# Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors

## TL;DR

This paper establishes new inequalities relating Chern classes for terminal threefolds with nef anti-canonical divisors, providing classifications, examples, and extending Miyaoka--Kawamata type results.

## Contribution

It introduces sharp bounds for Chern class intersections on such threefolds and offers a partial classification along with numerous examples.

## Key findings

- Proved that c_1(X)·c_2(X) ≥ 1/252 when non-zero.
- Established that for non-rationally connected X, c_1(X)·c_2(X) ≥ 4/5, which is sharp.
- Extended Miyaoka--Kawamata inequalities to terminal weak Fano 3-folds.

## 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.