Restriction theorems for homogeneous bundles

Abstract

We prove that for an irreducible representation $\tau :GL(n)\to GL(W)$, the associated homogeneous ${\mathbf{P}}_k^n$-vector bundle $W_{\tau }$ is strongly semistable when restricted to any smooth quadric or to any smooth cubic in ${\mathbf{P}}_k^n$, where $k$ is an algebraically closed field of characteristic $\ne 2,3$ respectively. In particular $W_{\tau }$ is semistable when restricted to general hypersurfaces of degree $\ge 2$ and is strongly semistable when restricted to the $k$-generic hypersurface of degree $\ge 2$.

Revised version contains a stronger result:strong semistability is proved for restrictions to generic hypersurfaces of arbitrary degree >1, instead of generic hypersurfaces of even degree