On Classification Of Higher-dimensional Algebraic Varieties With Ample Vector Bundles

Let X be a smooth projective variety of dimension n(n≥3)and Ean ample vector bundle with rank r=n kover X. We denoteΛ (E, K_X)=max{(K_X c1(E)) C|R=R+[C]∈Ω, and l (R)=K_X C}≥0, whereK_Xis the canonical bundle of X,c1(E)means the first Chern class of E, Ωdenotes the set of extremal rays R=R+[C]such that(K_X+c1(E)) R≤0,R+is theset of positive real number, and l (R)is the length of R. The classification of(X, E)will ba given when Λ (E, K_X)≥k1. Specifically，when Λ (E, K_X)=k+1,Xis a projective spacePn.