picard number 1. Moreover we prove that the automorphism group of any such variety acts with at most two orbits and we give a geometric characterisation of non-homogeneous ones.