标题： 有理主$GL_n$-丛在正特征中的半稳定性 显示英文标题

标题： Semistability of Rational Principal $GL_n$-Bundles in Positive Characteristic

摘要： 设$k$为特征为$p>0$的代数闭域，$X$为在$k$上的光滑射影概形，并且固定一个 ample 分量$H$。 设$E$是在$X$上的有理$GL_n(k)$-丛，且$\rho:GL_n(k)\rightarrow GL_m(k)$是一个次数至多为$d$的有理$GL_n(k)$-表示，使得$\rho$将$GL_n(k)$的根式$R(GL_n(k))$映射到$GL_m(k)$的根式$R(GL_m(k))$。 我们证明，如果对于某个整数$N\geq\max\limits_{0<r<m}C^r_m\cdot\log_p(dr)$，$F_X^{N*}(E)$是半稳定，则诱导的有理$GL_m(k)$-丛$E(GL_m(k))$是半稳定。 作为应用，如果$\dim X=n$，我们得到弗罗贝尼乌斯直接像的半稳定性的充分条件${F_X}_*(\rho_*(\Omega^1_X))$，其中$\rho_*(\Omega^1_X)$是通过有理表示$\rho$从$\Omega^1_X$得到的局部自由层。 显示英文摘要

摘要： 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 $GL_n(k)$-bundle on $X$, and $\rho:GL_n(k)\rightarrow GL_m(k)$ a rational $GL_n(k)$-representation at most degree $d$ such that $\rho$ maps the radical $R(GL_n(k))$ of $GL_n(k)$ into the radical $R(GL_m(k))$ of $GL_m(k)$. We show that if $F_X^{N*}(E)$ is semistable for some integer $N\geq\max\limits_{0<r<m}C^r_m\cdot\log_p(dr)$, then the induced rational $GL_m(k)$-bundle $E(GL_m(k))$ is semistable. As an application, if $\dim X=n$, we get a sufficient condition for the semistability of Frobenius direct image ${F_X}_*(\rho_*(\Omega^1_X))$, where $\rho_*(\Omega^1_X)$ is the locally free sheaf obtained from $\Omega^1_X$ via the rational representation $\rho$.