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Article: Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type
Title | Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type |
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Authors | |
Keywords | rational homogeneous spaces rigidity uniruled projective manifolds sub-VMRT structures |
Issue Date | 2019 |
Publisher | Springer Verlag, co-published with Science China Press. The Journal's web site is located at http://link.springer.com/journal/11425 |
Citation | Science China Mathematics, 2019, v. 62, p. 2335-2354 How to Cite? |
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 ω¯¯¯:C(S)→S on a uniruled projective manifold (X,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 Nℓ 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, Tx∣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. |
Persistent Identifier | http://hdl.handle.net/10722/282932 |
ISSN | 2023 Impact Factor: 1.4 2023 SCImago Journal Rankings: 1.060 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Mok, N | - |
dc.date.accessioned | 2020-06-05T06:23:11Z | - |
dc.date.available | 2020-06-05T06:23:11Z | - |
dc.date.issued | 2019 | - |
dc.identifier.citation | Science China Mathematics, 2019, v. 62, p. 2335-2354 | - |
dc.identifier.issn | 1674-7283 | - |
dc.identifier.uri | http://hdl.handle.net/10722/282932 | - |
dc.description.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 ω¯¯¯:C(S)→S on a uniruled projective manifold (X,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 Nℓ 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, Tx∣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. | - |
dc.language | eng | - |
dc.publisher | Springer Verlag, co-published with Science China Press. The Journal's web site is located at http://link.springer.com/journal/11425 | - |
dc.relation.ispartof | Science China Mathematics | - |
dc.rights | This is a post-peer-review, pre-copyedit version of an article published in [insert journal title]. The final authenticated version is available online at: http://dx.doi.org/[insert DOI] | - |
dc.subject | rational homogeneous spaces | - |
dc.subject | rigidity | - |
dc.subject | uniruled projective manifolds | - |
dc.subject | sub-VMRT structures | - |
dc.title | Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type | - |
dc.type | Article | - |
dc.identifier.email | Mok, N: nmok@hku.hk | - |
dc.identifier.authority | Mok, N=rp00763 | - |
dc.description.nature | link_to_subscribed_fulltext | - |
dc.identifier.doi | 10.1007/s11425-019-1592-6 | - |
dc.identifier.scopus | eid_2-s2.0-85074710803 | - |
dc.identifier.hkuros | 310106 | - |
dc.identifier.volume | 62 | - |
dc.identifier.spage | 2335 | - |
dc.identifier.epage | 2354 | - |
dc.identifier.isi | WOS:000491416700001 | - |
dc.publisher.place | China | - |
dc.identifier.issnl | 1869-1862 | - |