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