Geometry of horospherical varieties of Picard rank one

Abstract

We study the geometry of non-homogeneous horospherical varieties. 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. In particular, we describe the cohomology groups of these varieties as well as a Chevalley formula, and prove 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. The final section is devoted to the derived categories of coherent sheaves on horospherical varieties. We first discuss a general construction of exceptional bundles on these varieties. We then study 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.

71 pages

References in corpus (12)

• The intrinsic normal cone
• Derived categories of cubic fourfolds
• Exceptional collections for Grassmannians of isotropic lines
• Quantum Pieri rules for isotropic Grassmannians
• McKay equivalence for symplectic resolutions of singularities
• Enumerative geometry of hyperelliptic plane curves
• On the quantum cohomology of adjoint varieties
• On residual categories for Grassmannians
• Semisimple quantum cohomology of some Fano varieties
• Dubrovin's conjecture for $IG(2,6)$
• Conjecture $\mathcal{O}$ holds for the odd symplectic Grassmannian
• On the bounded derived category of $\mathsf{IGr}(3,7)$

Cited by in corpus (8)

• Conjecture $\mathcal{O}$ holds for the odd symplectic Grassmannian
• Fano manifolds and stability of tangent bundles
• On the bounded derived category of $\mathsf{IGr}(3,7)$
• Conjecture $\mathcal{O}$ holds for some Horospherical Varieties of Picard Rank 1
• Equivariant cobordism of smooth projective spherical varieties
• Greatest Ricci lower bounds of projective horospherical manifolds of Picard number one
• Smooth projective horospherical varieties of Picard group ${\mathbb{Z}}^2$
• Homological Projective Duality for the Plücker embedding of the Grassmannian