# Combinatorial identities and Chern numbers of complex flag manifolds

## TL;DR

This paper introduces new combinatorial identities derived from complex geometry, providing explicit formulas for Chern numbers of complex flag manifolds through circle actions and Bott's residue formula.

## Contribution

It presents a unified approach to compute Chern numbers of all complex flag manifolds using combinatorial identities and geometric methods.

## Key findings

- Derived new combinatorial identities via differential geometry.
- Explicit formulas for Chern numbers of complex flag manifolds.
- Constructed circle actions with isolated fixed points to apply Bott's residue formula.

## Abstract

We present in this article a family of new combinatorial identities via purely differential/complex geometry methods, which include as a speical case a unified and explicit formula for Chern numbers of all complex flag manifolds. Our strategy is to construct concrete circle actions with isolated fixed points on these manifolds and explicitly determine their weights. Then applying Bott's residue formula to these models yields the desired results.