Characterization of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1

Hong, Jaehyun; Mok, Ngaiming

doi:10.1090/S1056-3911-2012-00611-0

Characterization of smooth Schubert varieties in rational homogeneous manifolds of Picard number $1$

Authors: Jaehyun Hong and Ngaiming Mok
Journal: J. Algebraic Geom. 22 (2013), 333-362
DOI: https://doi.org/10.1090/S1056-3911-2012-00611-0
Published electronically: November 5, 2012
MathSciNet review: 3019452
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Abstract | References | Additional Information

Abstract:

In a series of works one of the authors has developed with J.-M. Hwang a geometric theory of uniruled projective manifolds basing on the study of varieties of minimal rational tangents, and the geometric theory has especially been applied to rational homogeneous manifolds of Picard number 1. In Mok [Astérisque 322, pp. 151–205] and Hong-Mok [J. Diff. Geom. 86 (2010), pp. 539–567] the authors have started the study of uniruled projective subvarieties, and a method was developed for characterizing certain subvarieties of rational homogeneous manifolds. The method relies on non-equidimensional Cartan-Fubini extension and a notion of parallel transport of varieties of minimal rational tangents.

In the current article we apply the notion of parallel transport to a characterization of smooth Schubert varieties of rational homogeneous manifolds of Picard number 1. Given a pair $(S, S_0)$ consisting of a rational homogeneous manifold $S$ of Picard number 1 and a smooth Schubert variety $S_0$ of $S$, where no restrictions are placed on $S_0$ when $S = G/P$ is associated to a long root (while necessarily some cases have to be excluded when $S$ is associated to a short root), we prove that any subvariety of $S$ having the same homology class as $S_0$ must be $gS_0$ for some $g \in \text {Aut}(S)$.

We reduce the problem first of all to a characterization of local deformations $S_t$ of $S_0$ as a subvariety of $S$. By Kodaira stability, $S_t$ is uniruled by minimal rational curves of $S$ lying on $S_t$. We establish a biholomorphism between $S_t$ and $S_0$ which extends to a global automorphism by reconstructing $S_t$ by means of a repeated use of parallel transport of varieties of minimal rational tangents along minimal rational curves issuing from a general base point. Our method is applicable also to the case of singular Schubert varieties provided that there exists a minimal rational curve on the smooth locus of the variety.

References

Sara Billey and Alexander Postnikov, Smoothness of Schubert varieties via patterns in root subsystems, Adv. in Appl. Math. 34 (2005), no. 3, 447–466. MR 2123545, DOI https://doi.org/10.1016/j.aam.2004.08.003
Michel Brion, Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, 2005, pp. 33–85. MR 2143072, DOI https://doi.org/10.1007/3-7643-7342-3_2
Michel Brion and Patrick Polo, Generic singularities of certain Schubert varieties, Math. Z. 231 (1999), no. 2, 301–324. MR 1703350, DOI https://doi.org/10.1007/PL00004729

R. Bryant, Rigidity and quasi-rigidity of extremal cycles in compact Hermitian symmetric spaces, math. DG/0006186.

Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88 (French). MR 354697

Hans Grauert, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135 (1958), 263–273 (German). MR 98199, DOI https://doi.org/10.1007/BF01351803

Hans Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368 (German). MR 137127, DOI https://doi.org/10.1007/BF01441136

Jaehyun Hong and Ngaiming Mok, Analytic continuation of holomorphic maps respecting varieties of minimal rational tangents and applications to rational homogeneous manifolds, J. Differential Geom. 86 (2010), no. 3, 539–567. MR 2785842

Jaehyun Hong, Rigidity of smooth Schubert varieties in Hermitian symmetric spaces, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2361–2381. MR 2276624, DOI https://doi.org/10.1090/S0002-9947-06-04041-4

Jun-Muk Hwang and Ngaiming Mok, Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation, Invent. Math. 131 (1998), no. 2, 393–418. MR 1608587, DOI https://doi.org/10.1007/s002220050209

