Advertisement

Science China Mathematics

, Volume 62, Issue 11, pp 2335–2354 | Cite as

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

  • Ngaiming MokEmail author
Articles
  • 12 Downloads

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 emanating from a general point xS, there exists an immersed neighborhood N of 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 ZX. 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 X0X. Under the additional hypothesis that all holomorphic sections in Γ(X0, Txx0) 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 X0X 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, Tx) → Γ(X0, TXx0) for (X0, X) = (GIII (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.

References

  1. 1.
    Bryant R. Rigidity and quasi-rigidity of extremal cycles in compact Hermitian symmetric spaces. ArXiv:math/0006186, 2000Google Scholar
  2. 2.
    Calabi E, Vesentini E. On compact, locally symmetric Kähler manifolds. Ann Math, 1960, 71: 472–507MathSciNetCrossRefGoogle Scholar
  3. 3.
    Debarre O. Higher-Dimensional Algebraic Geometry. New York: Springer-Verlag 2001CrossRefGoogle Scholar
  4. 4.
    Guillemin V. The integrability problem for G-structures. Trans Amer Math Soc, 1965, 116: 544–560MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hong J. Rigidity of singular Schubert varieties in Gr(m, n). J Differential Geom, 2005, 71: 1–22MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hong J. Rigidity of smooth Schubert varieties in Hermitian symmetric spaces. Trans Amer Math Soc, 2007, 359: 2361–2381MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hong J, Mok N. Analytic continuation of holomorphic maps respecting varieties of minimal rational tangents and applications to rational homogeneous manifolds. J Differential Geom, 2010, 86: 539–567MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hong J, Mok N. Characterization of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1. J Algebraic Geom, 2013, 22: 333–362MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hong J, Park K-D. Characterization of standard embeddings between rational homogeneous manifolds of Picard number 1. Int Math Res Not IMRN, 2011, 2011: 2351–2373MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hwang J, Mok N. Uniruled projective manifolds with irreducible reductive G-structures. J Reine Angew Math, 1997, 490: 55–64MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hwang J, Mok N. Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation. Invent Math, 1998, 131: 393–418MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hwang J, Mok N. Varieties of minimal rational tangents on uniruled manifolds. In: Schneider M, Siu Y-T, eds. Several Complex Variables. MSRI Publications, vol. 37. Cambridge: Cambridge University Press 1999, 351–389Google Scholar
  13. 13.
    Mok N. Geometric structures on uniruled projective manifolds defined by their varieties of minimal rational tangents. In: Proceedings of the Conference on Géométrie Différentielle, Physique Mathématique, Mathématique et Société. Astérisque, vol. 322. Paris: Soc Math France 2008, 151–205Google Scholar
  14. 14.
    Mok N. Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents. AMS/IP Stud Adv Math, 2008, 42: 41–61MathSciNetzbMATHGoogle Scholar
  15. 15.
    Mok N. Geometric structures and substructures on uniruled projective manifolds. In: Cascini P, McKernan J, Pereira J V, eds. Foliation Theory in Algebraic Geometry (Simons Symposia). Heidelberg-New York-London: Springer-Verlag 2016, 103–148CrossRefGoogle Scholar
  16. 16.
    Mok N, Zhang Y. Rigidity of pairs of rational homogeneous spaces of Picard number 1 and analytic continuation of geometric substructures on uniruled projective manifolds. J Differential Geom, 2019, 112: 263–345MathSciNetCrossRefGoogle Scholar
  17. 17.
    Robles C, The D. Rigid Schubert varieties in compact Hermitian symmetric spaces. Sel Math New Ser, 2012, 18: 717–777MathSciNetCrossRefGoogle Scholar
  18. 18.
    Walters M. Geometry and uniqueness of some extremal subvarieties in complex Grassmannians. PhD Thesis. Ann Arbor: University of Michigan 1997Google Scholar
  19. 19.
    Zhang Y. Admissible pairs of Hermitian symmetric spaces in the perspective of the theory of varieties of minimal rational tangents. PhD Thesis. Hong Kong: The University of Hong Kong 2014Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Hong KongHong KongChina

Personalised recommendations