Semistablity of syzygy bundles on projective spaces in positive characteristics

Abstract

In char $k = p >0$, A. Langer proved a strong restriction theorem (in the style of H. Flenner) for semistable sheaves to a very general hypersurface of degree $d$, on certain varieties, with the condition that `char $k > d$'. He remarked that to remove this condition, it is enough to answer either of the following questions affirmatively: {\it For the syzygy bundle $\sV_d$ of ${\mathcal O}(d)$, is $\sV_d$ semistable for arbitrary $n, d$ and $p = {char} k$?, or is there a good estimate on $\mu_{max}(\sV_d^*)$?} Here we prove that (1) the bundle $\sV_d$ is semistable, for a certain infinite set of integers $d\geq 0$, and (2) for arbitrary $d$, there is a good enough estimate on $\mu_{max}(\sV_d^*)$ in terms of $d$ and $n$. In particular one obtains Langer's theorem, in arbitrary characeristic.