Monoidal transformations and conjectures on algebraic cycles

摘要

We consider the following conjectures: Hodge(X), Tate(X) (over a perfect finitely generated field), Grothendieck's standard conjecture B(X) of Lefschetz type on the algebraicity of the Hodge operator *, conjecture D(X) on the coincidence of the numerical and homological equivalences of algebraic cycles and conjecture C(X) on the algebraicity of Kunneth components of the diagonal for smooth complex projective varieties. We show that they are compatible with monoidal transformations: if one of them holds for a smooth projective variety X and a smooth closed subvariety Y hooked right arrow X, then it holds for X', where f: X' X is the blow up of X along Y. All of these conjectures are reduced to the case of rational varieties.