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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 mth modified diagonals for a smooth connected projective variety, generalizing the Gross–Schoen modified small diagonal. These cycles Γm(X,a) depend on a choice of reference point a X (or more generally a degree-1 zero-cycle). We prove that for any X, a, the cycle Γm(X,a) vanishes for large m. We also prove the following conjecture of O’Grady: If X is a double cover of Y and Γm(Y,a) vanishes (where a belongs to the branch locus), then Γ2m1(X,a) vanishes, and we provide a generalization to higher-degree finite covers. We finally prove that Γn+1(X,oX) = 0 when X = S[m], 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
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