Some new results on modified diagonals

摘要

O'Grady studied m(th) modified diagonals for a smooth connected projective variety, generalizing the Gross-Schoen modified small diagonal. These cycles Gamma(m)(X,a) depend on a choice of reference point a is an element of 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 that Gamma(n+1) (X,o(X)) = 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.