arXiv 0907.11362009-07-07math.AG

关于定义在有限域上的projective variety上的主丛

On principal bundles over a projective variety defined over a finite field

Indranil Biswas

备注：: Final version of preprint (2007)

摘要：

设M是定义在有限域k上的几何不可约光滑射影簇，使得M存在k-有理点x_0.设\varpi（M，x_0）表示相应的由Nori引入的基本群-概型.设E_G是M上的主G-丛，G是定义在域k上的约化约化线性代数群。在M上固定一个偏振镜。本文证明了以下三个命题是等价的：M上的主G-丛E_G由同态\varpi（M，x_0）--> G给出。存在整数b > a > 0使得主G-丛（F^b_M）^*E_G同构于（F^a_M）^*E_G，其中F_M是M的绝对弗罗贝尼乌斯态射.主G-丛E_G是强半稳定的，其度（c_2（ad（E_G））c_1（n）^{d-2}）= 0，其中d = \dim M，且对G的每个特征标x，度（c_1（E_G（x））c_1（n）^{d-1}）= 0，其中E_G（x）是$M$上与x的$E_G$相关的线丛.第一个陈述和第三个陈述之间的等价性由S.在dim（M）= 1和$G$是半单的额外假设下的Subramanian。

英文摘要：

Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x_0. Let \varpi(M,x_0) denote the corresponding fundamental group--scheme introduced by Nori. Let E_G be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξon M. We prove that the following three statements are equivalent: The principal G-bundle E_G over M is given by a homomorphism \varpi(M,x_0) --> G. There are integers b > a > 0 such that the principal G-bundle (F^b_M)^*E_G is isomorphic to (F^a_M)^*E_G, where F_M is the absolute Frobenius morphism of M. The principal G-bundle E_G is strongly semistable, degree(c_2(ad(E_G))c_1(ξ)^{d-2}) = 0, where d = \dim M, and degree(c_1(E_G(χ))c_1(ξ)^{d-1}) = 0 for every character χof G, where E_G(χ) is the line bundle over $M$ associated to $E_G$ for χ. The equivalence between the first statement and the third statement was proved by S. Subramanian under the extra assumption that dim(M) = 1 and $G$ is semisimple.

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