# On Chern number inequality in dimension 3

## TL;DR

This paper proves an inequality involving Chern classes for threefold terminal flips, confirming a conjecture and extending results to divisorial contractions, advancing understanding in algebraic geometry.

## Contribution

It establishes a new inequality for Chern classes in threefold flips, providing an affirmative answer to a previously posed question and extending results to divisorial contractions.

## Key findings

- Proves Chern number inequality for threefold terminal flips
- Confirms a conjecture by Xie
- Extends results to divisorial contractions to curves

## Abstract

We prove that if $X---> X^+$ is a threefold terminal flip, then $c_1(X).c_2(X)\leq c_1(X^+).c_2(X^+)$ where $c_1(X)$ and $c_2(X)$ denote the Chern classes. This gives the affirmative answer to a Question by Xie \cite{Xie2}. We obtain the similar but weaker result in the case of divisorial contraction to curves.