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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Riccati flow and singularities of Schubert varieties


Author: James S. Wolper
Journal: Proc. Amer. Math. Soc. 123 (1995), 703-709
MSC: Primary 14M15; Secondary 58F25
MathSciNet review: 1221729
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Abstract: Let $ \operatorname{Gr}(m,n)$ be the Grassmannian of m-dimensional subspaces of an n-dimensional k-vector space, with $ k = {\mathbf{R}}$ or C. Fix an $ n \times n$ matrix R with coefficients in k. The Riccati Flow $ \Phi $ on $ \operatorname{Gr}(m,n)$ is the action of a one-parameter subgroup of $ {\text{GL}_n}(k)$, given by $ {\Phi _t}(\Lambda ) = {e^{Rt}}\Lambda $. We prove:

Theorem. Let X be a Schubert variety in $ \operatorname{Gr}(m,n)$. Then there exists a Riccati flow $ \Phi $ on X and a stable manifold W for $ \Phi $ such that W is the smooth locus of X.

Corollary (over C). X as above is smooth if and only if the cohomology of X satisfies Poincaré Duality.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1221729-X
PII: S 0002-9939(1995)1221729-X
Article copyright: © Copyright 1995 American Mathematical Society