Tangent bundle of hypersurfaces in G/P

摘要

Let G be a simple linear algebraic group defined over an algebraically closed field k of characteristic p ≥ 0, and let P be a maximal proper parabolic subgroup of G. If p > 0, then we will assume that dimG/P≤p. Let τ : H→G/P be a reduced smooth hypersurface in G/P of degree d. We will assume that the pullback homomorphism ?=Pic(G/P)→τ*Pic(H) is an isomorphism (this assumption is automatically satisfied when dimH≥3). We prove that the tangent bundle of H is stable if the two conditions τ(G/P)≠d and d>τ(G/P)(n-1)/2n-1 hold; here n = dimH, and τ(G/P)∈? is the index of G/P which is defined by the identity K~(-1)G/P=L?τ(G/P), where L is the ample generator of Pic(G/P) and K~(-1)G/P is the anticanonical line bundle of G/P. If d = τ(G/P), then the tangent bundle TH is proved to be semistable. If p > 0, and τ(G/P)>d>τ(G/P)(n-1)/2n-1, then TH is strongly stable. If p > 0, and d = τ(G/P), then TH is strongly semistable.