Monoidal transformations and conjectures on algebraic cycles

Abstract

We consider the following conjectures: , (over a perfect finitely generated field), Grothendieck's standard conjecture of Lefschetz type on the algebraicity of the Hodge operator , conjecture on the coincidence of the numerical and homological equivalences of algebraic cycles and conjecture on the algebraicity of Künneth 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 and a smooth closed subvariety , then it holds for , where is the blow up of along . All of these conjectures are reduced to the case of rational varieties.