Monoidal transformations and~conjectures on algebraic cycles

S. G. Tankeev

Izvestiya. Mathematics , Tome 71 (2007) no. 3, pp. 629-655

Résumé

We consider the following conjectures: $\operatorname{Hodge}(X)$, $\operatorname{Tate}(X)$ (over a perfect finitely generated field), Grothendieck's standard conjecture $B(X)$ of Lefschetz type on the algebraicity of the Hodge operator $\ast$, conjecture $D(X)$ on the coincidence of the numerical and homological equivalences of algebraic cycles and conjecture $C(X)$ 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 $X$ and a smooth closed subvariety $Y\hookrightarrow X$, then it holds for $X'$, where $f\colon X'\to X$ is the blow up of $X$ along $Y$. All of these conjectures are reduced to the case of rational varieties.