Rigidity of smooth Schubert varieties in Hermitian symmetric spaces

Hong, Jaehyun

doi:10.1090/S0002-9947-06-04041-4

Rigidity of smooth Schubert varieties in Hermitian symmetric spaces
HTML articles powered by AMS MathViewer

by Jaehyun Hong PDF

Trans. Amer. Math. Soc. 359 (2007), 2361-2381 Request permission

Abstract:

In this paper we study the space $\mathcal {Z}_k(G/P, r[X_w])$ of effective $k$-cycles $X$ in $G/P$ with the homology class equal to an integral multiple of the homology class of Schubert variety $X_w$ of type $w$. When $X_w$ is a proper linear subspace $\mathbb {P}^k$ $(k<n)$ of a linear space $\mathbb {P}^n$ in $G/P \subset \mathbb {P}(V)$, we know that $\mathcal {Z}_k(\mathbb {P}^n, r[\mathbb {P}^k])$ is already complicated. We will show that for a smooth Schubert variety $X_w$ in a Hermitian symmetric space, any irreducible subvariety $X$ with the homology class $[X]=r[X_w]$, $r\in \mathbb {Z}$, is again a Schubert variety of type $w$, unless $X_w$ is a non-maximal linear space. In particular, any local deformation of such a smooth Schubert variety in Hermitian symmetric space $G/P$ is obtained by the action of the Lie group $G$.

References

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.
Robert J. Baston and Michael G. Eastwood, The Penrose transform, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. Its interaction with representation theory; Oxford Science Publications. MR 1038279
Michel Brion and Patrick Polo, Generic singularities of certain Schubert varieties, Math. Z. 231 (1999), no. 2, 301–324. MR 1703350, DOI 10.1007/PL00004729
Insong Choe and Jaehyun Hong, Integral varieties of the canonical cone structure on $G/P$, Math. Ann. 329 (2004), no. 4, 629–652. MR 2076680, DOI 10.1007/s00208-004-0530-5
William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
A. B. Goncharov, Generalized conformal structures on manifolds, Selecta Math. Soviet. 6 (1987), no. 4, 307–340. Selected translations. MR 925263
Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 142696, DOI 10.2307/1970237
Bertram Kostant, Lie algebra cohomology and generalized Schubert cells, Ann. of Math. (2) 77 (1963), 72–144. MR 142697, DOI 10.2307/1970202
V. Lakshmibai and J. Weyman, Multiplicities of points on a Schubert variety in a minuscule $G/P$, Adv. Math. 84 (1990), no. 2, 179–208. MR 1080976, DOI 10.1016/0001-8708(90)90044-N
M. Walters, Geometry and uniqueness of some extreme subvarieties in complex Grassmannians, Ph.D. thesis, University of Michigan, 1997.