Let k be an algebraically closed field of characteristic p 〉 0, X a smooth projective variety over k with a fixed ample divisor H, FX：X → X the absolute Frobenius morphism on X. Let E be a rational GLn（k）-bundle ...Let k be an algebraically closed field of characteristic p 〉 0, X a smooth projective variety over k with a fixed ample divisor H, FX：X → X the absolute Frobenius morphism on X. Let E be a rational GLn（k）-bundle on X, and ρ：GLn（k） → GLm（k） a rational GLn（k）-representation of degree at most d such that ρ maps the radical R（GLn（k）） of GLn（k） into the radical R（GLm（k）） of GLm（k）. We show that if FXN＊（E） is semistable for some integer N ≥ max0 〈 r 〈 m （rm） · logp（dr）, then the induced rational GLm（k）-bundle E（GLm（k）） is semistable. As an application, if dim X=n, we get a sufficient condition for the semistability of Frobenius direct image FX＊（ρ＊（ΩX1））, where ρ＊（ΩX1） is the vector bundle obtained from ΩX1 via the rational representation ρ.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.11501418)Shanghai Sailing Program(Grant No.15YF1412500)
文摘Let k be an algebraically closed field of characteristic p 〉 0, X a smooth projective variety over k with a fixed ample divisor H, FX：X → X the absolute Frobenius morphism on X. Let E be a rational GLn（k）-bundle on X, and ρ：GLn（k） → GLm（k） a rational GLn（k）-representation of degree at most d such that ρ maps the radical R（GLn（k）） of GLn（k） into the radical R（GLm（k）） of GLm（k）. We show that if FXN＊（E） is semistable for some integer N ≥ max0 〈 r 〈 m （rm） · logp（dr）, then the induced rational GLm（k）-bundle E（GLm（k）） is semistable. As an application, if dim X=n, we get a sufficient condition for the semistability of Frobenius direct image FX＊（ρ＊（ΩX1））, where ρ＊（ΩX1） is the vector bundle obtained from ΩX1 via the rational representation ρ.
