Some new results on modified diagonals

Abstract

O’Grady studied $m^{th}$ modified diagonals for a smooth connected projective variety, generalizing the Gross–Schoen modified small diagonal. These cycles ${\Gamma }^m\left(X,a\right)$ 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 ${\Gamma }^m\left(X,a\right)$ vanishes for large $m$. We also prove the following conjecture of O’Grady: If $X$ is a double cover of $Y$ and ${\Gamma }^m\left(Y,a\right)$ vanishes (where $a$ belongs to the branch locus), then ${\Gamma }^{2m-1}\left(X,a\right)$ vanishes, and we provide a generalization to higher-degree finite covers. We finally prove that ${\Gamma }^{n+1}\left(X,o_X\right)=0$ when $X=S^{\left[m\right]}$, where $S$ is a $K3$ surface, and $n=2m$, which was conjectured by O’Grady and proved by him for $m=2,3$.