Rigidity of smooth Schubert varieties in Hermitian symmetric spaces

Author:
Jaehyun Hong

Journal:
Trans. Amer. Math. Soc. **359** (2007), 2361-2381

MSC (2000):
Primary 14C25, 32M15, 14M15

DOI:
https://doi.org/10.1090/S0002-9947-06-04041-4

Published electronically:
June 13, 2006

MathSciNet review:
2276624

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Abstract: In this paper we study the space of effective -cycles in with the homology class equal to an integral multiple of the homology class of Schubert variety of type . When is a proper linear subspace of a linear space in , we know that is already complicated. We will show that for a smooth Schubert variety in a Hermitian symmetric space, any irreducible subvariety with the homology class , , is again a Schubert variety of type , unless is a non-maximal linear space. In particular, any local deformation of such a smooth Schubert variety in Hermitian symmetric space is obtained by the action of the Lie group .

**[B]**R. Bryant, Rigidity and quasi-rigidity of extremal cycles in compact Hermitian symmetric spaces, to appear in Annals of Mathematics Studies 153, Princeton University Press.**[BE]**R. Baston and M. Eastwood, The Penrose Transform Its interaction with representation theory, Oxford Science Publication, 1989 MR**1038279 (92j:32112)****[BP]**M. Brion and P. Polo,*Generic singularities of certain Schubert varieties*, Math. Z., 231, 301-324 (1999) MR**1703350 (2000f:14078)****[CH]**I. Choe and J. Hong,*Integral varieties of the canonical cone structure on*, Math. Ann. 329, 629-652 (2004) MR**2076680 (2005e:32040)****[F]**W. Fulton, Young tableaux. With application to representation theory and geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, 1997 MR**1464693 (99f:05119)****[FH]**W. Fulton and J. Harris, Representation Theory; A First Course, Springer-Verlag, 1991 MR**1153249 (93a:20069)****[G]**A. B. Goncharov,*Generalized conformal structures on manifolds*, Selecta Mathematica Sovietica, vol. 6, No. 4, 1987. MR**0925263 (89e:53050)****[K1]**B. Kostant,*Lie algebra cohomology and the generalized Borel-Weil theorem*, Ann. of Math. (2) 74, No. 2, 329-387 (1961). MR**0142696 (26:265)****[K2]**B. Kostant,*Lie algebra cohomology and generalized Schubert cells*, Ann. of Math. (2) 77, No. 1, 72-144 (1963). MR**0142697 (26:266)****[LW]**V. Lakshmibai and J. Weyman,*Multiplicities of points on a Schubert variety in a minuscule*, Advances in Mathematics 84, 179-208 (1990) MR**1080976 (92a:14058)****[W]**M. Walters,*Geometry and uniqueness of some extreme subvarieties in complex Grassmannians*, Ph.D. thesis, University of Michigan, 1997.

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Additional Information

**Jaehyun Hong**

Affiliation:
Research Institute of Mathematics, Seoul National University, San 56-1 Sinrim-dong Kwanak-gu, Seoul, 151-747 Korea

Email:
jhhong@math.snu.ac.kr

DOI:
https://doi.org/10.1090/S0002-9947-06-04041-4

Keywords:
Analytic cycles,
Hermitian symmetric spaces,
Schubert varieties

Received by editor(s):
October 26, 2004

Received by editor(s) in revised form:
April 13, 2005

Published electronically:
June 13, 2006

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.