On the Lefschetz Standard Conjecture

Abstract

The subject of the present paper is Grothendieck's Lefschetz standard conjecture $B(X)$. Our main result is that, if $X$ is a projective smooth variety of dimension $n$ and the conjecture $B(\mathcal{Y})$ holds for the generic fibre $\mathcal{Y}$ (of dimension $n-1$ over the field $k(t)$) of a suitable Lefschetz fibration of $X$, then the operator ${\Lambda }_X-p_X^{n+1}$ is algebraic. If in addition $p_X^{n+1}$ is algebraic, then $B(X)$ is settled. Along the way we establish the algebraicity of the K\" unneth projectors ${\pi }_X^i$ for $i\ne n-1,n,n+1$ under the above hypotheses.