#### Volume 19, issue 6 (2015)

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 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
Some new results on modified diagonals

### Claire Voisin

Geometry & Topology 19 (2015) 3307–3343
##### 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 $m=2,3$.

##### Keywords
small diagonal, Chow groups, $K3$ surfaces
##### Mathematical Subject Classification 2010
Primary: 14C15, 14C25