Inequalities of Chern classes on nonsingular projective $n$-folds with ample canonical or anti-canonical line bundles

Abstract

Let $X$ be a nonsingular projective $n$-fold $(n \geq 2)$ which is either Fano or of general type with ample canonical bundle $K_X$ over an algebraic closed field $\kappa$ of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes $c_1, c_2, \dotsc , c_n$ by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. Moreover, we show that if the characteristic of $\kappa$ is $0$, then the Chern ratios $\left( \dfrac{c_{2,1^{n-2}}}{c_{1^n}} , \dfrac{c_{2,2,1^{n-4}}}{c_{1^n}} , \dotsc , \frac{c_n}{c_{1^n}} \right)$ are contained in a convex polyhedron 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 $d_1$, $d_2$, $d_3$ and $d_4$ depending only on $n$ such that $d_1 K^n_X \leq \chi_\mathrm{top} (X) \leq d_2 K^n_X$ and $d_3 K^n_X \leq \chi (X, \mathcal{O}_X) \leq d_4 K^n_X$ . If the characteristic of $\kappa$ is positive, $K_X$ (or $-K_X$) is ample and $\mathcal{O}_X (K_X) (\mathcal{O}_X(-K_X) \textrm{, respectively})$ is globally generated, then the same results hold.