The inequalities of Chern classes and Riemann-Roch type inequalities

TL;DR

This paper establishes universal bounds for Chern class intersection numbers on projective manifolds, generalizing Riemann-Roch inequalities and applying techniques from algebraic geometry such as Fujita conjecture and log-concavity.

Contribution

It introduces a universal polynomial bound for Chern class intersections depending only on the dimension, extending Riemann-Roch type inequalities to broader classes of manifolds.

Findings

Derived bounds for monomial Chern classes using universal polynomials.

Established inequalities for Chern classes when the canonical bundle or its inverse is ample.

Provided asymptotic versions of Riemann-Roch inequalities and extended results to logarithmic tangent bundles.

Abstract

Motivated by Koll\'{a}r-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 $\lambda$ of the positive integer $d$ there exists a universal bivariate polynomial $Q_\lambda(x, y)$ which has deg $Q \leq d$ and whose coefficients depend only on $n$, such that for any projective manifold $X$ of dimension $n$ and any ample line bundle $L$ on $X$, \begin{equation*} \left|c_\lambda(X)\cdot L^{n -d}\right|\leq \frac{Q_{\lambda}(L^{n}, K_X \cdot L^{n -1} )}{(L^{n})^{d-1}}, \end{equation*} where $K_X$ is the canonical bundle of $X$ and $c_\lambda(X)$ is the monomial Chern class given by the partition $\lambda$. 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 \begin{equation*} \left|\frac{c_\lambda(X)}{c_1 (X) ^{n}}\right|\leq c_n, \end{equation*} 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.