INEQUALITIES OF CHERN CLASSES ON NONSINGULAR PROJECTIVE n-FOLDS WITH AMPLE CANONICAL OR ANTI-CANONICAL LINE BUNDLES

Let X be a nonsingular projective n-fold (n ≥ 2) which is either Fano or of general type with ample canonical bundle K_X over an algebraic closed field κ of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes c₁, c₂, · · ·, c_n by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. Moreover, we show that if the characteristic of κ is 0, then the Chern ^c2,2,1ⁿ⁻⁴ ratios (Formula presented) are contained in a convex poly-c₁n hedron depending on the dimension of X only. So we give an affirmative answer to a generalized open question, that whether the region described by the Chern ratios is bounded, posted by Hunt [Hun] to all dimensions. As a corollary, we can get that there exist constants d₁, d₂, d₃ and d₄ depending only on n such that d₁K_Xⁿ ≤ χ_top(X) ≤ d₂K_Xⁿ and d₃K_Xⁿ ≤ χ(X, O_X) ≤ d₄K_Xⁿ . If the characteristic of κ is positive, K_X (or −K_X) is ample and O_X(K_X) (O_X(−K_X), respectively) is globally generated, then the same results hold.