Kawamata--Miyaoka type inequality for $\mathbb{Q}$-Fano varieties with canonical singularities

TL;DR

This paper establishes a Kawamata--Miyaoka type inequality for certain $Q$-Fano varieties with canonical singularities, providing new bounds on Chern classes that extend previous results to higher dimensions and specific singularity types.

Contribution

It proves a Kawamata--Miyaoka type inequality for $Q$-Fano varieties with canonical singularities, including a stronger version for threefolds with terminal singularities.

Findings

Established the inequality $c_1(X)^n< 4 c_2(X) imes c_1(X)^{n-2}$ for the class of varieties.

Derived a stronger inequality for threefolds with terminal singularities.

Extended classical inequalities to varieties with specific singularities and higher dimensions.

Abstract

Let $X$ be an $n$-dimensional normal $\mathbb{Q}$-factorial projective variety with canonical singularities and Picard number one such that $X$ is smooth in codimension two, $-K_X$ is ample and $n\geq 2$. We prove that $X$ satisfies the following Kawamata--Miyaoka type inequality: \[ c_1(X)^n< 4 c_2(X)\cdot c_1(X)^{n-2}. \] If additionally $X$ is a threefold with terminal singularities, then a stronger inequality is also obtained.