On the standard conjecture and the existence of a Chow-Lefschetz decomposition for complex projective varieties

摘要

We prove the Grothendieck standard conjecture B( X) of Lefschetz type on the algebraicity of the operators and of Hodge theory for a smooth complex projective variety X if at least one of the following conditions holds: X is a compactification of the N ' eron minimal model of an Abelian scheme of relative dimension 3 over an affine curve, and the generic scheme fibre of the Abelian scheme has reductions of multiplicative type at all infinite places; X is an irreducible holomorphic symplectic ( hyperk " ahler) 4- dimensional variety that coincides with the Altman- Kleiman compactification of the relative Jacobian variety of a family C ! P 2 of hyperelliptic curves of genus 2 with weak degenerations, and the canonical projection X ! P 2 is a Lagrangian fibration. We also show that a ChowLefschetz decomposition exists for every smooth projective 3- dimensional variety X which has the structure of a 1- parameter non- isotrivial family of K3- surfaces ( with degenerations) or a family of regular surfaces of arbitrary Kodaira dimension {with strong degenerations.