[Paper Review] Admissible pairs of Hermitian symmetric spaces in the perspective of the theory of varieties of minimal rational tangents

Yunxin Zhang|arXiv (Cornell University)|Dec 24, 2014

Advanced Algebra and Geometry10 references3 citations

TL;DR

This paper classifies admissible pairs of irreducible Hermitian symmetric spaces of compact type using the theory of varieties of minimal rational tangents (VMRT), establishing a sufficient condition for non-rigidity and proving that special pairs in the classification are algebraic. It extends the sub-diagram type embedding framework of Hong and Mok, showing that non-rigid pairs arise when degeneracy occurs via specific root vector conditions, with key examples including hyperquadrics and certain Grassmannians.

ABSTRACT

We study a pair (\mathcal{S}_0,\mathcal{S}) of irreducible Hermitian Symmetric Spaces of compact type (cHSS) in this paper, with the first aim being classifying all the admissible pairs (\mathcal{S}_0,\mathcal{S})). This notion is a natural generalization of the pairs of sub-diagram type originated by Jaehyun Hong and Ngaiming Mok ([HoM 10]). Based on this classification, we partially solve the rigidity problem for the admissible pairs (\mathcal{S}_0,\mathcal{S}) which was raised by Mok and Zhang (2014) ([MoZ 14]), culminating in determining a sufficient condition for the pairs being non-rigid and proving that special pairs, which show up in the classification procedure, are algebraic, as a weaker result than being rigid. However, whether special pairs are rigid or not remains unknown and needs further investigation in the framework of VMRT theory.

Motivation & Objective

• To classify all admissible pairs $(\mathcal{S}_0, \mathcal{S})$ of irreducible Hermitian symmetric spaces of compact type using VMRT theory.
• To extend the notion of sub-diagram type embeddings to a broader class of admissible pairs beyond those defined by Dynkin sub-diagrams.
• To investigate the rigidity problem for such pairs, particularly identifying sufficient conditions for non-rigidity.
• To determine whether special pairs identified in the classification are rigid or not, as a key open problem in VMRT theory.
• To establish that special pairs are algebraic, as a weaker result than full rigidity, under the framework of VMRT.

Proposed method

• Utilizes the theory of varieties of minimal rational tangents (VMRT) to analyze tangent structures of rational homogeneous spaces.
• Applies the formal definition of admissible pairs: equivariant embeddings $i: \mathcal{S}_0 \hookrightarrow \mathcal{S}$ that preserve VMRTs and induce isomorphism on $H_2(\cdot, \mathbb{Z})$.
• Employs Harish-Chandra coordinates dual to positive root vectors to analyze tangent space structures and degeneracy conditions.
• Identifies degeneracy via the vanishing of a bilinear form $\sigma'(E_{\gamma+\theta_{n+1}}, T_\alpha(\widetilde{\mathscr{C}}_o(\mathcal{S}_0))) = 0$.
• Constructs germs of submanifolds $S \subset \mathcal{S}$ via mappings $f: \mathbb{C}^n \to \mathbb{C}^n \times \mathbb{C}^{m-n}$ with $f(z_1,\dots,z_n) = (z_1,\dots,z_n, z_n^2, 0,\dots,0)$ to test sub-VMRT structure inheritance.
• Uses projective linear isomorphisms $\Lambda_x$ to verify sub-VMRT structure by matching $\Lambda_x(\mathscr{C}_x(S)) = \mathscr{C}_o(\mathcal{S}_0)$ and $\Lambda_x(\mathscr{C}_x(X)) = \mathscr{C}_o(X)$ locally.

Experimental results

Research questions

• RQ1 Which pairs $(\mathcal{S}_0, \mathcal{S})$ of irreducible Hermitian symmetric spaces of compact type are admissible under the VMRT-preserving condition?
• RQ2 What is the sufficient condition for an admissible pair to be non-rigid, based on degeneracy in the VMRT structure?
• RQ3 Are the special pairs identified in the classification process rigid, or do they admit non-trivial deformations?
• RQ4 How does the root correspondence $\Phi: \Delta_0 \to \Delta$ relate to the existence of standard embeddings in admissible pairs?
• RQ5 To what extent do submanifolds of $\mathcal{S}$ inherit a sub-VMRT structure modeled on $\mathcal{S}_0$, and when does this imply non-rigidity?

Key findings

• The classification of admissible pairs includes all sub-diagram type pairs and additional non-sub-diagram cases, such as $(Q^n, Q^m)$ with $n \equiv m \pmod{1}$, and $(Z^{n-1}_{\text{max}}, Q^{2n-1})$, $(Z^1_{\text{max}}, G^{\text{iii}}(n,n))$ for $n \geq 2$.
• A sufficient condition for non-rigidity is established: if the standard embedding $i_{\Phi}: \mathcal{S}_0 \hookrightarrow \mathcal{S}$ is not induced by root correspondence and degeneracy occurs via $\sigma'(E_{\gamma+\theta_{n+1}}, T_\alpha(\widetilde{\mathscr{C}}_o(\mathcal{S}_0))) = 0$, then the pair is non-rigid.
• The construction of a germ $S \subset \mathcal{S}$ via $f(z_1,\dots,z_n) = (z_1,\dots,z_n, z_n^2, 0,\dots,0)$ yields a submanifold that inherits a sub-VMRT structure modeled on $\mathcal{S}_0$, proving non-rigidity in degenerate cases.
• For the three exceptional degenerate pairs $(Z^{n-1}_{\text{max}}, Q^{2n-1})$, $(Z^1_{\text{max}}, G^{\text{iii}}(n,n))$, and $(Q^n, Q^m)$ with $n \equiv m \pmod{1}$, non-rigidity is confirmed via known examples (Hong–Choe, 2004) and general inheritance of sub-VMRT structures.
• Special pairs arising in the classification are proven to be algebraic, though their rigidity remains unresolved and requires further study within VMRT theory.
• The theory of sub-VMRT structures is formalized via local projective linear isomorphisms $\Lambda_x$ satisfying $\Lambda_x(\mathscr{C}_x(S)) = \mathscr{C}_o(\mathcal{S}_0)$ and $\Lambda_x(\mathscr{C}_x(X)) = \mathscr{C}_o(X)$, enabling the definition of sub-VMRT models on submanifolds.