Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors
Masataka Iwai, Chen Jiang, Haidong Liu
TL;DR
This work establishes Miyaoka-type inequalities for terminal projective 3-folds with nef anti-canonical divisor by combining Reid's basket formula with a structure theorem that passes to a finite cover étale in codimension 2 and yields an MRC fibration over a Calabi–Yau base. The authors perform a detailed case analysis of smooth versus singular and rationally connected versus non-rationally connected scenarios, deriving sharp lower bounds for $c_1(X)\cdot c_2(X)$ in each case (notably $\tfrac{1}{252}$ in general, $\tfrac{2}{5}$ when $-K_X$ is not big, and $\tfrac{4}{5}$ for non-rationally connected cases), supported by partial classifications and explicit examples. They also prove a Miyaoka–Kawamata type inequality for terminal weak Fano 3-folds and establish effective bounds of the form $c_1(X)^n\le b\,c_1(X)^{n-2}\cdot c_2(X)$ with $b$ depending only on the dimension. The results advance understanding of Chern class inequalities in the nef $-K_X$ setting and provide concrete structural and numerical constraints for such 3-folds, including a sharp lower bound realized by specific group actions on K3 and abelian surfaces.
Abstract
In this paper, we study the Miyaoka type inequality on Chern classes of terminal projective $3$-folds with nef anti-canonical divisors. Let $X$ be a terminal projective $3$-fold such that $-K_X$ is nef. We show that if $c_1(X)\cdot c_2(X)\neq 0$, then $c_1(X)\cdot c_2(X)\geq \frac{1}{252}$; if further $X$ is not rationally connected, then $c_1(X)\cdot c_2(X)\geq \frac{4}{5}$ and this inequality is sharp. In order to prove this, we give a partial classification of such varieties along with many examples. We also study the nonvanishing of $c_1(X)^{\dim X-2}\cdot c_2(X)$ for terminal weak Fano varieties and prove a Miyaoka--Kawamata type inequality for terminal weak Fano $3$-folds.