Some new results on modified diagonals

Voisin, Claire

doi:10.2140/gt.2015.19.3307

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Some new results on modified diagonals

Claire Voisin

Geometry & Topology 19 (2015) 3307–3343

DOI: 10.2140/gt.2015.19.3307

Abstract

O’Grady studied $m^{t h}$ modified diagonals for a smooth connected projective variety, generalizing the Gross–Schoen modified small diagonal. These cycles $Γ^{m} (X, a)$ 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 $Γ^{m} (X, a)$ vanishes for large $m$ . We also prove the following conjecture of O’Grady: If $X$ is a double cover of $Y$ and $Γ^{m} (Y, a)$ vanishes (where $a$ belongs to the branch locus), then $Γ^{2 m - 1} (X, a)$ vanishes, and we provide a generalization to higher-degree finite covers. We finally prove that $Γ^{n + 1} (X, o_{X}) = 0$ when $X = S^{[m]}$ , where $S$ is a $K 3$ surface, and $n = 2 m$ , 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

References

Publication

Received: 27 May 2014

Revised: 5 November 2014

Accepted: 23 December 2014

Published: 6 January 2016

Proposed: Lothar Göttsche

Seconded: Richard Thomas, Jim Bryan

Authors

Claire Voisin
Institut de Mathématiques de Jussieu CNRS 4 place Jussieu Case 247 75252 Paris Cedex 05 France

Volume 19, issue 6 (2015)