SEMISTABILITY AND NUMERICALLY EFFECTIVENESS IN POSITIVE CHARACTERISTIC

Let G be a connected reductive linear algebraic group defined over an algebraically closed field k of positive characteristic. Let Z(G) ⊂ G be the center, and , where each G_i is simple with trivial center. For i ∈ [1, m], let ρ_i : G → G_i be the natural projection. Fix a proper parabolic subgroup P of G such that for each i ∈ [1, m], the image ρ_i(G) ⊂ G_i is a proper parabolic subgroup. Fix a strictly anti-dominant character χ of P such that χ is trivial on Z(G). Let M be a smooth projective variety, defined over k, equipped with a very ample line bundle ξ. Let E_G → M be a principal G-bundle. We prove that the following six statements are equivalent:

(1) The line bundle E_G(χ) → E_G/P associated to the principal P-bundle E_G → E_G/P for the character χ is numerically effective.

(2) The sequence of principal G-bundles is bounded, where F_M is the absolute Frobenius morphism of M.

(3) The principal G-bundle E_G is strongly semistable with respect to ξ, and c₂(ad(E_G)) is numerically equivalent to zero.

(4) The principal G-bundle E_G is strongly semistable with respect to ξ, and [c₂(ad(E_G))c₁(ξ)^d-2] = 0.

(5) The adjoint vector bundle ad(E_G) is numerically effective.

(6) For every pair of the form (Y,ψ), where Y is an irreducible smooth projective curve and ψ : Y → M is a morphism, the principal G-bundle ψ*E_G → Y is semistable.