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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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