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

Abstract

We prove the Grothendieck standard conjecture of Lefschetz type on the algebraicity of the operators and of Hodge theory for a smooth complex projective variety if at least one of the following conditions holds: is a compactification of the Néron minimal model of an Abelian scheme of relative dimension over an affine curve, and the generic scheme fibre of the Abelian scheme has reductions of multiplicative type at all infinite places; is an irreducible holomorphic symplectic (hyperkähler) -dimensional variety that coincides with the Altman– Kleiman compactification of the relative Jacobian variety of a family of hyperelliptic curves of genus with weak degenerations, and the canonical projection is a Lagrangian fibration. We also show that a Chow– Lefschetz decomposition exists for every smooth projective 3-dimensional variety 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.