Semistability of Frobenius direct images over curves

Original · EN

Let X be a smooth projective curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p>0. Given a semistable vector bundle E over X, we show that its direct image F_*E under the Frobenius map F of X is again semistable. We deduce a numerical characterization of the stable rank-p vector bundles F_*L, where L is a line bundle over X.