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

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## Abstract

Recently, Mok and Zhang (2019) introduced the notion of admissible pairs (*X*_{0}, *X*) of rational homogeneous spaces of Picard number 1 and proved rigidity of admissible pairs (*X*_{0}, *X*) of the subdiagram type whenever *X*_{0} is nonlinear. It remains unsolved whether rigidity holds when (*X*_{0}, *X*) is an admissible pair **NOT** of the subdiagram type of nonlinear irreducible Hermitian symmetric spaces such that (*X*_{0}, *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 *X*_{0} 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 *X*_{0} ⊂ *X*. Under the additional hypothesis that all holomorphic sections in Γ(*X*_{0}, *T*_{x}∣*x*_{0}) lift to global holomorphic vector fields on *X*, we prove that the admissible pair (*X*_{0}, *X*) is rigid. As examples we check that (*X*_{0}, *X*) is rigid when *X* is the Grassmannian *G*(*n, n*) of *n*-dimensional complex vector subspaces of *W* ≅ ℂ^{2n}, *n* ⩾ 3, and when *X*_{0} ⊂ *X* is the La grangian Grassmannian consisting of Lagrangian vector subspaces of (*W, σ*) where *σ* is an arbitrary symplectic form on *W*.

## Keywords

rational homogeneous spaces rigidity uniruled projective manifolds sub-VMRT structures## MSC(2010)

32M15 14J45## Notes

### Acknowledgements

This work was supported by the GRF grant of the Hong Kong Research Grants Council (Grant No. 17335616). The author would like to thank the referee for carefully reading the manuscript and for many helpful comments. Especially, he would like to thank the referee for pointing out that the Statement (b) in the proof of the surjectivity of the restriction map *r*: Γ(*X, T*_{x}) → Γ(*X*_{0}, *T*_{X}∣*x*_{0}) for (*X*_{0}, *X*) = (*G*^{III} (*n,n*), *G*(*n, n*)) as given in Subsection (6.1) is a consequence of the Bott-Borel-Weil Theorem, thus allowing him to remove the laborious explicit checking of cohomological vanishing statements for the examples examined.

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