Rationality of the Instability Parabolic and Related Results

Abstract

In this paper we study the extension of structure group of principal bundles with a reductive algebraic group as structure group on a smooth projective variety defined over an algebraically closed field. Our main result is to show that given a finitedimensional representation ρ of a reductive algebraic group G, there exists an integer N which is quantifiable in terms of G and ρ such that any semistable G-bundle whose first N Frobenius pullbacks are semistable induces a semistable bundle on extension of structure group via ρ. We do this by quantifying the fields of definition of the instability parabolics associated to various parabolic reductions of the induced bundle.

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

“…F. Coiai and Y. I. Holla [2] generalized some results of [9] and showed that given a representation ρ : G → GL m (k), there exists a non-negative integer N , depending only on G and ρ, such that for any rational G-bundle E whose N -th Frobenius pull back F N * X (E) is semistable, then the induced rational GL m (k)bundle E(GL m (k)) is again semistable. S. Gurjar and V. Mehta [3] improved the result of [2] and obtain a explicit bound for N in terms of certain numerical data attached to ρ. The main ingredient of the proof in [2] and [3] is to give a uniform bound for the field of definition of the instability parabolics (Kempf' parabolic) associated to non-semistable points in related representing space.…”

“…S. Gurjar and V. Mehta [3] improved the result of [2] and obtain a explicit bound for N in terms of certain numerical data attached to ρ. The main ingredient of the proof in [2] and [3] is to give a uniform bound for the field of definition of the instability parabolics (Kempf' parabolic) associated to non-semistable points in related representing space.…”

“…In [2] and [3], the authors showed that there exists a uniform bound N , depending only on G and ρ, such that for all possible rational reductions to all maximal parabolic subgroups the instability parabolics of points corresponding to these rational reductions are actually defined over K p−N via different methods. This can be shown to imply that if E is a semistable rational G-bundle such that F N * X (E) is semistable, then the induced rational GL m (k)-bundle E(GL m (k)) is also semistable.…”

“…This can be shown to imply that if E is a semistable rational G-bundle such that F N * X (E) is semistable, then the induced rational GL m (k)-bundle E(GL m (k)) is also semistable. The major differences between the methods of [2] and that of [3] lie in the approach of estimating the field extension L of K such that a given K-scheme M has a L-rational point. F. Coiai and Y. I. Holla [2] proved the existence of the uniform bound by bounding the non-separability of the group action and the non-reducedness of the stabilizers of various unstable rational points.…”

“…However, the above estimation does not seen quantifiable. On the other hand, S. Gurjar and V. Mehta [3] directly estimated the field of definition of the instability parabolics which is probably weaker than the method of [2], but it is quantifiable.…”