Semistability of Rational Principal $G{L}_n$-Bundles in Positive Characteristic

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$, $X$ a smooth projective variety over $k$ with a fixed ample divisor $H$. Let $E$ be a rational $G{L}_n(k)$-bundle on $X$, and $\rho :G{L}_n(k)\to G{L}_m(k)$ a rational $G{L}_n(k)$-representation at most degree $d$ such that $\rho$ maps the radical $R(G{L}_n(k))$ of $G{L}_n(k)$ into the radical $R(G{L}_m(k))$ of $G{L}_m(k)$. We show that if $F_X^{N\ast }(E)$ is semistable for some integer $N\ge \underset{0<r<m}{\max }{C}_m^r\cdot {\log }_p(dr)$, then the induced rational $G{L}_m(k)$-bundle $E(G{L}_m(k))$ is semistable. As an application, if $\dim X=n$, we get a sufficient condition for the semistability of Frobenius direct image ${F_X}_{\ast }({\rho }_{\ast }({\Omega }_X^1))$, where ${\rho }_{\ast }({\Omega }_X^1)$ is the locally free sheaf obtained from ${\Omega }_X^1$ via the rational representation $\rho$.

14 pages. All comments are welcome

References in corpus (1)

• Direct images of bundles under Frobenius morphisms