[Paper Review] Rigidity of smooth Schubert varieties in a rational homogeneous manifold associated to a short root

Jaehyun Hong, Minhyuk Kwon|arXiv (Cornell University)|Jul 23, 2019

Advanced Algebra and Geometry12 references4 citations

TL;DR

This paper classifies non-linear smooth Schubert varieties in rational homogeneous manifolds associated to short roots, proving they are either homogeneous submanifolds associated to subdiagrams of the Dynkin diagram or horospherical varieties. The key result is that any subvariety with the same homology class as such a Schubert variety is induced by the action of the identity component of the automorphism group, establishing their rigidity except in specific linear cases.

ABSTRACT

We classify smooth Schubert varieties S_0 in a rational homogeneous manifold S associated to a short root, and show that they are rigid in the sense that any subvariety of S having the same homology class as S_0 is induced by the action of Aut_0(S), unless S_0 is linear.

Motivation & Objective

• To classify non-linear smooth Schubert varieties in rational homogeneous manifolds associated to short roots.
• To determine which such Schubert varieties are rigid, meaning any subvariety with the same homology class arises from the automorphism group action.
• To extend existing rigidity results beyond homogeneous submanifolds and odd symplectic Grassmannians to include horospherical varieties.
• To establish that for type $(F_4, α_4)$, all smooth Schubert varieties are linear, resolving a structural constraint.

Proposed method

• Use of Dynkin diagram subdiagram classification to identify homogeneous Schubert varieties as candidates for rigidity.
• Application of the theory of minimal rational tangents and isotropy groups to analyze tangent spaces of Schubert varieties.
• Embedding of Schubert varieties into larger homogeneous spaces and studying their hyperplane sections to reduce to known cases.
• Identification of horospherical varieties via representation-theoretic data, particularly highest weight representations and orbit closures.
• Use of the action of the identity component of the automorphism group $\operatorname{Aut}_0(S)$ to characterize subvarieties with the same homology class.
• Leveraging results from prior works on symplectic Grassmannians and $F_4$-type manifolds to unify classification across types.

Experimental results

Research questions

• RQ1 Which non-linear smooth Schubert varieties in rational homogeneous manifolds of short root type are rigid under homology class preservation?
• RQ2 What is the complete classification of such Schubert varieties beyond homogeneous submanifolds and odd symplectic Grassmannians?
• RQ3 How do horospherical varieties arise as Schubert varieties in $F_4$-type manifolds, and what is their geometric structure?
• RQ4 Why are all smooth Schubert varieties in $(F_4, \alpha_4)$-type manifolds linear, and what does this imply for rigidity?

Key findings

• Non-linear smooth Schubert varieties in $S = (F_4, \alpha_3)$ are either homogeneous submanifolds associated to subdiagrams or horospherical varieties of type $(C_2, \alpha_2, \alpha_1)$ or $(B_3, \alpha_2, \alpha_3)$.
• For $S = (C_n, \alpha_k)$, non-linear smooth Schubert varieties are of type $(C_m, \alpha_{i+1}, \alpha_i)$ with $n-k = m-i$, and are rigid under homology class preservation.
• In $S = (F_4, \alpha_4)$, all smooth Schubert varieties are linear, so no non-linear examples exist to consider for rigidity.
• Any subvariety of $S$ with the same homology class as a non-linear smooth Schubert variety $S_0$ is induced by the action of $\operatorname{Aut}_0(S)$, confirming rigidity in all non-linear cases except the excluded linear ones.
• The classification unifies previous results on odd symplectic Grassmannians and homogeneous submanifolds, showing they are special cases of the broader horospherical or subdiagram-based classification.