Let's put $m>n$ two nonnegative integers and $Gr:=Grass(n,k^m)$ the grassmanian of the subspaces of dimension $n$ in $k^m$. We have a natural immersion $Gr \subset P({\Lambda}^{n} k^m)$ and I call $C(Gr) \subset {\Lambda}^{n} k^m$ the affine cone above $Gr$. If we blow-up $C(Gr)$, we always obtain something smooth, but my question is: Is the blow-up a crepant resolution of the singularities of $C(Gr)$?
In fact the answer is no when $n=1$ or $n=m-1$, but what happens in the other cases ?