Modified diagonals and linear relations between small diagonals

TL;DR

This paper explores how the vanishing of modified diagonal cycles influences linear relations among small diagonals in the Chow ring of product varieties, extending to symmetric classes and revealing combinatorial structures of elementary symmetric polynomials.

Contribution

It establishes a general framework linking modified diagonal vanishings to linear relations among small diagonals and symmetric classes in algebraic geometry.

Findings

Proves that vanishings of modified diagonal cycles govern linear relations between small diagonals.

Generalizes results to arbitrary symmetric classes and different inclusion types.

Identifies combinatorial generators of relations among elementary symmetric polynomials.

Abstract

We prove that the vanishings of the modified diagonal cycles of Gross and Schoen govern the $\mathbb{Z}$-linear relations between small $m$-diagonals $\text{pt}^{\{1,\ldots,n\}\setminus A}\times\Delta_A$ in the rational Chow ring of $X^n$ for $A$ ranging over $m$-element subsets of $\{1,\ldots,n\}$. Our results generalize to arbitrary symmetric classes in place of the diagonal in $X^m$, and with different types of inclusions $A^\bullet(X^m)_{\mathbb{Q}}^{S_m} \hookrightarrow A^\bullet(X^n)_{\mathbb{Q}}$. The combinatorial heart of this paper, which may be of independent interest, is showing the $\mathbb{Z}$-linear relations between elementary symmetric polynomials $e_k(x_{a_1},\ldots,x_{a_m}) \in \mathbb{Z}[x_1,\ldots,x_n]$ are generated by the $S_n$-translates of a certain alternating sum over the facets of a hyperoctahedron.