The Riccati flow and singularities of Schubert varieties

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.