Inequalities of Chern classes on nonsingular projective $n$-folds of Fano and general type with ample canonical bundle

TL;DR

This paper develops new inequalities relating all Chern classes of nonsingular projective Fano or general type varieties with ample canonical bundle, showing the boundedness of certain Chern ratios and deriving bounds for topological and holomorphic Euler characteristics.

Contribution

It introduces a novel method using Schubert classes and the Gauss map to establish inequalities among Chern classes for these varieties, confirming boundedness of Chern ratios in characteristic zero.

Findings

Chern ratios form a convex polyhedron in characteristic zero.

Topological and holomorphic Euler characteristics are bounded by constants times the canonical bundle degree.

Results extend to positive characteristic under certain ampleness and generation conditions.

Abstract

Let $X$ be a nonsingular projective $n$-fold $(n\ge 2)$ of 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 $c_1, c_2, \cdots, c_n$ 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 $(\frac{c_{2,1^{n-2}}}{c_{1^n}}, \frac{c_{2,2,1^{n-4}}}{c_{1^n}}, \cdots, \frac{c_{n}}{c_{1^n}})$ are contained in a convex polyhedron for all $X$. 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 (\cite{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 $d_1K_X^n\le\chi_{top}(X)\le d_2 K_X^n$ and $d_3K_X^n\le\chi(X, \mathscr{O}_X)\le d_4 K_X^n$. If the characteristic of $\kappa$ is positive, $K_X$ (or $-K_X$) is ample and $\mathscr{O}_X(K_X)$ ($\mathscr{O}_X(-K_X)$, respectively) is globally generated, then the same results hold.