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We investigate the Lefschetz standard conjecture for degree $2$ cohomology ofhyper-K\"ahler manifolds admitting a covering by Lagrangian subvarieties. Inthe case of a Lagrangian fibration, we show that the Lefschetz standardconjecture is implied by the SYZ conjecture characterizing classes of divisorsassociated with Lagrangian fibration. In dimension $4$, we consider the moregeneral case of a Lagrangian covered fourfold $X$, and prove the Lefschetzstandard conjecture in degree $2$, assuming $\rho(X)=1$ and $X$ is general inmoduli. Finally we discuss various links between Lefschetz cycles and the studyof the rational equivalence of points and Bloch-Beilinson type filtrations,giving a general interpretation of a recent intriguing result of Marian andZhao.

On the Lefschetz standard conjecture for Lagrangian covered hyper-K\"ahler varieties