Jun-Muk Hwang and Ngaiming Mok, Varieties of minimal rational tangents on uniruled projective manifolds, Several complex variables (Berkeley, CA, 1995–1996) Math. Sci. Res. Inst. Publ., vol. 37, Cambridge Univ. Press, Cambridge, 1999, pp. 351–389. MR 1748609

Jun-Muk Hwang and Ngaiming Mok, Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1, J. Math. Pures Appl. (9) 80 (2001), no. 6, 563–575. MR 1842290, DOI https://doi.org/10.1016/S0021-7824%2800%2901200-9

Jun-Muk Hwang and Ngaiming Mok, Deformation rigidity of the rational homogeneous space associated to a long simple root, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 2, 173–184 (English, with English and French summaries). MR 1914930, DOI https://doi.org/10.1016/S0012-9593%2802%2901087-X

Jun-Muk Hwang and Ngaiming Mok, Birationality of the tangent map for minimal rational curves, Asian J. Math. 8 (2004), no. 1, 51–63. MR 2128297, DOI https://doi.org/10.4310/AJM.2004.v8.n1.a6

Jun-Muk Hwang and Ngaiming Mok, Deformation rigidity of the 20-dimensional $F_4$-homogeneous space associated to a short root, Algebraic transformation groups and algebraic varieties, Encyclopaedia Math. Sci., vol. 132, Springer, Berlin, 2004, pp. 37–58. MR 2090669, DOI https://doi.org/10.1007/978-3-662-05652-3_4

Jun-Muk Hwang and Ngaiming Mok, Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation, Invent. Math. 160 (2005), no. 3, 591–645. MR 2178704, DOI https://doi.org/10.1007/s00222-004-0417-9

Jun-Muk Hwang, Geometry of minimal rational curves on Fano manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000) ICTP Lect. Notes, vol. 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, pp. 335–393. MR 1919462

Jun-Muk Hwang, Rigidity of rational homogeneous spaces, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 613–626. MR 2275613

Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. MR 899071

Stefan Kebekus, Families of singular rational curves, J. Algebraic Geom. 11 (2002), no. 2, 245–256. MR 1874114, DOI https://doi.org/10.1090/S1056-3911-01-00308-3

K. Kodaira, On stability of compact submanifolds of complex manifolds, Amer. J. Math. 85 (1963), 79–94. MR 153033, DOI https://doi.org/10.2307/2373187

János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180

Joseph M. Landsberg and Laurent Manivel, On the projective geometry of rational homogeneous varieties, Comment. Math. Helv. 78 (2003), no. 1, 65–100. MR 1966752, DOI https://doi.org/10.1007/s000140300003

Ion Alexandru Mihai, Odd symplectic flag manifolds, Transform. Groups 12 (2007), no. 3, 573–599. MR 2356323, DOI https://doi.org/10.1007/s00031-006-0053-0

Ngaiming Mok, Geometric structures on uniruled projective manifolds defined by their varieties of minimal rational tangents, Astérisque 322 (2008), 151–205 (English, with English and French summaries). Géométrie différentielle, physique mathématique, mathématiques et société. II. MR 2521656

T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713

J. Tits, Groupes semi-simples complexes et geometrie projective, Seminaire Bourbaki, 7 (1954/1955), exposé 112

Maria F. Walters, Geometry and uniqueness of some extreme subvarieties in complex Grassmannians, ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)–University of Michigan. MR 2695855

Additional Information

Jaehyun Hong
Affiliation: Department of Mathematical Sciences, Research Institute of Mathematics, Seoul National University, 599 Gwanak-ro Gwanak-gu Seoul 151-742, Korea
Email: jhhong00@snu.ac.kr

Ngaiming Mok
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
MR Author ID: 191186
Email: nmok@hkucc.hku.hk

Received by editor(s): August 6, 2010
Received by editor(s) in revised form: January 7, 2011, February 3, 2011, June 2, 2011, and July 31, 2011
Published electronically: November 5, 2012
Additional Notes: The first author’s work was supported by Research Settlement Fund for the new faculty of College of Natural Sciences in SNU 301-20070006. The second author’s research was partially supported by GRF grant HK7039/06P of the Research Grants Council of Hong Kong.