Some new results on modified diagonals

TL;DR

This paper proves new vanishing results for modified diagonals on algebraic varieties, including conjectures by O'Grady, with implications for double covers and Hilbert schemes of K3 surfaces.

Contribution

It establishes the vanishing of modified diagonals for large m, confirms O'Grady's conjecture on double covers, and proves a new vanishing result for Hilbert schemes of K3 surfaces.

Findings

Vanishing of modified diagonals for large m.

Confirmation of O'Grady's conjecture on double covers.

Vanishing of diagonals for Hilbert schemes of K3 surfaces.

Abstract

O'Grady studied recently $m$-th modified diagonals for a smooth projective variety, generalizing the Gross-Schoen modified small diagonal. These cycles $\Gamma^m(X,a)$ depend on a choice of reference point $a\in X$ (or more generally a degree $1$ zero-cycle). We prove that for any $X,a$, the cycle $\Gamma^m(X,a)$ vanishes for large $m$. We also prove the following conjecture of O'Grady: if $X$ is a double cover of $Y$ and $\Gamma^m(Y,a)$ vanishes (where $a$ belongs to the branch locus), then $\Gamma^{2m-1}(X,a)$ vanishes, and we provide a generalization to higher degree finite covers. We finally prove the vanishing $\Gamma^{n+1}(X,o_X)=0$ when $X=S^{[m]}$, $S$ a $K3$ surface, and $n=2m$, which was conjectured by O'Grady and proved by him for $m=2,3$.