On the singularity of the irreducible components of a Springer fiber in sl(n)

Let ${ \mathcal B}_u$ be the Springer fiber over a nilpotent endomorphism $u\in End(\mathbb{ C}^n)$. Let $J(u)$ be the Jordan form of $u$ regarded as a partition of $n$. The irreducible components of ${ \mathcal B}_u$ are all of the same dimension. They are labelled by Young tableaux of shape $J(u)$. We study the question of singularity of the components of ${ \mathcal B}_u$ and show that all the components of ${ \mathcal B}_u$ are nonsingular if and only if $J(u)\in\{ (\lambda,1,1,...), (\lambda_1,\lambda_2), (\lambda_1,\lambda_2,1), (2,2,2)\}$.

Categories: math.AG