# Local Cohomology of Schubert Varieties

By Ulf Raben-Pedersen
PhD Dissertations
September 2005
Abstract:

Let $X$ denote the flag variety defined over an algebraic closed field $k$ and $Y\subset X$ a Schubert variety. Independently of the characteristic of $k$ there exists a simple sheaf of $\mathcal{D}_X$-modules $\mathcal{L}(Y,X)$ with support $Y$ and an inclusion $\mathcal{L}(Y,X)\subset \mathcal{H}_Y^{\mathrm{codim}(Y)}(\mathcal{O}_X)$. If the characteristic of $k$ is greater than zero, we prove, the inclusion is an isomorphism. If the characteristic of $k$ is zero, we prove relations between the irreducible components of Sing$(Y)$ and Supp$(\mathcal{H}_Y^{\mathrm{codim}(Y)}(\mathcal{O}_X)/\mathcal{L}(Y,X))$. Especially the inclusion above can be an isomorphism, even if $Y$ is not smooth.

Let Gr$(r,n)$ denote the set of of $r$ dimensional subspaces of $k^n$. The Schubert varieties are parameterized by $1\leq a_1 < a_2 < \dots < a_r\leq n$, $a_i\in \mathbb{N}$, and we denote it as $X(a_1,\dots ,a_r)$. If $k$ is of characteristic zero we find the largest $c$ such that $\mathcal{H}_{X(a_{s}-s+1,\dots ,a_s,a_{s+1},\dots ,a_r)}^c(\mathcal{O}_{\mathrm{Gr}(r,n)})\neq 0$ whenever $a_s\geq r$.