Some new results on modified diagonals

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 $m=2,3$.

Final version, to appear in Geometry and Topology

References in corpus (1)

• Computations with modified diagonals

Cited by in corpus (1)

• Vanishing results in Chow groups for the modified diagonal cycles