Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type

Abstract

Recently, Mok and Zhang (2019) introduced the notion of admissible pairs (X₀, X) of rational homogeneous spaces of Picard number 1 and proved rigidity of admissible pairs (X₀, X) of the subdiagram type whenever X₀ is nonlinear. It remains unsolved whether rigidity holds when (X₀, X) is an admissible pair NOT of the subdiagram type of nonlinear irreducible Hermitian symmetric spaces such that (X₀, X) is nondegenerate for substructures. In this article we provide sufficient conditions for confirming rigidity of such an admissible pair. In a nutshell our solution consists of an enhancement of the method of propagation of sub-VMRT (varieties of minimal rational tangents) structures along chains of minimal rational curves as is already implemented in the proof of the Thickening Lemma of Mok and Zhang (2019). There it was proven that, for a sub-VMRT structure $\overline{\omega} : \mathscr{C}(S) \rightarrow S$ on a uniruled projective manifold $(X,\,{\cal K})$ equipped with a minimal rational component and satisfying certain conditions so that in particular S is “uniruled” by open subsets of certain minimal rational curves on X, for a “good” minimal rational curve ℓ emanating from a general point x ∈ S, there exists an immersed neighborhood N_ℓ of ℓ which is in some sense “uniruled” by minimal rational curves. By means of the Algebraicity Theorem of Mok and Zhang (2019), S can be completed to a projective subvariety Z ⊂ X. By the author’s solution of the Recognition Problem for irreducible Hermitian symmetric spaces of rank ⩾ 2 (2008) and under Condition (F), which symbolizes the fitting of sub-VMRTs into VMRTs, we further prove that Z is the image under a holomorphic immersion of X₀ into X which induces an isomorphism on second homology groups. By studying ℂ*-actions we prove that Z can be deformed via a one-parameter family of automorphisms to converge to X₀ ⊂ X. Under the additional hypothesis that all holomorphic sections in Γ(X₀, T_x∣x₀) lift to global holomorphic vector fields on X, we prove that the admissible pair (X₀, X) is rigid. As examples we check that (X₀, X) is rigid when X is the Grassmannian G(n, n) of n-dimensional complex vector subspaces of W ≅ ℂ²ⁿ, n ⩾ 3, and when X₀ ⊂ X is the La grangian Grassmannian consisting of Lagrangian vector subspaces of (W, σ) where σ is an arbitrary symplectic form on W.