Products of Chern Classes and Chern Numbers on the Permutohedral Variety
Hideya Kuwata
TL;DR
The paper addresses expressing the product of Chern classes on the permutohedral variety $X_{A_n}$ as a scalar multiple of the top Chern class, giving an explicit closed form for the coefficient $\mu_k(n)$. It develops a purely combinatorial framework based on a fundamental block decomposition, reduction formulas, and non-vanishing criteria to reduce $c_k c_{n-k}$ to $c_n$ and extract the coefficient. The main result is $c_k c_{n-k}=\mu_k(n)c_n$ with $\mu_k(n)=\sum_{j=0}^{\lfloor k/2\rfloor} (\tfrac{1}{12})^j {k-j\choose j}{n-k-j\choose j}$, and the Chern numbers are $\langle c_k c_{n-k}, [X_{A_n}]\rangle=(n+1)!\mu_k(n)$. This advances the toric-geometry and algebraic-combinatorics interplay by providing explicit, computable invariants and suggesting a path to generalized products $c_\lambda$ via the invariant subring of the Weyl group action.
Abstract
A root system $Φ$ of rank $n$ determines an $n$-dimensional smooth projective toric variety $X(Φ)$ 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 $\langle c_k c_{n-k}, [X_{A_n}] \rangle$.