The inequalities of Chern classes and Riemann-Roch type inequalities

Motivated by Kollár-Matsusaka's Riemann-Roch type inequalities, applying effective very ampleness of adjoint bundles on Fujita conjecture and log-concavity given by Khovanskii-Teissier inequalities, we show that for any partition λ of the positive integer d there exists a universal bivariate polynomial $Q_{\lambda }(x,y)$ which has $\mathrm{deg}{Q}_{\lambda }\le d$ and whose coefficients depend only on n and λ, such that for any projective manifold X of dimension n and any ample line bundle L on X,$\left|c_{\lambda }\right(X)\cdot L^{n-d}|\le \frac{Q_{\lambda }(L^n,K_X\cdot L^{n-1})}{{\left(L^n\right)}^{d-1}},$ where $K_X$ is the canonical bundle of X and $c_{\lambda }\left(X\right)$ is the monomial Chern class given by the partition λ. As a special case, when $K_X$ or $-K_X$ is ample, this implies that there exists a constant $c_n$ depending only on n such that for any monomial Chern classes of top degree, the Chern number ratios satisfy the following inequality$\left|\frac{c_{\lambda }\left(X\right)}{c_1{\left(X\right)}^n}\right|\le c_n,$ which recovers a recent result of Du-Sun. The main result also yields an asymptotic version of the sharper Riemann-Roch type inequality. Furthermore, using similar method we also obtain inequalities for Chern classes of the logarithmic tangent bundle.