Modified diagonals and linear relations between small diagonals

Abstract:

Let $(X,\operatorname {pt})$ be a pointed smooth projective variety. We prove that the vanishings of the modified diagonal cycles of Gross and Schoen govern the $\mathbb {Z}$-linear relations between small $m$-diagonals $\operatorname {pt}^{\{1,\ldots ,n\}\setminus A}\times \Delta _A$ in the rational Chow ring of $X^n$ for $A$ ranging over $m$-element subsets of $\{1,\ldots ,n\}$.

The combinatorial heart of this paper, which may be of independent interest, shows that 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.