Products of Chern Classes and Chern Numbers on the Permutohedral Variety

TL;DR

This paper derives explicit formulas for products of Chern classes on the permutohedral variety, enabling precise computation of Chern numbers using combinatorial methods.

Contribution

It provides a new combinatorial approach to express products of Chern classes as multiples of the top Chern class on the permutohedral variety.

Findings

Explicit closed-form formulas for c_k c_{n-k} in terms of c_n.

Calculation of specific Chern numbers for the permutohedral variety.

Demonstration of combinatorial techniques in algebraic geometry.

Abstract

A root system $\Phi$ of rank $n$ determines an $n$-dimensional smooth projective toric variety $X(\Phi)$ associated with the fan of its Weyl chambers. For the root system of type $A_n$, this variety is the well-known permutohedral variety $X_{A_n}$. Using purely combinatorial methods, we obtain an explicit closed formula expressing the product of Chern classes $c_k c_{n-k}$ as a multiple of the top Chern class $c_n$ in the rational cohomology ring $H^*(X_{A_n};\mathbb{Q})$. The resulting coefficient, which depends only on $k$ and $n$, is given by a closed-form expression. As an application, we compute the Chern number $\langle c_k c_{n-k}, [X_{A_n}] \rangle$.